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Korteweg-de Vries Institute for Mathematics

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General Mathematics Colloquium — past lectures 2001–2011

Wednesdays, 11.15-12.15 hours
Science Park 904, room A1.04
1098XH Amsterdam

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Lectures in 2001-2002

Date Lecturer Affiliation Title More
September 12, 2001 Erik Koelink Delft University of Technology Special functions, the quantum dynamical Yang-Baxter equation and dynamical quantum groups Abstract
September 19, 2001 (canceled) Dennis Gaitsgory Harvard University An overview of the geometric Langlands conjecture. Abstract
September 26, 2001 Eva Hoogland University of Amsterdam A general picture of definability Abstract
October 10, 2001 Erik Balder Utrecht University New results in and applications of topological measure theory Abstract
October 17, 2001 Herman te Riele CWI On the distribution of class numbers of real quadratic number fields Abstract
October 24, 2001 Marc Yor Université Paris VI & Paris VII On W. Doeblin's sealed envelope deposited in the Academie des Sciences (Feb. 1940) about Kolmogorov's equation More info
October 31, 2001 Peter Paule Johannes Kepler Universität Linz Computer Algebra and Combinatorics: MacMahon's Partition Analysis Revisited Abstract
November 7, 2001 Rene Schoof Universita di Roma "Tor Vergata" Class numbers of cyclotomic fields Abstract
November 14, 2001 Arjan van der Schaft Universiteit Twente Hamiltonian formulation of network models of physical systems Abstract
November 21, 2001 Debby Lanser CWI Efficient numerical methods for atmospheric flow problems Abstract
November 28, 2001 Bernd Kuckert University of Amsterdam Thermodynamic equilibrium states of moving quantum systems Abstract
December 5, 2001 Roxana Ion Eurandom New Nonparametric Shewhart Control Charts and Sharp Chebyshev-Type Inequalities Abstract
January 9, 2002 Alexa van der Waall Universiteit Utrecht Lamé equations with finite monodromy Abstract
January 16, 2002 Stefan Steiner University of Waterloo Seven Habits of Highly Effective Industrial Problem Solvers
The Principles of Statistical Engineering
January 23, 2002 Ronald Meester Vrije Universiteit Amsterdam There is a phase-transition in the Bak-Sneppen evolution model Abstract
January 30, 2002 Kees Jan van Garderen University of Amsterdam & University of Bristol Statistical Geometry in Econometrics and the Exact Geometry of Explosive Autoregressive Models Abstract
February 13, 2002 Rob van der Vorst Vrije Universiteit Amsterdam & Georgiatech Braids and differential equations Abstract
February 20, 2002 Oscar Lemmers Umea Universitet A decomposition problem in complex analysis Abstract
February 27, 2002 Peter van Emde Boas Universiteit van Amsterdam Imperfect Information Games; looking for the right model Abstract
March 6, 2002 Elham Izadi Univ. of Georgia, Athens GA On the cohomology of hypersurfaces with automorphisms Abstract
March 13, 2002 Gerard Helminck University of Twente Integrable hierarchies and flag varieties Abstract
March 27, 2002 Jeroen de Mast IBIS Quality Improvement from the Viewpoint of Statistical Method Abstract
April 17, 2002 Paul Beneker AOT The Bergman space: strongly exposed points and the Bergman projection Abstract
April 24, 2002 Jan Willems University of Leuven, RUG The behavioral approach to systems and control Abstract
May 8, 2002 Klaas Landsman University of Amsterdam Quantization and the Baum-Connes conjecture Abstract
May 22, 2002 Peter Duren Ann Arbor Zeros of hypergeometric functions Abstract
May 29, 2002 Jan Karel Lenstra TUE Whizzkids: Two exercises in computational discrete optimization Abstract
June 19, 2002 Earl J. Taft Rutgers University Recursive Sequences and Combinatorial Identities Abstract

Lectures in 2002-2003
Date Lecturer Affiliation Title More
September 11, 2002 Peter Wakker Universiteit van Amsterdam/Universiteit van Maastricht How to Add up Uncountably Many Numbers? (Hint: Not by Integration) Abstract
September 25, 2002 Richard Cushman Universiteit Utrecht Geometric phases in the Euler top Abstract
October 2, 2002 David Iron Universiteit van Amsterdam Stability and dynamics of multi-spike solutions to a system of reaction-diffusion equations Abstract
October 9, 2002 Michael Müger Universiteit van Amsterdam From conformal to topological field theory: Equivariant results Abstract
October 16, 2002 Jan Willem Polderman Universiteit Twente A systems theoretic approach to list decoding of Reed Solomon codes Abstract
November 6, 2002 Wim Couwenberg Reflexis/Katholieke Universiteit Nijmegen Diophantine equations after an idea of Lehman Abstract
November 13, 2002 Sander Zwegers Universiteit Utrecht Mock Theta Functions Abstract
November 20, 2002 Andries Lenstra Eurandom/Universiteit van Amsterdam On information bounds Abstract
December 4, 2002 Chris Stolk École Polytechnique, Parijs Inversion of seismic data in complex media Abstract
January 15, 2003 Bas Edixhoven Universiteit Leiden Counting solutions of systems of equations over finite fields Abstract
January 29, 2003 Andreas Weiermann Universiteit Utrecht Some Hardy Ramanujan style counting problems Abstract
February 5, 2003 Gerton Lunter University of Oxford Statistical alignment of biological sequences Abstract
February 12, 2003 Hans van Duijn TU Eindhoven Mathematical issues in density driven porous media flow Abstract
February 26, 2003 Mark Peletier CWI/TU Eindhoven Continuum modelling of lipid bilayers Abstract
March 12, 2003 Hessel Posthuma Universiteit van Amsterdam Quantization and Topological Quantum Field Theory Abstract
March 26, 2003 Ton Levelt Katholieke Universiteit Nijmegen The other D.J. Korteweg: thermodynamics of binary mixtures Abstract
April 9, 2003 Remco Peters Universiteit van Amsterdam Some new insights into the volatility process Abstract
April 16, 2003 Peter Stevenhagen Universiteit Leiden Primes is in P Abstract
May 7, 2003 Rien Kaashoek Vrije Universiteit Amsterdam A lifting perspective to metric constrained interpolation Abstract
May 21, 2003 Floske Spieksma Universiteit Leiden Transient properties of random walk type processes Abstract
June 4, 2003 Michal Krížek Academy of Sciences, Prague From Fermat Numbers to Geometry Abstract

Lectures in 2003-2004
Date Lecturer Affiliation Title More
September 3, 2003 Hae-Won Uh Leids Universitair Medisch Centrum Kernel deconvolution Abstract
September 10, 2003 Klaas Slooten Universiteit van Amsterdam A combinatorial generalization of the Springer correspondence for classical type Abstract
October 1, 2003 Benedikt Löwe Universiteit van Amsterdam Large Cardinals and Foundations of Mathematics Abstract
October 15, 2003 Bart de Smit Universiteit Leiden Escher and elliptic curves Abstract
October 29, 2003 Jaap Kaandorp Universiteit van Amsterdam Modelling Developmental Regulatory Networks Abstract
November 12, 2003 Wessel van Wieringen Universiteit van Amsterdam Statistical models for the precision of categorical measurement systems Abstract
November 26, 2003 Johan van de Leur Universiteit Utrecht KP and a discrete family of rational solutions of Painlevé VI Abstract
December 10, 2003 Gunther Cornelissen Universiteit Utrecht Chess- and switchboards in arithmetic geometry Abstract
January 14, 2004 Jan Wiegerinck Universiteit van Amsterdam A question about exp and what came out of it Abstract
January 28, 2004 Leen Stougie Technische Universiteit Eindhoven A Linear Bound on the Diameter of the Transportation Polytope Abstract
February 11, 2004 Daniel Alpay Ben-Gurion University of the Negev Reproducing kernel spaces and the theory of linear systems Abstract
February 25, 2004 Paul Vitanyi CWI & UvA Statistics without probabilities (a la Kolmogorov) Abstract
March 10, 2004 Marius Crainic Universiteit Utrecht On rigidity results Abstract
March 17, 2004 John Kuiper Universiteit Utrecht Brouwer's road to intuitionism Abstract
March 24, 2004 Ute Ebert CWI en TU Eindhoven Branching sparks! -- sparking math? Abstract
April 7, 2004 Gerard Alberts CWI and UvA Aad van Wijngaarden and the ALGOL conspiracy - the battle on research agendas in computer science Abstract
May 12, 2004 Klaas Landsman Universiteit van Amsterdam The Atiyah-Singer index theorem Abstract
May 19, 2004 Misja Nuyens Universiteit van Amsterdam Queues, heavy tails and the Foreground-Background discipline Abstract
May 26, 2004 Odo Diekmann Universiteit Utrecht Population Dynamics, an impressionistic sketch Abstract

Lectures in 2004-2005
Date Lecturer Affiliation Title
September 8 Philip Holmes Princeton University Optimal decisions: From neural spikes, through stochastic differential equations, to behavior
September 15 Daniel Grunberg Max Planck Institut The quintic's quintessence: from 0 to 2875 in 60 minutes
September 22 Jan van Mill Vrije Universiteit Erdös spaces
October 6 Gerard van der Geer Universiteit van Amsterdam Congruences between Modular Forms of Genus One and Genus Two
October 20 Bernard Nienhuis Universiteit van Amsterdam Connections between problems in combinatorics, statistical mechanics and algebraic geometry
November 3 Elena Mantovan Berkeley The role of the geometry of Shimura varieties in the Langlands program
November 17 Torsten Ekedahl Stockholm University Polynomials with simple ramification and some p-adic volumes
December 1 Robin de Jong UvA A potential problem on compact Riemann surfaces
January 12 Michel Mandjes CWI and UvA Large deviations for Gaussian queues
(different time: 16.00-17.00, different room: B.240)
January 26 Nguyen Huu Khanh UvA Global bifurcation to strange attractors in a thermal convection model
February 9 Jasper Stokman UvA Hecke algebras and integrable systems
February 23 Jan van Neerven TU Delft Stochastic integration in UMD-spaces
March 9 Robbert Dijkgraaf UvA String Theory and Melting Crystals
March 23 Harry Buhrman CWI and UvA Quantum Information Processing
April 6 Marco Martens RUG Is the www alive?
April 20 Jan Aarts TU Delft Is there a bronze number?
May 11 Nicolas Guay UvA Variations on a theme of Schur and Weyl
May 25 Charles Dunkl University of Virginia Nonsymmetric Jack Polynomials and Calogero-Moser Models
June 1 Gail Letzter Virginia Tech q special functions and quantum symmetric spaces
July 6 Paulus Gerdes Maputo From the geometry of African sanddrawings to new symmetries and matrices

Lectures in 2005-2006
Date Lecturer Affiliation Title
September 7 Dion Gijswijt Universiteit van Amsterdam Semidefinite programming and coding bounds
September 21 Margit Rösler Universiteit van Amsterdam Bessel convolutions on matrix cones
September 28 Lev Aizenberg Bar-Ilan University Classical results of Bohr and Rogosinski on power series and their multivariate analogs
October 5 Wolter Groenevelt Universiteit van Amsterdam Non-symmetric Wilson polynomials
October 12 Bernhard Krötz Max Planck Institut, Bonn The complex crown of a Riemannian symmetric space
October 19 Bas Kleijn Vrije Universiteit Complementarity and quantum measures
November 2 Torsten Wedhorn Universität Bonn Classification of varieties and the wonderful compactification
November 16 Jan Bouwe van den Berg Vrije Universiteit Braided solutions of differential equations
November 30 Jaap Top Rijksuniversiteit Groningen Real Cubic Surfaces
December 7 Frank van der Meulen Universiteit van Amsterdam Estimation for Lévy processes and induced OU-processes
February 8 Sem Borst CWI/TUE/Bell Labs Flow-level performance in wireless data networks
February 15 Christoph Schweigert Universität Hamburg Frobenius algebras, topological and conformal field theory
February 22 Irene Bouw Heinrich-Heine-Universität Düsseldorf   Teichmüller curves and triangle groups
March 1 Emanuel Diaconescu Rutgers University Gromov-Witten Theory, Localization, and Large N Duality
March 8 Ton Dieker CWI Extremes and fluid queues
March 22 Mats Gyllenberg University of Helsinki Dual semigroups and Volterra functional equations
April 5 Hans Schumacher Universiteit van Tilburg Complementarity modeling of nonsmooth dynamical systems
April 26 Patrick Dehornoy Université de Caen From sets to braids
May 17 Johan Grasman Wageningen Universiteit Exploring persistence in stochastic models of biological populations and its application in chemostats
May 24 Lex Schrijver CWI/UvA Tensors, Invariants, and Combinatorics

Lectures in 2006-2007
Date Lecturer Affiliation Title
September 6 Alain Lascoux Université de Marne-la-Vallée Algebraic computations in several variables
September 20 Andrew Baker University of Glasgow Galois theory in a topological context
October 4 Erdal Emsiz Universiteit van Amsterdam Affine Weyl groups and integrable systems with delta-potentials
October 18 Alexis Kouvidakis University of Crete Symmetric products of curves and connections with the Nagata problem
November 1 Onno van Gaans Universiteit Leiden Invariant measures for infinite dimensional stochastic differential equations
November 22 Thijs Vermaat UvA/IBIS Statistical Process Control in Non-Standard Situations
November 29 Joost Batenburg Universiteit Antwerpen Discrete tomography: exploiting the fact that nature is discrete
December 13 Björn Sandstede University of Surrey Dynamics of spiral waves
February 7 Jaap Korevaar Universiteit van Amsterdam Prime pairs and the zeta function
February 21 Arnoud van Rooij Radboud Universiteit p-adic analysis
February 28 Maarten Solleveld Universiteit van Amsterdam Periodic cyclic homology and non-Hausdorff spaces
March 14 Thomas Quella Universiteit van Amsterdam Supersymmetry, supergeometry and physics
March 28 Antonios Zagaris CWI / Universiteit van Amsterdam Attracting invariant manifolds for systems with multiscale dynamics
April 25 Pieter Collins CWI Computable Analysis and Verification of Nonlinear Systems
May 2 Rob van der Waall UvA 1939-2004 : the development of classifying finite groups by means of n-isoclism of groups
May 30 Nitin Saxena CWI Isomorphism Problems and Cubic Forms Equivalence
June 6 Yde Venema UvA/ILLC Completions of lattice ordered algebras

Lectures in 2007-2008
Date Lecturer Affiliation Title
September 5 Harrie Willems   Wolfgang Doeblin, A Mathematician Rediscovered
September 19 Jonas Bergström UvA Counting points over finite fields and cohomology
October 3 Frank den Hollander Universiteit Leiden Random walk in random scenery
October 17 André Henriques Universiteit Utrecht   The string group
October 31 Ronald Cramer CWI Computing in the dark using algebraic geometry
November 14 Fokko van de Bult UvA Hyperbolic Hypergeometric Functions
November 28   Simon Ruijsenaars University of Leeds Integrable systems, analytic difference equations, special functions, Hilbert space: On the crossroads
December 12 Rob Stevenson UvA Adaptive wavelet methods for solving high dimensional PDE's
February 6 Nicolai Reshetikhin UC Berkeley / UvA On the Kasteleyn's solution of dimer models
February 20 Jochen Heinloth UvA An introduction to some aspects of the (geometric) Langlands program
March 5 Hicham Zmarrou Deloitte Dynamics and bifurcations of random circle diffeomorphisms
March 19 Monique Laurent CWI Real Solving Polynomial Equations with Semidefinite Programming
April 2 Rens Bod UvA (ILLC) Mathematical Universals in Music CANCELLED
April 16 Frits Beukers Universiteit Utrecht Algebraic hypergeometric functions
May 14 Dietrich Notbohm Vrije Universiteit Combinatorics of simplicial complexes and Stanley-Reisner algebras from a topological point of view CANCELLED
May 28 Kareljan Schoutens UvA (ITFA) Quantum hard squares, supersymmetry, and combinatorics

Lectures in 2008-2009
Date Lecturer Affiliation Title
September 3 Sergey Shadrin UvA Operads, PROPs, and graph complexes
September 17 Tilman Bauer Vrije Universiteit Finite loop spaces
October 1 Walter van Suijlekom Radboud Universiteit The structure of perturbative quantum gauge theories
October 15 Hessel Posthuma Universiteit van Amsterdam Index theory from the point of view of quantum mechanics
October 29 Bert Zwart CWI Fluid and diffusion approximations of bandwidth sharing networks
November 12 Bas Spitters Radboud Universiteit A computer-verified implementation of Riemann integration - an introduction to computer mathematics
November 26 Jonas Hartwig Universiteit van Amsterdam Generalized Weyl algebras
December 10 Robin de Vilder Universiteit van Amsterdam The causes and effects of the credit crisis    (download sheets)
January 21 Lawrence Zalcman Bar Ilan University Picard Theorems 1879-2009
February 4 Said El Marzguioui Universiteit van Amsterdam Fine aspects of pluripotential theory
February 18 Sabir Gusein-Zade Independent University of Moscow Poincare series of multi-index filtrations and their generalizations.
March 4 Kostas Skenderis Holography and Mathematics
March 18 Harry van Zanten Vrije Universiteit Amsterdam Prior and Posterior Stochastic Differential Equations
April 1 T.A. Springer Universiteit Utrecht On the work of Jacques Tits
April 15 Sebastian van Strien University of Warwick Games, Fictitious play and Chaos 4pm room P.017 !
April 29 Manfred Lehn Universität Mainz Holomorphic symplectic varieties.
May 13 Rob van der Waall KdV On the life and work of John Griggs Thompson Location: room A1.04, Science Park
May 27 Sander Bais UvA (Institute for Theoretical Physics) The physics of quantum groups and their breaking Location: room A1.04, Science Park

Lectures in 2009-2010
Date Lecturer Affiliation Title
September 2 Han Peters UvA Non-negative polynomials constant on a hyperplane
September 16 Stefan Kolb UvA Weyl group combinatorics and quantum groups
September 30 Remco van der Hofstad TU Eindhoven Critical behavior in inhomogeneous random graphs
October 14 Eduard Looijenga Universiteit Utrecht Some highlights of the work of Mikhail Gromov, Abel laureate of 2009
October 28 Henk Nijmeijer TU Eindhoven The electronic brain: does it synchronize?
November 11 Kees Oosterlee TU Delft / CWI The Heston model with stochastic interest rates and pricing options with Fourier-cosine expansions
November 25 Rob de Jeu Vrije Universiteit What is known about K2 of curves?
December 3 Christoph Schwab ETH Zürich Finite Element Methods for PDEs with Stochastic Coefficients (Odd room, day and time: A1.06, Thursday, 15:00 - 16:00)
December 9 Gunnar Klau CWI Combinatorial optimization and algorithmics for disease classification
February 3 Rob van der Vorst Vrije Universiteit Closed characteristics on non-compact manifolds
February 17 Frank Redig Radboud Universiteit Duality and bosonic particle systems
March 3 Urs Schreiber Universiteit Utrecht Differential geometry in an ∞-topos
March 17 Eric Cator TU Delft The Hammersley interacting particle process
March 31 Vivi Rottschäfer Universiteit Leiden Formation of singularities in natural systems
April 14 Tamás Hausel Oxford University Arithmetic harmonic analysis on character varieties
April 28 Jason Frank CWI / UvA A thermostat model for unresolved dynamics
May 19 Marjan Sjerps NFI / UvA Forensic Statistics: recent developments and brand new plans
May 26 Alessandra Palmigiano UvA, ILLC Dualities for noncommutative spaces
June 17 Matthias Kreck HIM, Bonn Codes, Arthmetic and Manifolds (Odd time and day: Thursday 14:30-15:30!)

Lectures in 2010-2011
Date Lecturer Affiliation Title
September 1 Roland van der Veen UvA Knot invariants: from the Jones polynomial to hyperbolic geometry (Odd room and time: C1.112, 11:00 - 12:00 !)
September 8 Peter Grünwald CWI / UL Statistics without Stochastics
September 22 Gert Vegter Rijksuniversiteit Groningen Geometric Approximation
October 6 no colloquium this week
October 20 Eric Verlinde UvA (ITFA) Emergence of Gravity
November 3 Jan Draisma TU/e Finiteness results in statistics using algebra
November 17 Jan Pieter van der Schaar UvA (KdVI) Cosmological inflation - Theory and Observations
December 1 Jan Brandts UvA (KdVI) Numerical Analysis meets Geometry: Acute and Nonobtuse Simplicial Partitions.
February 23 Tanja Eisner UvA (KdVI) Arithmetic progressions via ergodic theory.
March 2 Christoph Thiele UCLA / U Bonn Carleson's theorem, variations and applications.
March 9 Sindo Nunez Queija UvA / CWI Resource allocation in resource-sharing networks
March 23 Alexander Schönhuth CWI Complete identification of binary-valued hidden Markov processes
April 6 Neil Walton UvA (KdVI) Insensitive, maximum stable allocations converge to proportional fairness
April 20 Mai Gehrke Radboud Universiteit Profinite algebras as dual spaces
April 27 Joop Kolk Universiteit Utrecht Hans Duistermaat: the Man and his Mathematics
May 4 Lenny Taelman UL Believing in the Kummer-Vandiver conjecture
May 11 Thomas Ward University of East Anglia Group automorphisms from a dynamical point of view

Lectures in 2011-2012

Date Lecturer Affiliation Title
September 7 Wil Schilders TU Eindhoven Model Order Reduction: mathematical methods and applications
September 21 Igor Stojkovic TU Delft Gradient Flows, Product formulas, and Maximal Monotone Operators in Metric Spaces
October 5 Karen Aardal TU Delft and CWI Uncapacitated facility location: a problem we "almost" understand
October 12 Lars Diening Universität München A decomposition technique for John domains
October 19 Fokko van de Bult UvA A Mendeleev table for classical orthogonal polynomials: Obtaining the q-Askey scheme using elliptic hypergeometric functions
November 2 Karma Dajani Universiteit Utrecht Two special invariant ergodic measures for random beta transformations (Room D1.110)
November 16 Tanja Lange TU/e Advances in Elliptic-Curve Cryptography (Room D1.110)
November 30 Wieb Bosma Radboud Universiteit Some intriguing aspects of continued fractions (RoomG5.29)
December 14 Martijn Pistorius UvA / Imperial College Optimal dividend distribution in the presence of a penalty (RoomG5.29)
February 1 Bart Vlaar UvA Non-symmetric particle creation operators for the quantum nonlinear Schrodinger model (RoomC1.112 )
February 15 Krzysztof R. Apt CWI Choosing Products in Social Networks (Room C1.112)
February 29 Gil Cavalcanti UU Generalized geometry and T-duality ( Room C1.112)
March 14 Bert Zwart CWI, VU. An encounter with Erlang, Gauss, Poisson and Ramanujan ( Room C1.112)
March 28 Evgeny Verbitskiy UL Dimers, sandpiles and algebraic dynamics (Room C1.112)
April 11 Jose Blanchet Columbia University. Modeling and Efficient Rare Event Simulation of Systemic Risk in Insurance-Reinsurance Networks (Room C1.112)
April 18 Estate Khmaladze Victoria, University of Wellington. One approach to differentiation of set-valued functions and its applicatons to spatial statistical problems (Room C1.112)
April 25 Hans Maassen RU Quantum information, probability, and statistics. (Room C1.112)
May 9 Harrie Willems Late Style - Yuri Manin Looking Back on a Life in Mathematics: A film by Agnes Handwerk and Harrie Willems (Room C1.112)
May 23 Hans Zwart University of Twente Linear port Hamiltonian Systems (Room C1.112)

Lectures in 2012-2013

Date Lecturer Affiliation Title More
September 5 Sameer Murthy NIKHEF Mock theta functions and their appearance in physics (Room C1.112)
September 19 Ragnar Sigurdsson University of Iceland Growth estimates of entire functions and Paley-Wiener theorems (Room C1.112)
October 3rd Koen De Turck University of Gent Poisson's equation for Markov chains and its use in perturbations (Room C1.112)
October 17th Mehdi Tavakol University of Amsterdam Intersection theory on moduli spaces of curves. (Room C1.112)
October 31st WONDER-Afternoon at CWI University of Amsterdam and CWI Wonder-afternoon. (CWI, room Z009 )
November 14th David F. Anderson University of Wisconsin Madison Stochastic models of biochemical systems. (Room C1.112)
November 28th Bob Rink Vrije Universiteit Amsterdam Coupled cell networks: semigroups, Lie algebras and normal forms (Room C1.112)
February 6th Jop Briët CWI Amsterdam Grothendieck's inequality for quantum entanglement and combinatorial optimization (Room C1.112)
February 20th Daan Crommelin CWI Amsterdam Stochastic representation of unresolved scales in atmosphere models (Room C1.112)
March 6th Arthemy Kiselev University of Groningen On the geometry of the Batalin-Vilkovisky Laplacian (Room C1.112)
March 20th Roland van der Veen University of Amsterdam Knots, representations and some physics (Room C1.112)
April 3th Monique Laurent CWI Positive semidefinite matrix completion and geometric graph realizations (Room C1.112)
April 17th Raf Bocklandt University of Amsterdam A noncommutative glance through the mirror (Room C1.112)
May 15th (Room C1.112)
May 29th Tobias Mueller Utrecht University Logic and random graphs (Room C1.112)
Lectures in 2013-2014

Date Lecturer Affiliation Title
September 19 Hitoshi Murakami Tokyo Institute of Technology A survey of the volume conjecture and its generalization. (Room C1.112)
October 3 Harry van Zanten UvA The mathematics of nonparametric Bayesian statistics (Room A1.04)
October 24 Benno van den Berg (ILLC) UvA Nonstandard analysis and proof theory (Room C0.110)
November 21 Sander Dahmen Utrecht Diophantine problems and the modular method (Room C1.112)
December 5 Angela Ortega Humboldt Universität Prym-Tyurin varieties (Room G2.13)
Feburary 5 Miranda Cheng UvA Mock Modular Forms, Niemeier Lattices, and Umbral Moonshine (Room B0.201 )
Feburary 19 Guus Regts Uva Partition functions of edge- and vertex-coloring models: characterizations and connections to invariant theory and Hilbert spaces ( B0.201 )
March 5 Fabian Ziltener Utrecht Coisotropic Submanifolds of Symplectic Manifolds, Leafwise Fixed Points, and a Discontinuous Capacity ( B0.201 )
March 19 Rob van der Mei CWI,VU Towards a unifying theory on polling models: new results and challenges ( B0.201 )
April 2 Teun Koetsier and Jan van Mill VU, VU&UvA Irmgard Gawehn and the topological classification of manifolds. An attempt at rehabilitation. ( SP C1.112)
April 23 Lex Schrijver CWI, UvA GRAPHS, KNOTS, AND ALGEBRAS ( G0.05 )
May 28 Mingmin Shen UvA Differential equations from an algebraic point of view (Room C1.112 )


Abstracts of all lectures since Fall 2001

September 12, 2001  Erik Koelink  (Delft University of Technology )
Special functions, the quantum dynamical Yang-Baxter equation and dynamical quantum groups:
In the representation theory of concrete groups special functions often play an important role. We recall some classical results for the group SL(2,C) and its finite-dimensional irreducible representations related to special functions, as well as the decomposition of tensor products of irreducible representations. The last subject leads to Clebsch-Gordan coefficients (or 3j-symbols) and Racah coefficients (or 6j-symbols). From this we can obtain a solution of the quantum dynamical Yang-Baxter equation in terms of Racah coefficients, which can be traced back to an unpublished paper (1940) by E.P. Wigner (1902-1995). In the second part we discuss the quantum dynamical Yang-Baxter equation in more general terms, and for a specific solution we consider the related dynamical quantum group and its relation to special functions in more detail (joint work with Hjalmar Rosengren (G\"oteborg, Sweden)).
Note: more talks on closely related subjects take place in the workshop Applications of dynamical quantum groups and the KZ equation, 11-12 September

September 19, 2001  Dennis Gaitsgory  (Harvard University) (This lecture regrettably had to be canceled due to transportation problems as a consequence of the terroristic acts on September 11, 2001)
An overview of the geometric Langlands conjecture:
Let $G$ be a connected reductive group. The classical Langlands conjecture establishes a correspondence between the following two objects:
A) Automorphic representations of a group $G$ (with coefficients in adeles over some global field $K$) and
B) Homomorphisms of the Galois group $\rm{Gal}(\overline{K}/K)$ to the {\it Langlands dual} of $G$, denoted $\check G$.
In the geometric context we modify A) and B) as follows. We consider an algebraic curve $X$ and instead of B) we study
B') Homomorphisms of the fundamental group $\pi_1(X)$ to $\check G$.
The case of A') is more involved. It turns out that the "correct" notion to consider is the category of {\it constructible sheaves} on the moduli space $\rm{Bun}_G(X)$ if principal $G$-bundles on $X$.
In the talk we will discuss the motivation and formulation of the geometric Langlands conjecture and a description of results so far obtained in its direction.

September 26, 2001  Eva Hoogland  (University of Amsterdam)
A general picture of definability:
Definability is a property of logics such as compactness or decidability that has been established as a yardstick by which to measure the behavior of logics. In a slogan, the Beth (definability) property states that implicit definability equals explicit definability. These notions will be explained in full detail in the talk. The gist is that implicit definability is a semantic concept whereas explicit definability is a syntactic phenomenon. To say that the two forms of definability coincide (as the Beth property does) may therefore be regarded as an indication that there is a good balance between syntax and semantics of a logic.
Although the literature on definability is extensive, it is of a fragmentary nature. Investigations tend to concentrate on particular logics, and results are scattered throughout the literature. What is missing is an easily accessible introduction to the subject that sketches the general picture. In this talk I would like to present such a picture with an emphasis on giving simple examples.
In the second half of the talk the Beth property will be discussed from an algebraic perspective. I will discuss the relation between logics and algebras and more particular the connection between the Beth property and the algebraic property of having only surjective epimorphisms. An elegant example will be presented which shows the usefulness of this connection.
(Eva Hoogland recently completed a PhD in mathematical logic at the Institute for Logic, Language and Computation, University of Amsterdam. Her dissertation is entitled "Definability and Interpolation: model-theoretic investigations.'')

October 10, 2001  Erik Balder  (Utrecht University)
New results in and applications of topological measure theory:
Weak-strong convergence of product measures appears in various forms in the decision sciences (o.r., statistics, control). Here the product measure space is formed by an abstract measure space and a topological space. New results on this type of convergence will be presented; they are based on a joint generalization of two famous results by Komlos and Prohorov.

October 17, 2001  Herman te Riele  (CWI)
On the distribution of class numbers of real quadratic number fields:
Class groups of number fields play an important role in algebraic number theory. In some sense the class group of a number field D measures the extent to which factorization in D fails to be unique. In particular, factorization in D is unique if and only if the order of the class group, the class number, is 1. For real quadratic number fields of prime discriminant, it is not even known whether infinitely many of them have class number 1, but according to the so-called "Cohen-Lenstra heuristics", published in 1984, a substantial positive proportion of these fields, namely 76%, has class number 1. These heuristics also "predict" the distribution of class numbers > 1.
We shall discuss algorithms to compute class numbers, describe extensive computations of class numbers of real quadratic number fields and compare the frequency distributions which we found with the Cohen-Lenstra heuristics.
This is joint work with Hugh C. Williams of the University of Calgary in Canada.

October 31, 2001  Peter Paule  (Johannes Kepler Universität Linz)
Computer Algebra and Combinatorics: MacMahon's Partition Analysis Revisited:
The talk is directed to a general mathematics audience and introduces classical combinatorial themes along a method discovered by P.A. MacMahon (1854-1929). A significant portion (more than 100 pages) of his famous book `Combinatory Analysis' is devoted to the introduction of Partition Analysis as a method for solving problems in connection with linear diophantine inequalities and equations. But contrary to many other topics initiated by MacMahon, Partition Analysis has not found due attention. (Exception: R. Stanley, 1973.) However, recent joint work with G.E. Andrews (PennState) and A. Riese (RISC-Linz) shows that this `forgotten' method can be brought to new life when supplemented by computer algebra. In particular, new algorithms have been implemented in the form of the Mathematica package `Omega'. Its usage in practical problem solving is illustrated by a variety of examples ranging from additive number theory and counting problems to magic squares and generating functions.

November 7, 2001 Rene Schoof  (Universita di Roma "Tor Vergata")
Class numbers of cyclotomic fields
No good algorithms are known to compute class numbers of cyclotomic fields. In this lecture we describe an experimental approach to the problem.

November 14, 2001  Arjan van der Schaft  (Universiteit Twente)
Hamiltonian formulation of network models of physical systems
In this talk a geometric framework for the modelling and control of physical systems is presented, which is based on a combination of the network and the Hamiltonian approach. Historically, these two approaches have been developed separately from each other. The Hamiltonian approach has its roots in analytical mechanics and starts from the principle of least action, via the Euler-Lagrange equations, towards the Hamiltonian equations of motion. On the other hand, the network approach stems from electrical engineering, and constitutes a cornerstone of mathematical systems theory. While most of the analysis of physical systems has been performed within the Lagrangian and Hamiltonian framework, the network modeling point of view is prevailing in modeling and simulation of complex physical systems.
We introduce a geometric model structure, called port-Hamiltonian systems, which encompasses both the standard Hamiltonian systems as encountered in analytical mechanics, as well as the network type models as arising e.g. in electrical, electro-mechanical and complex mechanical systems. The key idea is to associate with the energetic interconnection structure (based on the ``port'' concept) a geometric object, called Dirac structure. Dirac structures encompass symplectic forms and Poisson brackets, and allow to describe Hamiltonian systems with constraints as arising from the interconnection of sub-systems. It will be indicated that the class of port-Hamiltonian systems is closed under power-conserving interconnection, and how this may be exploited for control and design. Finally we shall discuss the extension of this Hamiltonian description of actuated lumped-parameter systems to distributed parameter systems with energy flow through the boundary of the spatial domain. Key concept is the notion of the (infinite-dimensional) Stokes-Dirac structure based on the conservation laws of the system.

November 21, 2001  Debby Lanser  (CWI)
Efficient numerical methods for atmospheric flow problems
Today, weather and climate predction rely on so-called global circulation models for describing the evolution of the state of the atmosphere on a global scale. A circulation model consists of three main interacting parts, viz. the data assimilation part, the dynamical part and a physical parametrization part. We focus on the dynamical part, which usually contains the primitive equations of the hydrodynamics of the atmosphere and a numerical solution method for solving them.
The accuracy of a prediction depends on the applied numerical method, the resolution of the considered grid, the incorporated data and the physical parametrization scheme. Because the computations are known to be very time-consuming, much interest is directed at the development of efficient numerical methods on high-resolution grids. In this talk, we summarize our achievements in that direction. More precisely, we will discuss efficient numerical methods for solving the shallow water equations (SWEs) in spherical geometry, which serve as a first prototype of the horizontal dynamics in a global circulation model.
We spatially discretize the SWEs by a finite volume method, viz. Osher's scheme combined with a third-order upwind scheme for the constant state interpolation. This spatial discretization scheme is robust and second-order accurate. In addition, it has an excellent boundary treatment which proves useful on the grids considered. Furthermore, it is an upwind scheme of flux difference splitting type.
A common prejudice against finite volume methods concerns their expected inefficiency due to a severe step size restriction when applied on a standard uniform latitudinal-longitudinal (lat-lon) grid combined with an explicit time integration method to solve the resulting semi-discrete system. This prejudice has to do with the pole problem, which includes all problems related to the non-existence of the longitudinal unit vector in the poles and the convergence of the meridians when approaching them. We will discuss two ways to resolve this pole-problem: (1) a combined lat-lon reduced grid with two stereocaps in the polar area and (2) a linearly-implicit Rosenbrock time integration method (Ros3) combined with approximate matrix factorization (AMF) applied to the full Eulerian form of the shallow water equations on a uniform lat-lon grid.
Both remedies are investigated by numerical tests on a well-established test set from the literature. In addition, a comparison between Ros3 with AMF and Strang splitting will be discussed. The latter is also often applied to simplify a solution method and to make it cost effective. Their numerical dispersion relations and their local errors are analyzed.
In conclusion, Ros3 with AMF makes a good candidate for the efficient solution of the SWEs on a global fine lat-lon grid. It is even far more efficient than the solution of the semi-discrete SWEs on a combined grid, which already significantly reduces the step size restriction. Strang splitting is not advocated, in view of its inefficiency due to a large error in the polar area.

November 28, 2001  Bernd Kuckert  (University of Amsterdam)
Thermodynamic equilibrium states of moving quantum systems
A quantum dynamical system is described by a von Neumann algebra (describing the system's physical observables) with a one-parameter automorphism group (describing the time evolution of the observables). Thermodynamic equilibrium states are usually characterized by the KMS-condition, an analyticity property and boundary condition with respect to the time variable. This characterization is confined to one distinguished time evolution, which is unpleasant for both mathematical and physical reasons. Therefore, a criterion is given that tests whether a given state of a quantum dynamial system is a thermodynamic equilibrium state with respect to *some* (a priori unknown) time-evolution. A state that exhibits thermodynamic equilibrium with respect to any time-evolution can be thought of as a vacuum state, and this characterization can be used to derive standard properties of vacuum states such as the positivity of the energy and the Unruh effect.
The talk will be introductory for a broad mathematical audience, and special emphasis will be put on the coincidence of mathematically natural notions with crucial concepts of physics.

December 5, 2001  Roxana Ion  (Eurandom)
New Nonparametric Shewhart Control Charts and Sharp Chebyshev-Type Inequalities
Consider a training sample X1,...,Xn used to estimate the lower control limit (LCL) and upper control limit (UCL) of a Shewhart control chart.
In the first part of this talk several control charts for individual observations are compared. The traditional ones are the well-known Shewhart control charts with estimators for the spread based on the sample standard deviation and the average of the moving ranges. The alternatives are nonparametric control charts, based on empirical quantiles (which are related to the bootstrap method), and some new control charts based on kernel estimators, and extreme-value theory. It will be seen that the performance of our Alternative Empirical Quantile control chart is excellent for all distributions considered.
In the second part of this talk we will study, asymptotically, the probability that a new random variable X, independent of this training sample, is lower than LCL or higher than UCL. Chebyshev's inequality provides an upper bound for this probability under the special case of UCL = -LCL. However, in practice this condition does not always hold. For this reason we have calculated an upper bound on this probability over four different classes of distributions, namely arbitrary, symmetric, unimodal and symmetric unimodal distributions.

January 9, 2002  Alexa van der Waall  (Universiteit Utrecht)
Lamé equations with finite monodromy
The Lamé differential equation L_n(y)=0 is the linear differential equation p(z)y''(z)+{1/2}p'(z)y'(z)-(n(n+1)z+B)y(z)=0 in which p(z)=z3-g2z-g3 is a square-free polynomial in z and n and B are constants. The general question we try to answer is for which n, g2, g3 and B the Lamé equation has an algebraic basis of solutions. In the work of F. Baldassarri and B. Chiarellotto a systematic approach for solving this question was described for the first time. Some of their results will be mentioned in this talk. We extend their ideas by using a classification of the finite monodromy groups of the Lamé equation. This leads to a general algorithm that, given n, B and finite monodromy group, decides for which g2, g3 the Lamé equation has an algebraic basis of solutions.

January 16, 2002  Stefan Steiner  (University of Waterloo) (This lecture regrettably had to be canceled due to transportation problems)
Seven Habits of Highly Effective Industrial Problem Solvers
The Principles of Statistical Engineering

Statistical Engineering (SE), or Shainin method, is the name given to a problem solving and quality improvement system widely used and promoted in industry. Much of the SE methodology, however, is not well-documented or discussed in peer-reviewed journals. The goal of this talk is to provide an overview and critical assessment of the SE approach. The emphasis is on discussion of what we see as the seven guiding principles of SE. The SE approach is also compared and contrasted with some other quality improvement systems. In our assessment, the principles underlying the SE problem solving approach are valuable for many types of problems. However, many of the specific tools promoted in conjunction with the SE approach are not novel, or necessarily the best.

January 23, 2002  Ronald Meester  (Vrije Universiteit Amsterdam)
There is a phase-transition in the Bak-Sneppen evolution model
The Bak-Sneppen evolution model can be viewed as a toy model for evolutionary phenomena. Physicists have studied it in order to see whether or not one can describe such phenomena with self-organising behaviour. In this lecture, I will first describe the underlying paradigma of self-organised criticality, and explain how the Bak-Sneppen model fits in this framework. I will explain why physicists believe that this model shows self-organisational behaviour, and finally I will sketch our contribution, which should be seen as a first step in a mathematical theory of models of this type. (joint work with Dmitri Znamenski)

January 30, 2002  Kees Jan van Garderen  (University of Amsterdam & University of Bristol)
Statistical Geometry in Econometrics and the Exact Geometry of Explosive Autoregressive Models
Statistical curvature of econometric models can have serious consequences for inference. We shall briefly review differential geometry in statistics and show how we have developed and implemented some of the general ideas behind it in econometrics. As a particular application we shall discuss the first order autoregressive model with stable and unit roots, as well as explosive roots larger than unity. We will derive exact expressions for statistical curvature and related geometric quantities in the AR(1) model. We develop a method for deriving exact moments of arbitrary order in general autoregressive models. Of particular interest is the Efron curvature which is continuous and bounded in finite samples, but increases rapidly when the autoregressive parameter changes from stable to explosive values.

February 13, 2002  Rob van der Vorst  (Vrije Universiteit Amsterdam & Georgiatech)
Braids and differential equations
Certain types of ordinary and partial differential equations can be regarded as evolutions of two and three dimensional curves. As such two and three dimensional curves possess interesting topological properties such as braiding and knotting. If the right types of differential equations are considered then there exists a strong relationship between the topological properties of two and three dimensional curves and the evolutionary properties of the differential equations. In particular this relationship will enable us to get a good understanding of all kinds of long term behavior such as periodic orbits, stationary states, chaotic motions, etc. We will develop a Morse type theory to quantify these ideas.

February 20, 2002  Oscar Lemmers  (Umea Universitet)
A decomposition problem in complex analysis
In this talk, we shall discuss a decomposition problem in complex analysis. No special knowledge about this field is required to be able to understand it.
We start with a point p in a domain G in the complex plane, and a bounded function f that vanishes at p. Is there a bounded holomorphic function g such that f(z)= (z-p)g(z) ? The answer is yes; one can simply divide out a factor (z-p) in the power series of f.
We now step into the world of several complex variables : let G be a domain in Cn. We wonder if we can find bounded holomorphic functions f1,...,fn such that f(z) = (z1-p1)f1(z) + ... + (zn-pn)fn(z). Simply dividing out factors does not always work - thus the problem is much harder.
This type of problem is known as the Gleason problem. We shall present the history of the problem and present some recent results.

February 27, 2002  Peter van Emde Boas  (Universiteit van Amsterdam)
Imperfect Information Games; looking for the right model
The theory relating the endgame analysis of "reasonable games" with the complexity class PSPACE, which was developed 25 years ago breakes down for imperfect information games. This creates also aon open slot in an, in other regards, smooth relation between game models and modes of computation. Analysis of Imperfect information games is relevant grace to the fact that the recent applications of game models in computer science all involve imperfect information as one of their essential ingredients. Similarly, imperfect information versions of games used in the logic and games field have been introduced as well. The forthcoming NWO supported InIGMA project which will start this summer is an attempt to extend the existing theory to include imperfect information games as well. In the talk I will indicate the few results available in the literature, what we know about this problem, and how we hope to solve it.

March 6, 2002  Elham Izadi  (Univ. of Georgia, Athens GA)
On the cohomology of hypersurfaces with automorphisms
The study of the cohomology groups of a complex algebraic variety is simpler when the degree of the cohomology group is lower. Lower degree cohomolgy groups form simpler Hodge structures with simpler classifying spaces. For instance, Alcock, Carlson and Toledo exhibited a complex hyperbolic structure on the moduli space of cubic surfaces by associating to each such surface a Hodge structure of weight 1, replacing the Hodge structure of weight 2 obtained as the primitive cohomology of the cubic surface itself.
A different construction was done earlier by Kuga and Satake which to a K3 surface associates a weight one Hodge structure.
Van Geemen generalized these constructions in what he calls a "half-twist": to the data of a Hodge structure with the action of a CM-field, he associates a Hodge structure of lower weight, its half-twist (under certain hypotheses).
In this work (joint with Van Geemen), we consider the case where the Hodge structure is the primitive cohomology of an n-dimensional hypersurface of degree d, cover of Pn totally ramified along an (n-1)-dimensional hypersurface of degree d. In this case the CM-field is a field of roots of unity. We determine when the half-twist exists, and, when it does, embed it into the cohomology of a "nice" algebraic variety. We then prove a Torelli theorem for the half-twist when d=3. Finally, we prove the existence of a "Kuga-Satake" correspondence (a special case of the Hodge conjecture) in the case where it is applicable.

March 13, 2002  Gerard Helminck  (University of Twente)
Integrable hierarchies and flag varieties
In a finite dimensional space, increasing chains of subspaces of a fixed size, so-called flags, form varieties that play an important role in various parts of mathematics like the representation theory of Lie groups and in physics. Hilbert versions of such flags occur naturally in the context of Korteweg de Vries type systems of nonlinear equations and related areas. In the talk an exposition will be given of the role the flag varieties play for these integrable systems.

March 27, 2002  Jeroen de Mast  (IBIS)
Quality Improvement from the Viewpoint of Statistical Method
In the course of the twentieth century statistical methods have come to play an ever more important role in quality improvement in industry. The current research strives after the formulation of a methodological framework for this application of statistics.
Improvement projects that follow statistical method aim at the identification of relations between factors in the production process and the quality characteristic under study. The proposed framework provides among other things definitions of relevant concepts, a phasing of improvement projects, and a number of heuristics and methodological rules for the identification and verification of opportunities for improvement. Well-known statistical techniques are placed in this framework.
For the corroboration of the proposed framework it is studied whether the proposed framework gives an accurate reconstruction of popular improvement strategies such as the Six Sigma programme. Furthermore, the frameworks ability to generate effective improvement approaches is illustrated from two case studies.

April 17, 2002  Paul Beneker  (AOT)
The Bergman space: strongly exposed points and the Bergman projection
In Banach space theory one often seeks to determine the geometry of the unit ball of a given Banach space. A common way to distinguish ``round" and ``flat" parts of the boundary of the unit ball is through extreme and non-extreme points. Among the extreme points, or ``round parts of the boundary, further refinements can be made, for example exposed and strongly exposed points. In this talk, we discuss these notions for the Bergman space of unit disc in C. This space consists of all holomorphic functions which are are-integrable on the unit disc. In particular, we answer the question which polynomials are strongly exposed in the unit ball of the Bergman space. For this the Bergman projection is ideally suited. (Joint work with Jan Wiegerinck, UvA.)

April 24, 2002  Jan Willems  (University of Leuven, RUG)
The behavioral approach to systems and control
The aim of this colloquium is to put forward a mathematical framework that aims at systems in interaction with its environment. Special attention will be paid to linear time-invariant systems. We will discuss some specific issues that arise in this setting, notably

  • Elimination of latent variables
  • Controllability and image representations
  • Observability
The talk will be light on heavy mathematics, and inspired by issues that arise in modeling.

May 8, 2002  Klaas Landsman  (University of Amsterdam)
Quantization and the Baum-Connes conjecture
Quantization, in the sense of passing from classical to quantum mechanics, or, mathematically, from functions on phase space and their Poisson brackets to operators on Hilbert space and their commutators, has always been a somehwat mysterious procedure. A very simple reformulation in terms of groupoids removes the mystery.
The so-called Baum-Connes conjecture is the main issue in noncommutative geometry, a program, developed by Alain Connes, in which quantum mechanics is applied to mathematics itself. In its original formulation, even the statement of this conjecture is almost inpenetrable. However, using the above reformulation of quantization, it is possible to explain to a general mathematical audience what the Baum-Connes conjecture says.

May 22, 2002  Peter Duren  (Ann Arbor)
Zeros of hypergeometric functions
The standard Gauss hypergeometric function is denoted by F(a,b;c;z), and is expressed by a power series convergent in the unit disk |z| < 1 . If a is a negative integer -n , the series terminates and reduces to a polynomial of degree n , called a hypergeometric polynomial. The classical orthogonal polynomials provide important examples. Recent studies, aided by Mathematica graphics, have produced new information about the zeros of hypergeometric polynomials and their behavior as parameters vary, as well as related information about zeros of more general hypergeometric functions. This talk will survey some of those results.

May 29, 2002  Jan Karel Lenstra  (Technische Universiteit Eindhoven)
Whizzkids: Two exercises in computational discrete optimization
In 1996 and 1997 the Department of Mathematics and Computer Science at the Technische Universiteit Eindhoven organized two contests in cooperation with the software firm CMG Nederland and the newspaper De Telegraaf. The purpose of these contests was to increase interest in mathematics and computer science among highschool students. The participants had to construct a newspaper delivery scheme in 1996 and a timetable for a parents' evening at a high school in 1997. Both times they faced an optimization problem which was easy to formulate but hard to solve, and which caused exciting evenings and sleepless nights to both the puzzler at the kitchen table and the advanced algorithm designer.
I will discuss the background of the "Whizzkids contests" and describe how the tools of combinatorial optimization can be applied in finding good solutions and in attempting to prove that no better solutions exist. These tools include upper bounding techniques based on local search and lower bounding techniques using linear programming and constraint satisfaction.

June 19, 2002  Earl J. Taft  (Rutgers University)
Recursive Sequences and Combinatorial Identities
We consider sequences of scalars from a field of characteristic zero, which satisfy a linear homogeneous difference equation in the shift operator D with polynomial coefficients, where a polynomial p(x) acts by Hadamard (pointwise) multiplication by the sequence (p(n)) for n non-negative. These sequences have the structure of a topological bialgebra under Hadamard product. We give algorithms for computing the coproduct of such a sequence. Such a coproduct formula has an interpretation in the dual algebra of polynomials as a combinatorial identity on the terms of the sequence. We give several examples of such identities. All this can be extended to a more general action of a polynomial on a sequence, including Gaussian recursion (i.e., q-recursive sequences). (This is joint work with Carl A. Futia and Eric F. Muller.)

September 11, 2002  Peter Wakker  (Universiteit van Amsterdam/Universiteit van Maastricht)
How to Add up Uncountably Many Numbers? (Hint: Not by Integration)
Using Mathematical Tools to Convince Economists of the Appropriateness of Models.

Being trained as a mathematician, I have tried for 20 years to communicate with economists. I haven't yet succeeded in finding all the maths-entrances to their thinking. This lecture shows part of the path I went so far. It demonstrates how mathematical tools (embeddings of binary relations in the reals through generalized integral functionals) can serve to convince economists of the appropriateness of specific decision models. Along the way, a mathematical hurdle has to be taken: how to add up uncountably many numbers? Integration does not work, because integration requires a natural measure, which need not exist on a general set.

September 25, 2002  Richard Cushman  (Universiteit Utrecht)
Geometric phases in the Euler top.
This talk is about geometric phases in the Euler top, namely the force free rigid body with fixed center of mass. I will start by giving an elementary introduction to the Lie group model of the Euler top, which results in the equations of motion of the top in physical space. I then integrate these equations (without using Euler angles) and find an explict expression for the rotation number of the flow on an Liouville torus of constant energy and length of angular momentum. The rest of the talk shows how to split the rotation number into a sum of a dynamic and geometric phase. Previous authors have only been able to do this modulo 2 pi. I give an exact answer.

October 2, 2002  David Iron  (Universiteit van Amsterdam)
Stability and dynamics of multi-spike solutions to a system of reaction-diffusion equations.
The Gierer-Meinhardt equations are a system of reaction-diffusion equations of activator-inhibitor type. It is known that this system has solutions with highly localized structures or spikes. These spikes, which represent locally elevated levels of concentration of a chemical. These locally elevated levels of concentration are used to model the development of localized structures during fetal development. I will present a brief overview of the history of this model. Then I will discuss my work which uses the method of matched asymptotic expansions to investigate the stability of spike solutions and the dynamics of spike interactions.

October 9, 2002  Michael Müger  (Universiteit van Amsterdam)
From conformal to topological field theory: Equivariant results.
That certain conformal quantum field theories (QFTs) (the rational chiral ones) give rise to topological QFTs (Witten) in 3 dimensions is an old idea in theoretical physics. This can be made rigorous based on
1. Atiyah's definition of a TQFT,
2. Turaev's construction of d=3 TQFTs from `modular categories' and
3. the fact that the representation category of rational chiral conformal field theories is modular (due to Y. Kawahigashi, R. Longo, and myself).
After introducing these matters we explain the quite recent equivariant version, where one replaces 1-3 by
1. a rational chiral CQFT with a finite symmetry group G,
2. a modular crossed G-category, and
3. a K(G,1)-homotopy TQFT in 2+1 dimensions.

October 16, 2002  Jan Willem Polderman  (Universiteit Twente)
A systems theoretic approach to list decoding of Reed Solomon codes.
Let F be a finite field of cardinality at least n. An (n,k) Reed Solomon code is a k dimensional subspace C of the n dimensional vector space over the field F. C is defined as the space of evaluations of all polynomials over F of degree not exceeding k-1. The elements of C are called codewords. Given any vector r the problem of decoding is to find a vector c in C that is closest to c. More generally, given a vector r the problem of list decoding is to find all codewords c in C that are within a given distance of r. Typically c is associated with the transmitted word and r with the received word. The received word may contain errors with respect to the transmitted word.
Recently it has been shown that list decoding may be translated into a bivariate interpolation problem. The interpolation problem is to find a bivariate polynomial of minimal weighted degree that interpolates the n pairs formed from c and r.
We present a systems theoretic approach to this interpolation problem. With the data points we associate a set of time series, also called trajectories. For this set of trajectories we construct the Most Powerful Unfalsified Model (MPUM). This is the smallest possible model that explains these trajectories. The bivariate polynomial is then derived from a specific polynomial representation of the MPUM.
The talk consist of three parts. In the first part we briefly explain the problem of Reed Solomon coding. The second part gives an overview of the systems theoretic ingredients that we use. Particularly we explain the theory of behaviors over finite fields. Finally, in the third part we present the systems theoretic solution to the interpolation problem.

November 6, 2002  Wim Couwenberg  (Reflexis/Katholieke Universiteit Nijmegen)
Diophantine equations after an idea of Lehman.
In 1974 R. Lehman published a deterministic algorithm to factor a positive integer N in O(N1/3log N) steps. His method can be clarified by a simple geometric principle. This principle of "normal approximation" can be readily applied to other diophantine equations besides xy = N. The talk will discuss some quadric and cubic equations and touches topics such as sums of squares and factorization of "squareful" integers.

November 13, 2002  Sander Zwegers  (Universiteit Utrecht)
Mock Theta Functions.
The mock theta functions were ''invented'' by the Indian mathematician S. Ramanujan in 1920. In the last letter he wrote to Hardy, he explained the concept of a mock theta function and provided a list of 17 examples. In this talk I will explain what Ramanujan means by a mock theta function and mention some of the results that were found by Watson, Selberg, Andrews and others, concerning the 17 examples. In the second half of the talk I will give some of my own results.

November 20, 2002  Andries Lenstra  (Eurandom/Universiteit van Amsterdam)
On information bounds.
In empirical sciences, the question at hand is often not a question regarding the available data themselves, but a question regarding the (at least partially) random mechanism that, one thinks, produced these data. In such situations statistical procedures are invoked to extract an answer from the data. Information bounds are bounds for the precision with which this can be done; as such, they provide optimality criteria for statistical procedures.
The best known classical information bound is the Cramér-Rao inequality. It has a history of eighty years, but the customary proofs do little to reconcile us to its truth. We present a viewpoint from which it is as obvious as the observation that in a right-angled triangle the hypotenuse is the longest side. Another famous bound, the van Trees inequality, then follows from Pythagoras' theorem.

December 4, 2002  Chris Stolk  (École Polytechnique, Parijs)
Inversion of seismic data in complex media.
In a seismic experiment one generates acoustic waves in the earth using sources at the surface. The wavefield is recorded by an array of receivers, also located at the surface. The purpose is to construct an image of the subsurface from the reflections present in the data. This leads to a reconstruction problem for the coefficient function in an acoustic partial differential equation. Usually the data is modeled by doing a linearization in the medium coefficient around a smooth background, and using high frequency asymptotics. Inversion for both the medium perturbation and the background must be done, which results in a strongly nonlinear problem. I will first discuss the standard approach to this problem, and then some recent results concerning the case in which wave fronts develop singularities and self intersections. A combination of analytic and geometrical methods is used (microlocal analysis).

January 15, 2003  Bas Edixhoven   (Universiteit Leiden)
Counting solutions of systems of equations over finite fields.
I will first explain what the problem means (i.e., what is a finite field, what kind of equations do we consider), give some examples, and explain how it is related to cryptography. Then I will discuss the currently known algorithms and their complexity and limitations. Finally, I will describe my research plan for the next years concerning the problem of getting rid of at least one limitation: that the characteristic of the field should be small.

January 29, 2003  Andreas Weiermann  (Universiteit Utrecht)
Some Hardy Ramanujan style counting problems
Proof theory and analytical number theory are usually considered to be completely separated fields. In this talk I will discuss problems of a purely analytic character which arose naturally in the context of proof theory. For this purpose we use Hardy's orders of infinity to give an intuitive (ordinal free) presentation of counting very modestly into the transfinite.
The Skolem class is the least set of functions from the natural numbers into the natural numbers such that this class contains the constant zero function and such that the class contains with two functions their sum and with a function its exponential with respect to a base function consisting of the identity function. This class is ordered by eventual domination. For a function in the Skolem class let its norm be the (uniquely determined) number of applications of exponentiations which are used in its generation process. Given a function f in the Skolem class we are interested in the function mapping a natural number n to the number of functions in the Skolem class having a norm equal to n and which are below f with respect to the relation of eventual domination. Our goal consists in classifying the asymptotic behaviour of these count functions with respect to large arguments.
For certain specific functions built up via iterated exponentiation these count functions are well known. The count function for one iterated exponential of the identity is the partition function for which Hardy and Ramanujan published a celebrated asymptotic formula.
We explain where the count functions show up naturally and we present recent results about them and related functions. The (non-logical) methods which are used are from real and complex analysis (Tauberian theorems, singularity analysis and the saddle point method).
If time is left we indicate some applications to probabilistic properties of `randomly' chosen elements from the Skolem class.

February 5, 2003  >Gerton Lunter  (University of Oxford)
Statistical alignment of biological sequences
Alignment of DNA or protein sequences is of everyday importance in biological research, e.g. to find homologues to new genes for functional prediction, or to establish phylogenetic relationships between species based on their DNA. A widely used tool for this is BLAST (Basic Local Alignment Search Tool), a heuristic and score-based algorithm. One drawback of a score-based approach is that parameters must be chosen by `eyeballing' the resulting alignments, which involves much biological expertise. A statistical approach has many advantages, one being the possibility of recovering parameters from the data by maximum likelihood. In this talk I discuss an existing statistical model for sequence evolution, and an efficient implementation on phylogenetic trees, involving a state reduction of the underlying Markov chain. This reduction has connections with Systems Theory, in particular with the concept of observability. Finally I discuss a new model, which is a proper statistical counterpart of the BLAST scoring method.

Febuary 12, 2003  Hans van Duijn  (TU Eindhoven)
Mathematical issues in density driven porous media flow
We will consider various cases of density induced groundwater flow. In the stable case, with respect to gravity, there is a sharp transition between fluids of different density. This case is often described in terms of a multi-dimensional free boundary. We present some examples and discuss the mathematical techniques for analysing them. In the unstable case, salt fingers may appear. We use the method of linearised stability and the energy method to derive stability bounds.

February 26, 2003  Mark Peletier  (CWI/TU Eindhoven)
Continuum modelling of lipid bilayers
One can view lipid bilayers, biological membranes, as a group of molecules that together form a coherent structure, a two-dimensional curved surface, without being chemically bonded to each other. This sentence contains the main question in my talk: why do they do that?
The classical answer, that the hydrophobic tails of the composing lipid molecules group together, does not explain why the resulting structure is planar. Nor does it explain why this planar structure turns out to resist deformation: if you bend the structure, you need to apply a force, as if it were elastic.
In this talk I will take a purely mathematical approach. I will take a simple model, that is only remotely reminiscent of actual lipid bilayers. For this model I will argue that it gives rise to planar structures, and that one may even identify the pseudo-elasatic behaviour.

March 12, 2003  Hessel Posthuma  (Universiteit van Amsterdam)
Quantization and Topological Quantum Field Theory
We give an introduction to the symplectic geometry of moduli spaces of flat connections. In particular we discuss a "classical" manifestation of the axioms of Topological Quantum Field Theory, which suggests a specific approach to the problem of quantization of such spaces.

March 26, 2003  Ton Levelt 
The other D.J. Korteweg: thermodynamics of binary mixtures
In 1891 two papers by D.J. Korteweg appeared in Archives Néerlandaises: "Sur les points de plissement" and "La théorie générale des plis". They contain Korteweg's mathematical research on pleats or "plaits" of surfaces in three-dimensional space and the application to phase equilibria of mixtures of two substances.
As J.D. van der Waals's Ph.D. student Kortweg was well versed in thermodynamics and the deep mathematical understanding of van der Waals's model for this situation must have been a challenge to him. By restricting himself to the so-called "symmetric case" he was able to give a detailed description of the phase equilibria in binary mixtures.
It was known to Gibbs, Maxwell and van der Waals that the phase equilibria in question correspond to bitangent planes to the Helmholtz free energy surface. But it was Korteweg who developed the relevant mathematical theory. In spite of the beauty and completeness of Korteweg's results his papers faded into oblivion. But about 1990 Paul Meijer (CUA) unearthed them and judged them important.
An overview of Korteweg's work on binary mixtures has appeared in Physics Today, December 2002 ("Diederik Korteweg, Pioneer of Criticality" by Johanna Levelt Sengers and Antonius H.M. Levelt). In that paper the emphasis is on physics, basic notions, history and rediscovery in the last century of some results of Korteweg. It also contains biographical information and some of Korteweg's nice diagrams summarizing the results of his research, but little of his mathematics.
In my talk the full accent will be on mathematics, though I'll start with a short introduction to the relevant thermodynamics.

Theorie generale des plis

Illustration taken from "La théorie générale des plis"

April 9, 2003  Remco Peters  (Universiteit van Amsterdam)
Some new insights into the volatility process
It is well known that the distributions of daily log-returns of stock indices, such as the AEX index, display heavy tails and are asymmetric. These characteristics are in contradiction with the assumption that the underlying model is a geometric Brownian motion. Large amounts of intra-day data of the U.S. S&P 500 stock index are used to test the hypothesis that the log-return process may be a time-changed Brownian motion. The hypothesis cannot be rejected. Some of the consequences for modeling financial processes will be discussed. The time change is determined by the quadratic variation, which can be made visible by using the data. There is a close relation between the time-change and the volatility (the variability of the price process). We observe that volatility is very unstable on small time-intervals. However, on a larger time-scale this appears not to be the case. Several regimes in the volatility process may be distinguished. We shall point out some of the consequences.

April 16, 2003  Peter Stevenhagen  (Universiteit Leiden)
Primes is in P
In August 2002, the Indian computer scientists Agrawal, Kayal and Saxena proved that primality of an integer can be tested by means of a deterministic algorithm that runs in polynomial time. For several decades, this had been an outstanding problem. We discuss the importance of the result in theory and practice, and give an impression of the mathematics that goes into it.

May 7, 2003  Rien Kaashoek  (Vrije Universiteit Amsterdam)
A lifting perspective to metric constrained interpolation
In the theory of non-selfadjoint operators the idea to lift an operator to one with a rich spectral theory has proved to be very useful. On the basis of this idea new techniques have been developed for solving interpolation problems of Schur and Nevanlinna-Pick type, with the commutant lifting theorem as one the main abstract results. In this talk this development will be reviewed. Also two new additions to the commutant lifting theorem will be presented. The first is motivated by relaxed versions of the classical interpolation problems, when the usual H-infinity norm condition is replaced by a weaker one. The second is a robust version of the lifting theorem, and solves an old problem, proposed by B. Sz.-Nagy, about extending the commutant lifting theorem to the case when the underlying operators do not intertwine. The talk is based on joint work with Ciprian Foias (Texas A&M) and Art Frazho (Purdue University).

May 21, 2003  Floske Spieksma  (Universiteit Leiden)
Transient properties of random walk type processes
It is a well-known fact that the symmetric random walk on the integer lattice Zd is recurrent in dimensions d=1 and 2. In higher dimensions (d >2) it is transient, that is, the probability of coming back to a state is smaller than 1. Moreover, for Zd the matrix of jump probabilities, the unique bounded solution f to the linear system Pf=f, is the constant function. So, the constant function is the unique bounded harmonic function associated with this walk. This is still the case, when symmetry is lost and there is a non-zero drift.
In applications like queueing, one often has to deal with slightly more general processes called `face-homogeneous random walks': there is a finite partition of the state space Zd, within which the jumps are identically distributed. In the transient case, with each direction D into which into the walk may disappear, one can associate a (bounded, non-negative) harmonic function fD. The value fD(i), with i a state, has the probabilistic interpretation of being the probability of direction D being chosen by the walk starting at i.
In this talk, for some simple examples I will discuss how to identify the disappearance directions as well as the construction of the associated harmonic functions. This involves the construction of suitable Lyapunov functions on the state space.

June 4, 2003  Michal Krížek  (Czech Academy of Sciences, Prague)
From Fermat Numbers to Geometry
The purpose of this lecture is to provide an overview of several of the fascinating properties of Fermat numbers and to demonstrate their numerous appearences and applications in areas such as number theory, probability theory, signal processing, etc. A special emphasis is the employement of geometric interpretations of many number-theoretic results.

September 3, 2003  Hae-Won Uh  (Leids Universitair Medisch Centrum)
Kernel deconvolution
Let Y and Z be independent random variables with probability density functions f and k. Then the random variable X=Y+Z has the density g=f*k where * denotes convolution. Under the assumption that Z is a random noise variable with known distribution, the probability density function f of Y can be estimated from observations X_1,...,X_n. Stefanski and Carroll proposed a deconvolution kernel density estimator, depending on a kernel function w to estimate f.
Usually we distinguish two cases: ordinary smooth and super smooth deconvolution problems. If the tail of the characteristic function of k decreases algebraically, then we are in the ordinary smooth case. In the super smooth case this tail decreases exponentially.
We will mainly discuss asymptotic normality results in super smooth deconvolution problems. In particular we consider deconvolution for the symmetric lambda-stable densities k.
The main results are that, if one uses for w the sinc kernel, the asymptotic distribution is always normal, but that the rates of convergence are quite different for the different values of lambda as well as the asymptotic variance. For 1< lambda <= 2 we will see that the estimator is asymptotically distribution free.

September 10, 2003  Klaas Slooten  (Universiteit van Amsterdam)
A combinatorial generalization of the Springer correspondence for classical type
Affine Hecke algebras appear in several mathematical contexts, for example in the representation theory of p-adic groups, which makes it desirable to study their representation theory. In particular, one would like to know the tempered representations (which are the ones occurring in the Plancherel formula). In some specific cases, their parametrization has been obtained by Kazhdan and Lusztig. This description is related to the so-called Springer correspondence and can be given in terms of Green functions, functions which are originally defined in the context of finite groups of Lie type. For the general affine Hecke algebra, its set of central characters of irreducible tempered representations is known, but several representations might have the same central character. In my thesis I considered the affine Hecke algebra attached to a root system of type B. For this algebra, the specific case covered by Kazhdan and Lusztig admits a combinatorial description. I generalize these combinatorics, and conjecture that they describe the general type $B$ affine Hecke algebra, in the sense that they lead to a parametrization of the irreducible tempered representations with real central character for the affine Hecke algebra with arbitrary root labels. In addition they also describe the decomposition upon restriction to the finite dimensional Hecke algebra. This generalization is essentially obtained by combinatorics on Young tableaux of partitions. In low rank examples which can be checked by hand (and computer), the conjectures have been confirmed.

October 1, 2003  Benedikt Löwe  (Universiteit van Amsterdam)
Large Cardinals and Foundations of Mathematics
In the everyday experience of the average mathematician, there are two infinite cardinalities: countable (like the natural numbers) and uncountable (like the real numbers). It is just an empirical fact that whenever you pick an infinite set of real numbers, it's either countable or there is a rather easily definable bijection with the entire set of real numbers. And yet, set theorists know an infinitude of cardinalities, and logicians claim that so-called large cardinals have an influence on the foundations of mathematics, and even on concrete mathematical questions about concrete mathematical objects (e.g., the real numbers). What are large cardinals? What is the correlation between them and the theory of the real numbers? And why do we rarely (if ever) see those sets of real numbers that are influenced by the existence or nonexistence of these huge objects?

October 15, 2003  Bart de Smit  (Universiteit Leiden)
Escher and elliptic curves
One of M.C. Escher's most intriguing works depicts a man standing in a gallery who looks at a print of a city that contains the building that he is standing in himself. This picture, with the title Print Gallery, contains a mysterious white hole in the middle. It turns out that basic theory of elliptic curves over the complex numbers tells us how to complete the picture. In the course of a 3 year project at the University of Leiden the hole has been filled, and many variations were made. In this talk the mathematics will be explained and illustrated with computer animations.

October 29, 2003  Jaap Kaandorp  (UvA)
Modelling Developmental Regulatory Networks
A model is discussed for simulating regulatory networks that is capable of quantitatively reproducing spatial and temporal expression patterns in developmental processes. The model is a generalization of the standard connectionist model used for modelling genetic interactions, where the terms for the regulation of gene products and the diffusion term have been separated. This model can be coupled with biomechanical models of cell aggregates and can be used to study the formation of spatial and temporal patterns of gene products during development in cellular systems.

November 12, 2003  Wessel van Wieringen  (Universiteit van Amsterdam)
Statistical models for the precision of categorical measurement systems
A measurement system is the collection of instruments and activities that lead to the assigment of values to properties of objects in such a way as to characterize and preserve emperical relationships among objects. One recognizes several types of measurement systems depending on the set type from which assigned values are selectected. If this set consists of two values, say, 0 and 1, one speaks of a binary measurement system. The result of measuring two objects with a binary measurement system is that we can only establish whether the objects - with respect to measured properties - are either equal or different. An ordinal measurement system assigns values from a set consisting of more than two values. This set is supplied with an ordening. As an illustration: consider an ordinal measurement system that measures the quality of coffee. This measurement system assigns the values 'good', 'mediocore' and 'bad'. These three values represent the quality of coffee, where the value 'good' is better than the other values, and the value mediocore is better than the value 'bad'. Such a measurement system enables us to distinquish between the different qualities of coffee. By precision is meant the extent to which a measurement system yields comparable results when a property of an object is measured repetitively. Precision of measurement system is investigated by means of a measurement system experiment. In my thesis methods and statistical models are developed that assess the precision of binary and ordinal measurement systems.

November 26, 2003  Johan van de Leur  (Universiteit Utrecht)
KP and a discrete family of rational solutions of Painlevé VI
The Painlevé VI equation is a second order differential equation with 4 parameters for which the solutions have no essential movable singularities. In this talk I will construct certain rational solutions of this equation by using the geometry of some Grassmannian.

December 10, 2003  Gunther Cornelissen  (Universiteit Utrecht)
Chess- and switchboards in arithmetic geometry
Everyone is familiar with taking the reduction of a number modulo another number. Something similar can be done in geometry, at least with suitable metrics, and one can for example speak of a curve and its reduction. That reduction can look like a graph, sometimes with miraculous information transmission properties (which one can only prove studying the original curve) - and conversely, it is sometimes possible to prove a property of the original curve by looking at its reduction (for example the fact that it has many symmetries).

January 14, 2004  Jan Wiegerinck  (Universiteit van Amsterdam)
A question about exp and what came out of it
This lecture is really about a property that graphs of functions like exp(1/z) may or may not have. The property is that our graph be precisely the minus-infinity set of a plurisubharmonic function.
Plurisubharmonic functions are the several variable analogue of subharmonic functions in the complex plane. They are the subject of study in pluripotential theory.
While exploring the aforementioned property, we will make short visits into pluripotential theory, function theory, and classical potential theory.

January 28, 2004  Leen Stougie  (Technische Universiteit Eindhoven)
A Linear Bound on the Diameter of the Transportation Polytope
The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [Hitchcock, 1941] and independently by Koopmans in 1947 [Koopmans, 1948], and appears in any standard introductory course on operations research. The m x n TP has m supply points and n demand points. Total supply equals total demand. The objective is to minimize total transportation costs. The set of feasible solutions of TP, is called the transportation polytope. The 1-skeleton (edge graph) of this polytope is defined as the graph with vertices the vertices of the polytope and edges its 1-dimensional faces. In 1957 W.M. Hirsch stated his famous conjecture (cf. [Dantzig, 1963]) saying that any d-dimensional polytope with n facets has diameter at most n-d. So far the best bound for any polytope is O( nlog d+1) [Kalai and Kleitman, 1992]. Any strongly polynomial bound is still lacking. We will give a simple proof that the diameter of the transportation polytope is less than 8(m+n-2). The proof is constructive: it gives an algorithm that describes how to go from any vertex to any other vertex on the transportation polytope in less than 8(m+n-2) steps along the edges.

February 11, 2004  Daniel Alpay  (Ben-Gurion University of the Negev)
Reproducing kernel spaces and the theory of linear systems
We review the relationships between the theory of linear systems and reproducing kernels. Then we can discuss related inverse scattering problem. We also explain briefly how the reproducing kernel approach allows one to tackle more general situations such as non-stationary systems and situations where complex numbers are replaced by points on a compact real Riemann surface or by the quaternions.

February 25, 2004  Paul Vitanyi  (CWI & UvA)
Statistics without probabilities (a la Kolmogorov)
As perhaps the last mathematical innovation of an extraordinary scientific career, Kolmogorov in 1974 proposed to found statistical theory on finite combinatorial principles independent of probabilistic assumptions, as the relation between the individual data and its explanation (model), expressed by Kolmogorov's structure function.
In classical probabilistic statistics the goodness of the selection process is measured in terms of expectations over probabilistic ensembles. For current applications, average relations are often irrelevant, since the part of the support of the probability density function that will ever be observed has about zero measure. This may be the case in, for example, complex video and sound analysis. There arises the problem that for individual cases the selection performance may be bad although the performance is good on average, or vice versa. There is also the problem of what probability means, whether it is subjective, objective, or exists at all. Kolmogorov's proposal outlined strives for the firmer and less contentious ground expressed in finite combinatorics and effective computation.
This Kolmogorov's structure function, its variations and its relation to model selection, have obtained some notoriety (many papers and Cover and Thomas textbook on Information Theory) but have not before been comprehensively analyzed and understood. It has always been questioned why Kolmogorov chose to focus on the a mysterious function denoted as hx, rather than on a more evident function denoted as βx (for details see paper referred to below). Our main result, with the beauty of truth, justifies Kolmogorov's intuition. One easily stated consequence is: For all data, minimizing a two-part code consisting of one part model description and one part data-to-model code (essentially the celebrated MDL code), subject to a given model-complexity constraint, as well as minimizing the one-part code consisting of just the data-to-model code (essentially the maximum likelihood estimator), in every case (and not only with high probability) selects a model that is a ``best explanation'' of the data within the given constraint. In particular, when the ``true'' model that generated the data is not in the model class considered, then the ML or MDL estimator still give a model that ``best fits'' the data. This notion of ``best explanation'' and ``best fit'' is understood in the sense that the data is ``most typical'' for the selected model in a rigorous mathematical sense that is discussed below. A practical consequence is as follows: While the best fit (minimal randomness deficiency under complexity constraints on the model) cannot be computationally monotonically approximated, we can monotonically minimize the two-part code, or the one-part code, and thus monotonically approximate implicitly the best fitting model. But this should be sufficient: we want the best model rather than a number that measures its goodness. These results open the possibility of model selection and prediction that are best possible for individual data samples, and thus usher in a completely new era of statistical inference that is *always* best rather than *expected*. Based on joint work with Nikolai Vereshchagin presented at the 47th IEEE Symp on Foundat. Comput. Sci., 2002, Vancouver, Canada.
Nikolai Vereshchagin & Paul Vitanyi, Kolmogorov's Structure Functions and Model Selection
Nikolai Vereshchagin & Paul Vitanyi, Kolmogorov's Structure Functions with an Application to the Foundations of Model Selection.

March 10, 2004  Marius Crainic  (Universiteit Utrecht)
On rigidity results
The plan of this talk is to describe several techniques for proving rigidity results: cohomological methods, analytical methods (Nash-Moser), or more geometrical ones.

March 17, 2004  John Kuiper  (Universteit Utrecht)
Brouwer's road to intuitionismp
In the beginning of the twentieth century a new movement was added to the existing two that attempted to lay a solid foundation for the mathematical building. After Frege, Russell and Couturat, who viewed logic as the ultimate basis for mathematics, and Hilbert's formalist approach in which mathematics is just a manipulation with meaningless signs and symbols, Brouwer worked out earlier ideas by Poincaré and Borel: mathematics has an extra-logical content too.
For Brouwer, the ultimate basis for all mathematics is the ur-intuition of `the move of time', that is, the experience of the fact that two not-coinciding mental events are connected by a time continuum. Departing from this ur-intuition, the whole of mathematics, hence including set theory and geometry, can be constructed. In is early years as an active mathematician (in his own terms: his `first intuitionistic period', between 1907 and, say, 1914; note that most of his time during those years was spent on topology) his constructivistic requirements were very strict: only that what is constructed by the individual mind (mathematics is essentially languageless) counts as a mathematical object. In this lecture we will work this out for the logical figure of the hypothetical judgement in a mathematical context, and we will see that, in hindsight, Brouwer went too far in his constructivism.

March 24, 2004  Ute Ebert  (CWI and TU Eindhoven)
Branching sparks! - sparking math?
The dynamics of propagation, branching and interaction of sparks can now be accessed by ultrafast cameras, and high altitude lightning has been discovered only recently - but can we proceed beyond observations towards a quantitative description?

I will introduce a minimal model for the process that consists of two reaction-advection-diffusion equations with an interesting nonlinear coupling through the Poisson equation of electrostatics. Computations show that solutions of the model can exhibit a multiscale structure where already a single spark channel consists of a thin front region surrounding a rather inert interior. Catching the mathematical essence of the process proceeds through a number of steps: investigation of the ionization fronts, moving boundary approximations for these fronts, solutions of the moving boundary problem with conformal mapping methods and further steps of upscaling. I will discuss exact results for the ionization fronts and exact solutions of the moving boundary problem as well as steps of approximation and open problems.

April 7, 2004  Gerard Alberts  (CWI and Universiteit van Amsterdam)
Aad van Wijngaarden and the ALGOL conspiracy - the battle on research agendas in computer science
Agenda fights in computer science all but burst into bodily conflict in 1962. Edsger Dijkstra and Klaus Samelson succeeded in keeping their masters Aad van Wijngaarden and Friedrich Bauer apart.

Van Wijngaarden
Aad van Wijngaarden (1916-1987) grew to be the founding father of computer science in The Netherlands, and he left his traces in the international development of the field. By formation Van Wijngaarden was engineer. The style of Applied Mechanics in which he was trained involved daring efforts in computing, not only tedious and long but also insightful and innovative, e.g. designing novel schemes of calculation. Thus, when in 1947 he was called to be the head of the Computing Department of the newly founded Mathematical Center in Amsterdam, he knew what computing was about. With the advent of automatic computers in the 1940's and 1950's computing evolved and Van Wijngaarden evolved with it. From Numerical Analysis to Programming, to the Design of Programming Languages and to Software Engineering were big leaps in the development of the discipline of computer science.

ALGOL conspiracy
The Applied Mechanics background provided international connections. Van Wijngaarden attended several European conferences in the late forties and early fifties of which the 1955 meeting in Darmstadt was the most important one. In the wake of that conference a group of German and Swiss computers scientists took up the aspiration to develop a programming language allowing to directly formulate problems in mathematical, algorithmic, terms. Here started what Friedrich Bauer, accepting a term coined by Peter Läuchli, called the ALGOL conspiracy, Die ALGOL-Verschwörung. At that point Van Wijngaarden, although almost present at its inception, was not involved. He entered the scene in 1959, when ALGOL was on its way to international recognition. More than an internal transition from ALGOL 58 to ALGOL 60, this process involved setting an agenda for research in the field of computer science.

Van Wijngaarden and his team readily accepted this agenda for a universal -as to the mathematics- and machine-independent language. Soon, however, he propounded his own particular views on what universality and independence might mean. This divergence of taste brought him into conflict with the original conspiracy. The 1962 culmination of their conflict coincided with IFIP's adoption of the ALGOL-group as its Working Group 2.1. Suffice it to combine Bauer's statement that in 1962 he lost faith in the ALGOL agenda, with Van Wijngaarden's chief authorship of ALGOL 68. By 1968 software engineering became the new research agenda, defined once more by Friedrich Bauer in convening the Nato conference in Garmisch Partenkirchen. This was the leap Van Wijngaarden did not take. Finishing his own ALGOL 68 job, he did not attend the conference and never got "into" software engineering.

May 12, 2004  Klaas Landsman  (Universiteit van Amsterdam)
The Atiyah-Singer index theorem
The 2004 Abel prize in mathematics has recently been awarded to M.F. Atiyah and I.M. Singer in recognition of their so-called index theorem. This theorem combines analysis, topology, and geometry, and some proofs have even used probability theory. The index theorem has very wide applications, ranging from number theory to quantum field theory. This talk is an introduction to the Atiyah-Singer index theorem and its significance, starting from elementary linear algebra.

May 19, 2004  Misja Nuyens  (Universiteit van Amsterdam)
Queues, heavy tails and the Foreground-Background discipline
A large number of real world phenomena may be modelled by a mathematical queue. One should not only think of the counter in the post office, but also of an internet router transmitting files. Such a mathematical queue consists of three ingredients. The time between two successive arrivals to the queue, and the amount of service that a customer needs (his `service time'), are described by probability distributions.
The third ingredient is the way in which the customers are served. This is called the (service) discipline. By means of the discipline one may influence the behaviour of the queue. The question which discipline to use is therefore a crucial one.
Recently, distributions of files sizes in internet traffic were shown to have so-called heavy tails. This means that very large values show up relatively often. For service-time distributions with heavy tails, classical queueing disciplines fail. For example, in the queue with the FIFO (first in first out) discipline, one very large customer may block the server for a long time, and the queue length will grow dramatically.
A good alternative is the Foreground-Background discipline, which will be introduced in the talk. The queue with this discipline works effectively, even for heavy-tailed distributions of the service times.

May 26, 2004  Odo Diekmann  (Universiteit Utrecht)
Population Dynamics, an impressionistic sketch
The aim of this lecture is to give a mathematical-bird's-eye view of population dynamics (including the spread of infectious diseases).
A unifying theme will be that individuals interact with each other via an environmental feedback mechanism : their survival probability, physiological development and reproductive output depend on the environmental conditions (like food availability and predation pressure or, in the context of diseases, force of infection) and these, in turn, are affected by the individuals.
We pay some attention to the "art" of averaging and to spatial heterogeneity (the problem of pattern and scale). In addition we briefly touch upon phenotypic evolution by natural selection.

September 8, 2004  Philip Holmes  (Princeton University)
Optimal decisions: From neural spikes, through stochastic differential equations, to behavior
There is increasing evidence from in vivo recordings in monkeys trained to respond to stimuli by making left- or rightward eye movements, that firing rates in certain groups of `visual' neurons mimic drift-diffusion processes, rising to a (fixed) threshold prior to movement initiation. This supplements earlier observations of psychologists, that human reaction time and error rate data can be fitted by random walk and diffusion models, and has renewed interest in optimal decision-making ideas from information theory and statistical decision theory as a clue to neural mechanisms.
I will review some results from decision theory and stochastic ordinary differential equations, and show how they may be extended and applied to derive explicit parameter dependencies in optimal performance that may be tested on human and animal subjects. I will then describe a biophysically-based model of a pool of neurons in a brainstem organ - locus coeruleus - that is implicated in widespread norepinephrine release. This neurotransmitter can effect transient gain and response threshold changes in cortical circuits of the type that the abstract drift-diffusion analysis requires. I will argue that, in spite of many gaps and leaps of faith, a rational account of how neural spikes give rise to simple behaviors is beginning to emerge.
This work is in collaboration with Eric Brown, Rafal Bogacz, Jeff Moehlis and Jonathan Cohen (Princeton University), and Ed Clayton, Janusz Rajkowski and Gary Aston-Jones (University of Pennsylvania). It is supported by the National Institutes of Mental Health.

September 15, 2004  Daniel Grunberg  (Max Planck Institut)
The quintic's quintessence: from 0 to 2875 in 60 minutes
We'll count the number of lines in a quintic hypersurface of CP4. This goes along the gentle slopes of enumerative geometry: How many lines in space intersect 4 given lines ? How many lying in a plane and meeting a 2 lines ? How many lying in a cubic surface ? We'll explain the problem of excess intersection and use it to solve the puzzle of counting conics through 5 points in the plane - or tangent to 5 lines. It's easy, fun, and can win you the lucky number.
September 22, 2004  Jan van Mill  (Vrije Universiteit)
Erdös spaces
Let M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary countable dense subset of M. Consider the topological group H(M,D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M,D) as follows. If M is a one-dimensional topological manifold then H(M,D) is homeomorphic to Q, the countable power of the space of rational numbers. In all other cases we found that H(M,D) is homeomorphic to the famed Erdös space, which consists of the vectors in Hilbert space l2 with rational coordinates. We obtain the second result by developing a topological characterization of Erdös space. This is joint work with Jan Dijkstra.

October 6, 2004  Gerard van der Geer  (Universiteit van Amsterdam)
Congruences between Modular Forms of Genus One and Genus Two
Modular forms for the group SL(2,Z) form a classical topic and occur in almost all branches of mathematics, from number theory, topology to mathematical physics. Siegel modular forms form a natural generalization, but remain rather mysterious. In joint work with Carel Faber we used curves over finite fields to learn much about Siegel modular forms of genus 2. We could provide a lot of support for a conjecture of Harder that predicts congruences between modular forms of genus 1 and genus 2. The talk will give an introduction to this topic.

October 20, 2004  Bernard Nienhuis  (Universiteit van Amsterdam)
Connections between problems in combinatorics, statistical mechanics and algebraic geometry
The problem of percolation on a strip or cylinder can be translated into properties of an eigenvector of a representation of a simple operator in the Temperley-Lieb algebra. The components of this eigenvector have been observed to be equal to specific counts of the so-called alternating sign matrices. This relation has now been studied for some five years, but remains unproven to this day. More recently in collaboration with Jan de Gier we studied the analogous eigenvector associated with the Brauer algebra. Some components of this vector turn out to be equal to the degree of the variety of a pair of commuting matrices. Again an unproven observation, which, if true, gives an efficient way to calculate this degree.

November 3, 2004  Elena Mantovan  (Berkeley)
The role of the geometry of Shimura varieties in the Langlands program
In a letter to André Weil in 1967, Langlands suggested the existence of a relation between two seemingly unrelated mathematical objects: Galois representations and automorphic representations. Since then, the work of many mathematicians focused on isolating and constructing algebraic varieties whose geometry is supposed to explain the existence of such correspondences. For correspondences defined over a number field, this role is played by Shimura varieties. In my talk I will discuss some aspects of the geometry of the Shimura varieties and how they reflects Langlands' conjectures.

November 17, 2004  Torsten Ekedahl  (Stockholm University)
Polynomials with simple ramification and some p-adic volumes
The density of a subset of the lattice of integer vectors often appears as product of factors at each prime. In a particular case, the density of monic integral polynomials a root of which generate a number field with only simple ramification, this turns out to be the case and each factor is a p-adic volume. Starting with the definition of density I will show how the p-adic volumes appear and how they can be computed using some simple changes of variables.

December 1, 2004  Robin de Jong  (UvA)
A potential problem on compact Riemann surfaces
We pose and solve a certain potential problem on compact Riemann surfaces. The motivation for this potential problem stems from Arakelov theory and its solution has been used recently by Edixhoven to estimate the complexity of an algorithm of his to compute the coefficients of the Fourier expansion of the discriminant modular form. No prior exposure to Riemann surfaces is required: we introduce all concepts necessary to state the problem.

January 12, 2005  Michel Mandjes  (CWI and UvA)
Large deviations for Gaussian queues
In this talk I'll discuss Gaussian queues, or, more precisely, the large deviations of queues with Gaussian inputs. Research on Gaussian processes has a substantial tradition, but its relevance to telecommunication engineering has been recognized only recently. The inherent flexibility of the Gaussian traffic model enables the analysis of both long-range and short-range dependent input models in a single mathematical framework. We show how any traffic model has its Gaussian counterpart, and how the analysis of the resulting Gaussian queueing model can be done.
Emphasis is on rare-event analysis: what is the probability of buffer overflows, or extremely long delays? Exact computations being too complex, we present a collection of asymptotic techniques, usually referred to as 'large deviations'. I'll review the main results on large-buffer asymptotics (that were mainly derived during the 1990s). Relatively new are the so-called many sources asymptotics: relying on Schilder's theorem, I've found elegant explicit results, not only for single-node FIFO queueing systems, but also for queues operating under more complex scheduling disciplines as well as queueing networks.
A part of my talk is devoted to applications from communication engineering. We focus on procedures for resource provisioning, and the required measurement procedures (which can be done elegantly in our Gaussian setting). Another application relates to the weight setting problem in Generalized Processor Sharing (GPS).

January 26, 2005  Nguyen Huu Khanh  (UvA)
Global bifurcation to strange attractors in a thermal convection model
We consider a model for the development of spatio-temporal structures in Rayleigh-Benard convection. The model consists of ODEs obtained by projection to four modes. It possesses a (Z/2Z) x (Z/2Z) symmetry and depends on the Rayleigh number R and the Prandtl number P. A numerical investigation identifies global bifurcations that organise the bifurcation diagram. The bifurcations, in particular heteroclinic cycles with a double principal stable eigenvalue at the origin, two pairs of heteroclinic cycles with a resonance condition among eigenvalues, and an inclination flip of a pair of homoclinic loops, are analysed. The unfoldings of these bifurcations are shown to give Lorenz type strange attractors.

February 9, 2005  Jasper Stokman  (UvA)
Hecke algebras and integrable systems
Symmetries underlying integrable particle systems on the line are governed by Hecke algebras. I will discuss this fundamental insight for particles with pair-wise delta-function interactions. I will shortly review the role of Hecke algebras in more advanced particle systems.

February 23, 2005  Jan van Neerven  (TUD)
Stochastic integration in UMD-spaces
It is well known that the theory of stochastic integrals can be extended to Hilbert space-valued processes in a very satisfactory way. The reason for this is that the Itô isometry is an L2-isometry. At the same time this indicates that serious obstructions may be expected for setting up a theory of stochastic integration for Banach space-valued processes.
Recent work by Brzezniak, Veraar, Weis and the speaker has shown that such an extension is nevertheless possible if one reinterprets the Itô isometry in a suitable operator-theoretic framework. The rôle of L2-spaces is then replaced by the operator-ideal of space of radonifying operators, which in the Hilbert space case coindices with the operator ideal of Hilbert-Schmidt operators.
In this talk we outline the main features of this theory, first for functions with values an an arbitrary Banach space and then, by a decoupling approach, to processes with values in a UMD Banach space.

March 9, 2005  Robbert Dijkgraaf  (UvA)
String Theory and Melting Crystals
Gromov-Witten theory studies quantum invariants of Kaehler manifolds, obtained from the moduli space of maps of Riemann surfaces into the manifold. These invariants are usually very difficult to compute. Recently, ideas from string theory have suggested a very different approach to these invariants, that leads to a remarkable simplification. Crucial objects in this so-called Donaldson-Thomas theory are vector bundles, or more generally sheaves. In the case of toric three-folds this leads to a suggestive picture of a melting crystal that captures the quantum manifold.

March 23, 2005  Harry Buhrman  (CWI and UvA)
Quantum Information Processing
The new paradigm of quantum computing makes use of quantum mechanical effects to speed up computation. It has been shown by Shor that factorization of a number M can be done in polynomial time on a quantum computer. In comparison the best known classical algorithms take close to exponential time. Whereas classical computers operate on bits, quantum algorithms make essential use of bits in superposition: qubits.
Qubits can---just as classical bits---be used to code information. A fundamental result in quantum information theory by Kholevo (1973) shows that k qubits can not contain more information than k classical bits. Nevertheless it can be shown that communication via qubits can drastically reduce the communication cost in the setting of Distributed Quantum Computations.
We will give an introduction and overview of results obtained in this area.

April 6, 2005  Marco Martens  (RUG)
Is the www alive?
Alan Turing was aware that artificial intelligence can not be understood in terms of a discrete theory of computers. Half a century later, computer science has still not yet been able to construct artificial intelligence. Our brain, as the neurologists say, is just a set of neurons. However, we are intelligent. Apparently very large discrete systems show "phase transitions". The colloquium will discuss surprising phenomena at the transition from discrete mathematics to analysis, continuous mathematics.

April 20, 2005  Jan Aarts  (TU Delft)
Is there a bronze number?
The silver number is defined as the real root of the equation: the cube of X equals X plus 1. It was christened by Midhat J. Gazale in his book "Gnomon, from pharaohs to fractals", Princeton University Press 1999. The silver number shares many properties with the golden number. Both numbers play an important role in arts and architecture. In this talk we shall discuss several properties that the golden and silver numbers have in common. These properties are both algebraic and geometric in nature. We shall present a proof of the following conjecture of Gazale. Suppose that V and W are polygons in the plane which have no interior points in common. Then W is called a gnomon of V if the union of V and W is similar to V (So adding W to V does not change the shape of V). The square is a gnomon for the golden rectangle and the equilateral triangle is a gnomon for the silver pentagon. Gazale conjectured that if the gnomon W is a regular polygon then V must be the golden rectangle or the silver pentagon.

May 11, 2005  Nicolas Guay  (UvA)
Variations on a theme of Schur and Weyl
During the first movement (andante), I will present a classical result of I. Schur and H. Weyl which provides an equivalence of categories between representations of the symmetric group Sl and certain representations of the Lie algebra sln. Decades later, when quantum groups were introduced, similar equivalences were proved by Jimbo, Drinfeld and others, with the symmetric group replaced by a Hecke algebra and sln by a quantum group. The second movement (allegro moderato) will give an overview of these different generalizations, and one application, due to M. Varagnolo and E. Vasserot, to the construction of Fock spaces, which are important objects in mathematical physics. In the last movement (prestissimo), I will describe recent work on a Schur-Weyl type of equivalence involving Cherednik algebras and Yangians of affine type.

May 25, 2005  Charles Dunkl  (University of Virginia)
Nonsymmetric Jack Polynomials and Calogero-Moser Models
This is an overview of the technique of differential-difference operators associated with finite reflection groups in the context of complete integrability of certain Hamiltonians. The symmetric and hyperoctahedral groups are the main symmetry groups to be discussed. The wave functions for some Calogero-Moser models on the circle and on the line can be expressed in terms of nonsymmetric Jack polynomials, a family of orthogonal polynomials of several variables.

June 1, 2005  Gail Letzter  (Virginia Tech)
q special functions and quantum symmetric spaces
There are many interconnections between three ares of mathematics: special functions, Lie theory, and symmetric spaces. There are now q versions of all three theories. The subject of q special functions is quite old. The introduction of quantum groups in the 1980's provided q analogs of Lie groups and Lie algebras. This talk is an overview of the q version of the last subject, namely, quantum symmetric spaces. Their construction, using quantum groups, is discussed. Zonal spherical functions on quantum symmetric spaces are defined and identified with Macdonald's family of orthogonal polynomials.

July 6, 2005  Paulus Gerdes  (Maputo)
From the geometry of African sanddrawings to new symmetries and matrices
An introduction to the tradition of 'sona' sanddrawings from East Angola and neighbouring areas of Congo and Zambia will be presented, as well as a discussion of the reconstruction of mathematical aspects of the 'sona' (for instance, algorithms and chain rules for 'sona'). It will be shown how the analysis of 'sona' led to the discovery of new concepts as Lunda-designs and Liki-designs that display interesting symmetries and conducts to the introduction of several types of matrices with attractive visual properties.
[1] Paulus Gerdes: Une tradition géométrique en Afrique - Les dessins sur le sable, L'Harmattan, Parijs, 1995 (3 volumes)
[2] Paulus Gerdes: Ethnomathematik am Beispiel der Sona Geometrie, Spektrum Verlag, Heidelberg, 1997
[3] Paulus Gerdes: Geometry from Africa, The Mathematical Association of America, Washington DC, 1999

September 7, 2005  Dion Gijswijt  (Universiteit van Amsterdam)
Semidefinite programming and coding bounds
Semidefinite programming has become an important tool in combinatorial optimization. We will give a short introduction to semidefinite programming, and discuss some applications to combinatorial optimization problems. In particular, we will show how semidefinite programming can be used to improve existing bounds for error-correcting codes.

September 21, 2005  Margit Rösler  (Universiteit van Amsterdam)
Bessel convolutions on matrix cones
Multivariable hypergeometric functions are an important tool in the analysis on symmetric cones. Much of the interest in them is driven by applications in number theory and multivariate statistics, and the modern theory of hypergeometric functions associated with root systems has parts of its origin in the early work of Herz, James and Constantine for matrix cones.
In this talk we shall focus on Bessel functions on cones of positive definite matrices which occur naturally in the study of radial problems on Euclidean matrix spaces. We shall explain how the geometric interpretation of the Bessel functions with half-integer indices can be used to obtain convolution structures and explicit product formulas for Bessel functions within the full index range. These results have interesting applications to the theory of Dunkl operators which are also outlined.

September 28, 2005  Lev Aizenberg  (Bar-Ilan University)
Classical results of Bohr and Rogosinski on power series and their multivariate analogs
Classical Bohr's theorem (1914) asserts that if the modulus of the sum of a power series in the unit disc is less than 1, then the sum of the absolute values of the terms of this series is less than 1 in the disk of radius 1/3, and this constant 1/3 is sharp. A new very simple proof of this theorem will be given in the lecture.
Rogosinski's theorem (1923) asserts that in the same situation, each partial sum of the series is less than one in modulus in the disc of radius 1/2, and here 1/2 is also sharp. This first part of the lecture is available even to students. In the second part multivariate analogs of these results will be discussed.

October 5, 2005  Wolter Groenevelt  (Universiteit van Amsterdam)
Non-symmetric Wilson polynomials
The Hankel transform is a self-dual Fourier transform with a Bessel function as a kernel. A "non-symmetric" version of the Hankel transform can be studied using a degeneration of a double affine Hecke algebra. We use a similar approach to study non-symmetric versions of the Wilson polynomials and the corresponding Fourier transform. In this way several properties, such as the orthogonality relations and the duality property, of the (non-)symmetric Wilson polynomials are obtained.

October 12, 2005  Bernhard Krötz  (Max Planck Institut für Mathematik, Bonn)
The complex crown of a Riemannian symmetric space
Our concern is with Riemannian symmetric spaces X =G/K of the non-compact type, i.e. the Lie-theoretic generalizations of the upper half plane H=SL(2,R)/SO(2). For such a space X we introduce its natural complexification, the so-called complex crown of X. The crown domain arises naturally in several contexts, for example when one considers Stein extensions of X or analytic continuation of representations of the symmetry group G. Guided by the example of the upper half plane we will will explain some basic features of the crown domain. Finally we will give some interesting applications to number theory.

October 19, 2005  Bas Kleijn  (Vrije Universiteit)
Complementarity and quantum measures
The complementarity principle refers to the interpretation of quantum physics by classical observers. In the present context, it gives physical meaning to theorems concerning the relations that exist between quantum and ordinary probability theory. From a mathematical perspective, they allow one to identify the building blocks suitable for an approach to quantum stochastics analogous to measure-theoretic probability.
Guided by complementarity, we define quantum measures (with appropriately restricted domain), measurability of observables, integrals with respect to a quantum measure, etc and demonstrate that they have the properties they should have. Appropriate generalizations of the theorems on which measure-theory hinges are also given (for instance, the monotone class theorem and monotone convergence for quantum integrals).
To demonstrate usefulness, we conclude with a non-commutative generalization of the Radon-Nikodym theorem (which is also valid in non-separable Hilbert spaces), that covers (a version of) Gleason's theorem, Schrödinger's time-evolution, the Radon-Nikodym theorem in ordinary probability theory and conditioning in quantum stochastics.

November 2, 2005  Torsten Wedhorn  (Universität Bonn)
Classification of varieties and the wonderful compactification
We use a semi-linear variant of a compactification of the projective linear group PGL(n) (the so-called wonderful compactification) to classify invariants associated to algebraic varieties.

November 16, 2005  Jan Bouwe van den Berg  (Vrije Universiteit)
Braided solutions of differential equations
The comparison principle for scalar second order parabolic partial differential equations (PDEs) admits a topological interpretation: after lifting the graphs to Legendrian braids, the curves evolve as to decrease the algebraic length of the braid. Via discretization we define a suitable Conley index, which gives a toolbox of purely topological methods for finding invariant sets of scalar parabolic PDEs. There is a close connection with twist maps and in this context it applies to (variational) fourth order ordinary differential equations.

November 30, 2005  Jaap Top  (Rijksuniversiteit Groningen)
Real Cubic Surfaces
Cubic surfaces belong to the classical subjects of algebraic geometry. There exist precisely 27 straight lines in any smooth cubic surface, and any such surface is obtained by blowing op the plane in six points. In the talk, we discuss examples and applications, and show that some classical results on cubic surfaces over the complex numbers, do not hold over the reals.

December 7, 2005  Frank van der Meulen  (Universiteit van Amsterdam)
Estimation for Lévy processes and induced OU-processes
Many models within the field of financial mathematics are defined by stochastic differential equations. Whereas traditionally these equations are driven by a Brownian motion, nowadays the driving process is often assumed to be a more general process. One possibility which has been proposed consists of replacing the Brownian motion by a Lévy process, a process with independent and stationary increments. The introduction of more difficult models has resulted in new statistical problems.
In this talk I will first motivate the use of these models. Then I will turn attention to Levy processes and their induced Ornstein-Uhlenbeck processes. The latter is a type of stochastic processes that has gained some popularity in finance. After an introduction of these processes and their characterization, I will discuss related statistical problems. The focus will be on estmation of an infinite dimensional parameter.

February 8, 2006  Sem Borst  (CWI/TUE/Bell Labs)
Flow-level performance in wireless data networks
Channel-aware scheduling algorithms provide an effective mechanism for improving throughput performance in wireless data networks by exploiting multi-user diversity. In this talk, we focus on the flow-level performance of channel-aware scheduling algorithms in a dynamic setting with random finite-size data transfers. We present simple conditions for flow-level stability, and show that in certain cases the flow-level performance may be evaluated by means of a Processor-Sharing model where the service rate varies with the total number of users. Time permitting, we conclude with a discussion of capacity issues and flow-level performance in network scenarios with several interacting base stations.

February 15, 2006  Christoph Schweigert  (Universität Hamburg)
Frobenius algebras, topological and conformal field theory
We first recall that every semi-simple symmetric Frobenius algebra allows to construct a two-dimensional topological field theory in the sense of Atiyah. We then present a generalization of this construction in the framework of braided tensor categories which allows to construct the correlation functions of rational conformal field theories. This leads to a dictionary between physical concepts and algebraic notions in braided categories.

February 22, 2006  Irene Bouw  (Heinrich-Heine-Universitüt Düsseldorf)
Teichmüller curves and triangle groups
Teichmüller curves are algebraic curves isometrically embedded in the moduli space of curves Mg. They arise from the study of the dynamics of billiards. In this talk we construct a Teichmüller curve uniformized by the Fuchsian triangle group Δ(n, m, infty), for every m,n. The construction relies on properties of hypergeometric differential equations.

March 1, 2006  Emanuel Diaconescu  (Rutgers University)
Gromov-Witten Theory, Localization, and Large N Duality
Gromov-Witten theory is an area of algebraic geometry concerning enumerative aspects of curves on algebraic varieties. Many important developments in this field, such as mirror symmetry and integrability results, are based on the Atiyah-Bott localization theorem. In this talk we will present a new approach to localization in Gromov-Witten theory based on large N string duality. We will focus on the interplay between localization, Chern-Simons theory and extremal transitions, emphasizing recent developments for non-toric Calabi-Yau threefolds.

March 8, 2006  Ton Dieker  (CWI)
Extremes and fluid queues
One of the cornerstones of queueing theory is the single-server queue, in which customers arrive to a system (counter, call center, elevator, or traffic light), possibly wait, and subsequently leave the system. However, for some applications (e.g., modern communication networks), individual customers are so small that they can hardly be distinguished. It is then easier to imagine a continuous stream of work that flows into the system. The resulting queueing model is called a fluid queue. In my talk, I will first show that fluid queues are closely related to extremes (e.g., the maximum) of a stochastic process. The advantage of studying fluid queues through extremes is that the theory is also relevant for risk theory and financial mathematics. To give an impression of the techniques that can be used to study extremes, a few examples will be worked out in somewhat detail. I'll also treat networks of fluid queues.

March 22, 2006  Mats Gyllenberg  (University of Helsinki)
Dual semigroups and Volterra functional equations
Volterra functional equations (including the well-known integral equations) induce semi-flows (or semi-groups) on the history space. The fully nonlinear equations are notoriously difficult. However, a certain trick transforms the problem into a semilinear problem for which there exists a variation-of-constants-formula. The price one has to pay is that the nonlinear perturbation maps into a space bigger than the originally chosen history space.
In this talk I show how the theory of adjoint (or dual) semigroups provides a canonical method for constructing this bigger space and for embedding the original problem into the bigger space. The variation-of-constants formula is a convenient tool for proving the the principle of linearize stability and instability as well as the Hopf bifurcation theorem.
The abstract framework leads to simplified proofs of known theorems, but also completely new results.

April 5, 2006  Hans Schumacher  (Universiteit van Tilburg)
Complementarity modeling of nonsmooth dynamical systems
Many phenomena in engineering as well as in biology and in economics are most naturally described by dynamic models that exhibit regime-switching behavior. Examples include stick-slip models in mechanics, ideal diode models in circuit theory, models of bacterial growth subject to limited supply of nutrients, and models of queueing networks with limited buffer sizes. The most popular approach to a systematic description of nonsmooth behavior makes use of differential inclusions. In the talk I will discuss an alternative modeling framework, which is based on a combination of ideas from systems theory and from operations research.

April 26, 2006  Patrick Dehornoy  (Université de Caen)
From sets to braids
Large cardinals in Set Theory are objects whose existence is, and will remain, an unprovable assumption. We show how studying such strange objects naturally led to considering algebraic systems of a new type, and, eventually, to discovering some canonical linear ordering of braids. The latter has now received a number of geometrical and topological constructions, and it is the basis for new very efficient braid algorithms, with possible cryptographical applications. Thus, in this case, a continuous path connects very abstract objects from Set Theory to quite concrete questions of applied mathematics.

May 17, 2006  Johan Grasman  (Wageningen Universiteit)
Exploring persistence in stochastic models of biological populations and its application in chemostats
We consider dynamical models of biological populations with a stochastic input, e.g. environmental noise. If the corresponding deterministic system has a stable equilibrium, then the state of the stochastic system will fluctuate around this equilibrium. These fluctuations may be so large that one of the participating populations may die out. This will occur with probability one in finite time. A system of interacting biological populations is highly resilient if after a perturbation the deterministic system rapidly returns to its equilibrium state. For stochastic models this qualification does not apply. We will introduce the notion of local persistence to make up for this incompatibility. Global persistence is related to the problem of extinction described above. We will give examples of both in two different chemostat models. The first one models an experimental sewage treatment system consisting of three trophic levels: sewage-bacterium-worm. Worms are introduced to reduce the amount of remaining sludge: they transform bacterium biomass into their own biomass for one part and use it as an energy source for another part. The worm population fluctuates strongly and gets extinct from time to time. In a second application we study a closed nutrient-limited system of two interacting populations and analyse the local persistence at an internal equilibrium over a parameter interval bounded by a saddle-node bifurcation point at one side and a Hopf bifurcation point at the other side.
J. Grasman, O.A. van Herwaarden and T.J. Hagenaars, Resilience and persistence in the context of stochastic population models, in Current Themes in Theoretical Biology: A Dutch Perpective, T.A.C. Reydon and L. Hemerik, eds., Springer, 2005, p.267-280.
J. Grasman, M. de Gee and O.A. van Herwaarden, Breakdown of a chemostat exposed to stochastic noise, J. Engrg. Math. 53 (2005), 291-300.
S.A.L.M. Kooijman, J.Grasman and B.W.Kooi, A new class of non-linear stochastic population models with mass conservation, preprint.

May 24, 2006  Lex Schrijver  (CWI/UvA)
Tensors, Invariants, and Combinatorics
We give a characterization of those tensor algebras that are invariant rings of a subgroup of the unitary group. The theorem has as consequences several "First Fundamental Theorems" (in the sense of Weyl) in invariant theory.
Moreover, the theorem gives a bridge between invariant theory and combinatorics. It implies some known theorems on self-dual codes, and it gives new characterizations of graph parameters coming from mathematical physics, related to recent work with Michael Freedman and Laszlo Lovasz and of Balazs Szegedy.
In the talk we give an introduction to and explanation of these results.

September, 2006  Alain Lascoux  (Marne-la-Vallee)
Algebraic computations in several variables
Compared to the case of functions of one variable, or to the case of symmetric functions, there are not many algebraic tools to manipulate polynomials in several variables. I mean simple tools which could figure in a textbook of algebra. The symmetric group can greatly help in that matter. In fact, many new developments that I will mention in my talk directly stem from operators that Newton introduced to solve another problem (which was how to transform a discrete set of data into an algebraic function).

September 20, 2006  Andrew Baker  (University of Glasgow)
Galois theory in a topological context
The classical Galois theory of fields has been extended and reinterpreted in a variety of ways in algebra, geometry and differential equations. I will begin by reviewing the Galois theory of commutative rings (largely developed in the 1960's) then give an overview of the `brave new Galois theory' first systematically investigated by Rognes. This applies to topological objects called commutative S-algebras which arise in connection with multiplicative cohomology theories. The basic theory is now well understood and I will also mention joint work with Birgit Richter on realizability issues that requires powerful obstruction theory machinery based on André-Quillen cohomology for commutative algebras.

October 4, 2006  Erdal Emsiz  (UvA)
Affine Weyl groups and integrable systems with delta-potentials
I will talk about affine Weyl groups generalizations of the one-dimensional quantum Bose-gas on the ring with pair-wise delta-function interaction. Outline of the talk:
* Symmetry: By considering Dunkl-type differential-reflection operators associated to the integrable system, we show that the fundamental object controlling the algebraic relations between the Dunkl-type operators and the natural Weyl group action is the associated graded of Cherednik's (suitably filtered) degenerate double affine Hecke algebra.
* Spectrum: The allowed spectral parameters are controlled by certain transcendental equations called the Bethe ansatz equations. I shall sketch why the spectrum of the system satisfies (an affine Weyl group version of) the exclusion principle of Pauli.
* Completeness: I indicate why (the linear span of) the eigenfunctions of the quantum system are complete in the Hilbert space of symmetric functions (with respect to the affine Weyl group) with respect to ordinary Lebesgue measure.
If time permits we will also discuss the following.
* What I really would like to prove: Some conjectures about orthogonality and norms of the eigenfunctions of the quantum system will be made.
* Discuss briefly affine Weyl group generalizations of quantum spin-particles with pair-wise delta-function interactions.
We shall define all concepts necessary to state the problem. In particular no prior exposure to affine Weyl groups and Hecke algebras is required. (The major part of this talk is joint work with Eric Opdam and Jasper Stokman)

October 18, 2006  Alexis Kouvidakis  (University of Crete)
Symmetric products of curves and connections with the Nagata problem
The Nagata problem for plane algebraic curves can be phrased as: given a number of points p_1,..., p_n in the plane, how many curves of given degree d pass through these points with given singularity multiplicities? In this talk we will discuss the various aspects of this problem and questions about symmetric products of algebraic curves related to that.

November 1, 2006  Onno van Gaans  (Universiteit Leiden)
Invariant measures for infinite dimensional stochastic differential equations
If a deterministic system is perturbed by noise, it will not settle to a steady state. Instead, there may exist invariant measures. Existence of an invariant measure requires tightness of a solution, which is a compactness condition. A solution of a finite dimensional stochastic differential equation is tight if it is bounded. Boundedness is not sufficient in the case of an infinite dimensional state space. We will discuss several conditions on infinite dimensional stochastic differential equations that provide existence of tight solutions and invariant measures.

November 22, 2006  Thijs Vermaat  (UvA/IBIS)
Statistical Process Control in Non-Standard Situations
In Statistical Process Control (SPC) we monitor a quality characteristic for a certain process. An example is the monitoring of the temperature in a glass furnace. When a large change in the temperature is signaled by a control chart, corrective actions in the process have to be made.
Traditionally, we assume that this quality characteristic follows a normal distribution and the subsequent observations in time are independent. The first problem we will discuss is that of a quality characteristic that is not normally distributed. We present two possible solutions: a non-parametric, and a semi-Bayesian approach. The second problem is that the observations are serial correlated. We have developed a modified Exponentially Weighted Moving Average (EWMA) control chart which is adapted for serial correlation, and will be discussed during the talk.

November 29, 2006  Joost Batenburg  (Universiteit Antwerpen)
Discrete tomography: exploiting the fact that nature is discrete
In recent years electron microscopes have evolved to a point where it is now possible to observe crystalline structures at the atomic scale. However, the images as seen by the microscope are two-dimensional projection images of the three-dimensional crystals. By recording many projections images, each from a different angle, one may hope to get enough information to reconstruct the crystal. This leads to a very interesting combinatorial inverse problem, known as the discrete tomography problem. Discrete tomography is a young research field that has links with several other fields from Mathematics and Computer Science. In this talk I will describe the basic problems from discrete tomography and discuss several known results. In the second part of the talk I will explain how network flow methods from Operation Research can be used for the solution of discrete tomography problems.

November 13, 2006  Björn Sandstede  (University of Surrey)
Dynamics of spiral waves
Spiral waves are captivating patterns that have been observed in many biological and chemical systems, for instance, in chemical reactions, during the aggregation of amobae, and as electrochemical waves in the heart. Part of their fascination is due to the intriguing instabilities such as meandering and drifting, core and far-field breakup, and spatio-temporal period doubling, that they exhibit. Among the challenges for theoretical studies of spirals is the task of explaining and predicting these instabilities using, for instance, symmetry groups, spectral theory, and dynamical-systems techniques. In this talk, I will give an overview of spiral instabilities and discuss recent results as well as open problems in this area.

February 7, 2007  Jaap Korevaar  (Universiteit van Amsterdam)
Prime pairs and the zeta function
Are there infinitely many prime pairs with given even difference? Most mathematicians believe it, just as they accept Riemann's Hypothesis (RH) on zeta's complex zeros. On the internet one can find prime twins (difference two) of more than fifty thousand digits! Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen.
There is strong numerical support for the prime-pair conjectures (PPC's) of Hardy and Littlewood (1923), on the number of prime pairs (p,p+2r) with p =< x. Assuming RH, Montgomery and others have studied the pair correlation of zeta's complex zeros, mentioning connections with the PPC's. Using a Tauberian approach, speaker has shown that under RH, the PPC's are equivalent to specific boundary behavior of certain functions involving zeta's zeros. For real s these functions resemble pair-correlation expressions. A hypothetical supplement to Montgomery's work would imply that there is an abundance of prime pairs for some differences 2r.

February 21, 2007  Arnoud van Rooij  (Radboud Universiteit)
p-adic analysis
In Functional Analysis a monopoly is held by the real and the complex numbers. However, other scalar fields are possible. The use of such fields to build an alternative theory creates an extra dimension, giving some insight in the question which properties of the real and complex numbers (such as local compactness or separability) are crucial for certain classical results (such as the Hahn-Banach Theorem or the Gelfand Representation Theorem). Furthermore, it may help understand the relations between the results. The alternative theory is also interesting in its own right, and part of it does not even have an analogue in the classical situation. In the lecture we consider some of these scalar fields, in particular the p-adic number fields, where p is a prime number. By way of an introduction we treat the 10-adic integers: “reversed” infinite sequences of digits such as ... 6, 2, 9, 5, 1, 4, 1, 3, with natural addition an multiplication. It turns out that the sequence 1!, 2!, 3!, ... is summable, and the number 1 has four square roots.
The lecture is elementary. Functional Analysis makes an appearance only in the epilogue.

February 28, 2007  Maarten Solleveld  (Universiteit van Amsterdam)
Periodic cyclic homology and non-Hausdorff spaces
In noncommutative geometry one uses geometric techniques to study noncommutative algebras. Typically one considers an element of any algebra as a function on some weird topological space, the spectrum of that algebra. Periodic cyclic homology can be regarded as the noncommutative generalization of De Rham cohomology. We will discuss this, and use it as the motivation to define a new cohomology theory on certain non-Hausdorff spaces.

March 14, 2007  Thomas Quella  (Universiteit van Amsterdam)
Supersymmetry, supergeometry and physics
Supersymmetry is one of the key concepts in modern high energy physics, and in particular in string theory. It is probably less known that it also had a significant influence on mathematical subjects, especially topology. Mirror symmetry is one example, the determination of topological invariants another.
In this talk we will first broadly discuss the concept of supersymmetry in the simple setup of quantum mechanics. This context already allows to get a glimpse of the links to Hodge theory and index theorems. Based on the structures we found in the quantum mechanical setup we motivate the existence of geometric objects called supergroups. In the spirit of Connes' non-commutative geometry the definition of supergroups can be given in terms of its super Hopf algebra of functions. We explain that the harmonic analysis on supergroups exhibits a lot of surprises which have no analogue in the case of ordinary groups. Finally, we outline why physicists are interested in analyzing the propagation of strings on supergroups and their quotients. No physical background is required in order to be able to follow the seminar.

March 28, 2007  Antonios Zagaris  (CWI/Universiteit van Amsterdam)
Attracting invariant manifolds for systems with multiscale dynamics
Invariant manifolds are (typically lower-dimensional) subsets of the space in which the evolution of a dynamical system takes place which remain invariant under the dynamics of this system. Classic examples include the stable, unstable, and center manifolds corresponding to a point fixed under the dynamics. The notion of an invariant manifold has proved to be a most fruitful one, since such manifolds serve (locally) as the backbone on which the full-dimensional dynamics are organized. In this talk, we will focus on attractive invariant manifolds for systems with multiple timescales. We will discuss the existence and uniqueness (or often non-uniqueness) of such manifolds, as well as the way in which the full system dynamics quickly become slaved to the dynamics on these manifolds. Then, we will proceed to present the ideas underlying some of the most common methods used in applications to compute attractive manifolds for finite-dimensional systems (ODEs). We will conclude this talk with a short discussion of invariant sets for infinite dimensional parabolic semiflows generated by reaction-diffusion PDEs and highlight the challenges that have to be met in developing methods to identify such sets.

April 11, 2007  Pieter Collins  (CWI)
Computable Analysis and Verification of Nonlinear Systems
An important problem in control theory is to compute a controller satisfying some objective, and to verify that the controller does indeed satisfy the control objective. A simple objective is the safety property; given an initial set and a safe set, the system evolution starting from any point in the initial set must remain in the safe set. Although mathematically simple to state, it is unclear whether it is possible to computationally verify this property for a given nonlinear system, or even how to express the input data for arbitrary sets and systems! In this talk I will show how to use the theory of computable analysis developed by Weihrauch and co-workers (which is related to Scott domain theory and formal topology) to give a rigorous yet computationally tractable semantics to the problem, and discuss what is possible (and what is impossible) to compute using these semantics.

May 2, 2007  Rob van der Waall  (UvA)
1939-2004 : the development of classifying finite groups by means of n-isoclism of groups
Up to isomorphism there are 2 groups of order 4, 2 groups of order 6 , 5 groups of order 8, 14 groups of order 16 , 51 groups of order 32, 267 of order 64. All groups of order at least up up to 1000 are known now, certainly due to the cumbersome classification of all the groups of order of order 128, of order 256, of order 512. The number of groups of order 1024 is known as well .. From the year 1939 on, thinking on how to classify finite groups either by developing theoretical methods, and by computational methods as well, took place. One important tool in that respect, is to apply what is called, 1-isoclinsm of groups. It is a concept of a certain equivalence relation much general than that of isomorphism. For instance, the quaternion group of order 8 and the dihedral group of order 8 are 1-isoclinic to each other, but unfortunately all abelian groups are 1-isoclic to each other. It was Philip Hall who started to study equivalence classes of 1-isoclinic groups. In Amsterdam and Leiden three dissertations dealt with the subject and with the more general notion of n-isoclinsm. In this talk I will say something about the development of properties of n-isoclinsm classes of groups, and mention some numerical results. If time permits,I shall close the talk by indicating the solution of the so-called n-isoclism embedding subgroup problem, as obtained in 2003 and published in 2004. Today, properties of n-isoclisms of groups and of so-called n-isologisms of groups are pursued mainly in Iran, notably in Mashad, by several people there..

May 30, 2007  Nitin Saxena  (CWI)
Isomorphism Problems and Cubic Forms Equivalence
Cubic forms equivalence is the problem of checking whether two given homogeneous degree 3 polynomials can be made equal by applying an invertible linear transformation on the variables. This is a natural algebraic problem with no known general algorithm. In this talk we will show interesting connections of cubic forms equivalence with the isomorphism problems of commutative algebras and graphs. These results also give the feeling that the case of cubic forms is the hardest of all r-forms.

June 6, 2007  Yde Venema  (Universiteit van Amsterdam)
Completions of lattice ordered algebras
Lattice ordered algebras, or lattices expanded with additional algebraic structure, have been widely studied in general algebra. They are of particular importance in the algebraic semantics of logics, where the meet and join operation of the lattice correspond to the logical conjunction and disjunction, respectively. A common theme concerning these structures is that one wants to embed a given lattice ordered algebra into one of which the underlying order is a *complete* lattice, i.e., with all meets and joins existing. Examples include the MacNeille completion, which generalizes Dedekind's construction of the reals out of the rationals, and the canonical extension, which naturally arises as a `double dual' in the context of Stone dualities.
In the talk I will address the issue, which properties of an algebra are preserved when moving to the completion (of the kind under investigation). I will also discuss some recent results which establish some connections between various kinds of completions.

September 5, 2007  Harrie Willems 
Wolfgang Doeblin, A Mathematician Rediscovered (film)
Wolfgang Doeblin, one of the great probabilists of the 20th century, was already widely known in the 1950s for his fundamental contributions to the theory of Markov chains. His coupling method became a key tool in later developments at the interface of probability and statistical mechanics. But the full measure of his mathematical stature became apparent only in 2000 when the sealed envelope containing his construction of diffusion processes in terms of a time change of Brownian motion was finally opened, 60 years after it was sent to the Academy of Sciences in Paris.
This film documents scientific and human aspects of this amazing discovery and throws new light on the startling circumstances of Doeblin's death at the age of 25.

Agnes Handwerk and Harrie Willems (2007), Wolfgang Doeblin, A Mathematician Rediscovered (DVD), Springer, VideoMATH
Harrie Willems (2002), Verzegelde formules, Nieuw Archief voor Wiskunde, 5 (3)

September 19, 2007  Jonas Bergström 
Counting points over finite fields and cohomology
Recall that an algebraic variety locally is given by a finite set of polynomials and that its points are the common zeros of these polynomials. An algebraic variety considered over a finite field thus has a finite number of points. An important invariant of an algebraic variety is its cohomology, and if the algebraic variety is considered over the complex numbers the cohomology can be defined, in the same way as in algebraic topology, using its complex topology.
There is actually a striking connection between the two phenomena. That is, there is cohomological information to be found by counting the number of points over finite fields of an algebraic variety. This was established by Grothendiecks construction of étale cohomology. I will describe this connection, and exemplify by some countings of points of the moduli space of curves.

October 3, 2007  Frank den Hollander 
Random walk in random scenery
In this talk we consider random walks on randomly coloured lattices. We are interested in the sequence of colors seen by the walk in the course of time, called the color record. We will show that this color record is an intricate random process, with remarkable properties. This talk is for a general mathematical audience.

October 17, 2007  André Henriques 
The string group
The group String(n) is the 3-connected cover of Spin(n). By definition, this means that it is a topological group with π0=0,   π1=0,  π2=0,  π3=0, and that it is equipped with a group homomorphism String(n) → Spin(n) which induces isomorphisms on all homotopy groups πi, for i>3. The above definition defines it uniquely up to homotopy equivalence. In this talk, I will discuss various concrete models of String(n), and also what it means for a manifold to have a string structure. I will recall the role of spin structure in the study of K-theory, and will use it to explain why it is interesting to consider string manifolds, and why it is important to have good geometric models for the string group.

October 31, 2007  Ronald Cramer 
Computing in the dark using algebraic geometry
Cryptology provides mathematical techniques for digital security in a malicious environment. Encryptions and digital signatures protect legitimate parties against eavesdropping and tampering by malicious outsiders, i.e., uni-lateral security. Secure computation focuses instead on multi-lateral security, i.e., secure cooperation among mutually distrusting parties or parties with conflicting interests. Potential applications are myriad, and include privacy protection, negotiations, and simulation of an incorruptible mediator. A fundamental theorem from the 1980s says in essence that ``all multi-lateral security problems solvable with a trusted host are securely solvable without.''
In 2000 it was proved by Cramer, Damgaard and Maurer that secure computation can be realized from mathematical devices called linear secret sharing schemes with (strong) multiplication. In 2006 it was shown by Chen and Cramer that such devices can be constructed using algebraic function fields. Using in addition a well-known theorem of Garcia-Stichtenoth on curves with many rational points they showed how to perform secure computation with improved efficiency. This is the first non-trivial connection between secure computation and algebraic geometry. After an introduction to the concept of secure computation, some of the mathematical details behind this result are discussed.

November 14, 2007  Fokko van de Bult 
Hyperbolic Hypergeometric Functions
q-difference equations are discretized versions of differential equations, involving the operator Q, defined by (Qf)(x) = f(qx) (for q in C* = C\{0}), instead of derivatives. q-difference equations are deformations of differential equations, as the formal limit q → 1 turns q-difference equations into differential equations. When |q|=1 the behaviour of the solution space of q-difference equations changes radically. In particular, there often exist no non-zero solutions in this case. However, in such cases we can consider the 'doubled version' of the q-difference equation. This doubled version is a particular system of two ordinary difference equations on C canonically associated to the original q-difference equation on C*. A solution to the doubled system can now serve as a solution to the original q-difference equation. Even in cases where we start with a q-difference equation with |q|=1 we can, in important examples, find solutions to the doubled system. These solutions are given in terms of Barnes's double gamma function, and are examples of hyperbolic hypergeometric functions.
In this talk I will explain this process of doubling. Subsequently I will discuss some properties of hyperbolic hypergeometric functions. In particular I will discuss a hyperbolic hypergeometric function which satisfies an E7-symmetry.

November 28, 2007  Simon Ruijsenaars 
Integrable systems, analytic difference equations, special functions, Hilbert space: On the crossroads.
In this lecture we aim to survey the four areas mentioned above, with a bias towards the study of special classes of analytic difference operators. The problem of understanding their Hilbert space features centers around the issue of orthogonality and completeness of suitable eigenfunction transforms, which generalize various previously known transforms (the most well-known being Fourier transformation). The analytic difference operators arise from certain integrable systems, including nonlocal soliton equations and relativistic N-particle systems of Calogero-Moser and Toda type.

December 12, 2007  Rob Stevenson 
Adaptive wavelet methods for solving high dimensional PDE's
We present a non-standard numerical method for solving (linear) PDE's. Such equations can be written in the form Lu=f, where L is a boundedly invertible operator between Hilbert spaces. By equipping these spaces with (wavelet) bases, an equivalent infinite matrix vector equation is obtained. We will show that the adaptive wavelet scheme produces a sequence of approximations to the solution vector that converges with the best possible rate, where the cost of producing these approximations is proportional to their length. Finally, we discuss the application of this scheme to problems on product domains, where we will obtain rates that are independent of the space dimension.

February 6, 2008  Nicolai Reshetikhin 
On the Kasteleyn's solution of dimer models
Perfect matching on a planar graph is a perfect matching between vertices of the graph which are connected by edges. A dimer configuration is another name for such matching. A dimer model is a certain probability measure. The main object of study in such models are expectation values of certain local function on dimer configurations. They are called correlation functions. In 1963 P.W. Kasteleyn discovered the determinant formula for such correlation functions. He also predicted how the solution should look like for surface graphs of higher genus. It took about 30 years to verify this prediction.
In this talk I will explain the Kasteleyn's solution and will outline how it works for higher genus surfaces.

February 20, 2008  Jochen Heinloth 
An introduction to some aspects of the (geometric) Langlands program
In this introductory lecture we will begin with the classical quadratic reciprocity law and recall how this is explained by class field theory. We will then try to explain why this theory gets much more tractable when one considers fields of functions on curves instead of number fields.
If time is left, we will try to indicate briefly what the general (geometric) conjectures are about.

March 5, 2008  Hicham Zmarrou 
Dynamics and bifurcations of random circle diffeomorphisms
We consider iterates of parametric circle diffeomorphisms. The parameter is a random variable with value in a bounded interval. We give precise conditions under which orbits converge to a random attracting fixed point or random attracting periodic orbits exist.
We discuss bifurcations leading to an explosion of the support of a stationary measure from a union of intervals to the circle. We show that this typically involves a transition from a unique random attracting periodic orbit to a unique random attracting fixed point. (Joint work with Ale Jan Homburg)

March 19, 2008  Monique Laurent 
Real Solving Polynomial Equations with Semidefinite Programming
While good methods exist for computing complex roots to polynomial equations (assuming there are finitely many), the problem of computing all real solutions is less well understood. We propose a numerical method for finding the real solutions to a system of polynomial equations, assuming their number is finite (while the number of complex roots could be infinite). Our method relies on expressing the real radical ideal of the ideal generated by the given polynomials as the kernel of a suitable positive semidefinite moment matrix. We use semidefinite optimization for finding such a matrix, combined with linear algebra techniques for computing the real roots as well as a (border or Gröbner) basis of the real radical ideal. It turns out that our stopping criterion (based on some flat assumption for moment matrices) is closely related to the stopping criterion used by Zhi and Reid in their algorithm for complex roots inspired by involutive methods for systems of linear PDE's. This is joint work with J.B. Lasserre (LAAS, Toulouse) and P. Rostalski (ETH, Zurich).

April 2, 2008  Rens Bod 
Mathematical Universals in Music
The mathematical study of music is one of the oldest applications of mathematics in the history of science. In my talk, which is based on joint work with Aline Honingh, I will show how notions from topology have recently led to new insights in the structure of music. When notes of a musical scale are represented as points in a two-dimensional grid (using n-limit just intonation), they form convex or star-convex regions. We found that this holds for all scales in 5-limit just intonation from the Scala Database (3,500 scales), ranging from the Indian Shruti and Chinese Zhou scales to scales proposed by modern western composers. This generalization seems to point at a cognitive universal of music. The property of convexity is not limited to scales, but also holds for classes of chords and it can be used in a practical application known as `pitch spelling'. Time permitting, I will discuss how the notion of convexity is integrated in a pattern-based music-analysis program, which is part of my current Vici-project.

April 16, 2008  Frits Beukers 
Algebraic hypergeometric functions
Hypergeometric functions (in one and several variables) are classical functions which occur at many places in mathematics and mathematical physics. We shall be interested in hypergeometric functions which are at the same time algebraic in their arguments. This subject started in 1873 with the work of H.A. Schwarz. In this lecture we discuss a number of examples, introduce a general class of several variable hypergeometric functions, the so-called GKZ-functions, and describe a combinatorial criterion for their algebraicity.

May 14, 2008  Dietrich Notbohm 
Combinatorics of simplicial complexes and Stanley-Reisner algebras from a topological point of view
Given a finite simplicial complex there are several associated algebraic and topological invariants, e.g. the Stanley-Reisner algebra and Davis-Januszkiewics spaces which are topological realizations of these algebras. The homotopy theory of these spaces is reflected by combinatorial properties of the complexes. We will discuss two different applications and will relate coloring question of the complex with splitting properties of vector bundles over the Davis-Janucskiewicz space and depth of the Stanley-Reisner algebra with the combinatorics of the complex.

May 28, 2008  Kareljan Schoutens 
Quantum hard squares, supersymmetry, and combinatorics
Motivated by the physics of electrons in low-dimensional condensed matter systems, we study quantum mechanical models for fermionic particles (`hard squares') on two-dimensional grids. The quantum ground states of a particular supersymmetric model are in 1-to-1 correspondence with homology cycles of the independence complex of the grid. For a generic two-dimensional grid, the number of independent quantum ground states (homology cycles) grows exponentially with the size (area) of the grid. For the square grid, the ground state counting problem has been fully solved through a remarkable correspondence with specific rhombus tilings of the plane.

September 3, 2008  Sergey Shadrin 
Operads, PROPs, and graph complexes
I am going to give a survey of some recent results on graph formulas for homotopy algebra structures, with a special emphasize on basic concepts of the theory of operads and its extensions and applications of arising graph complexes in various part of geometry.

September 17, 2008  Tilman Bauer 
Finite loop spaces
A finite loop space is the homotopy theorist's version of a compact Lie group -- no manifold structure, just a finite homotopy type with a multiplication. How different are these two concepts from each other? Although there are many more finite loop spaces than there are compact Lie groups, there is still a strong structural analogy, and finite loop spaces can be classified in a ``local'' setting, which I will explain. One striking consequence of this classification is that every finite loop space can be realized as a smooth manifold -- albeit at the cost of losing the multiplication map.

October 1, 2008  Walter van Suijlekom 
The structure of perturbative quantum gauge theories
Quantum gauge theories are the building blocks of the extremely well-tested Standard Model of high-energy physics. It describes all known elementary particles and their interactions. Mathematically, however, these theories are not so well understood, in sharp contrast to their classical counterparts. We try to unravel the structure of quantum gauge theories by taking a perturbative (i.e. formal power series) approach. In particular, we try to understand two aspects of quantum gauge theories: renormalization and quantum gauge symmetries.
After a brief introduction to these two physical ideas, we describe them mathematically in terms of a Hopf algebra and a Gerstenhaber algebra, respectively. We also establish their connection by (co)representing the Hopf algebra on the Gerstenhaber algebra. The so-called master equation satisfied by the physical Lagrangian due to gauge invariance implies the existence of the quantum symmetries on the level of the Hopf algebra. In this way, we rigorously show that renormalization is compatible with the quantum symmetries.

October 15, 2008  Hessel Posthuma 
Index theory from the point of view of quantum mechanics
I will start by giving an introduction to index theory and explaining how ideas from quantum mechanics and noncommutative geometry play a role. In particular, these ideas yield a promising approach for extending the index theorem to certain singular geometric situations.

October 29, 2008  Bert Zwart 
Fluid and diffusion approximations of bandwidth sharing networks
Bandwidth sharing networks are stochastic networks that model congestion in computer and communication networks in which multiple users are served simultaneously, sometimes by several servers. We give a brief introduction to the modeling and analysis of this type of networks, and then focus on a specific problem: We analyze a bandwidth sharing network that is overloaded. Using limit theorems for measure valued stochastic processes and convex analysis techniques, we identify the growth rates of the number of users in the system, which allows us to obtain a functional law of large numbers. Time permitting, we also discuss extensions to models with impatient users as well as limited versions of bandwidth sharing.

November 12, 2008  Bas Spitters 
A computer-verified implementation of Riemann integration - an introduction to computer mathematics
The use of floating point real numbers is fast, but may cause incorrect answers due to overflows. These error can be avoided by hand. Better, exact real arithmetic allows one to move this bookkeeping process entirely to the computer allowing one to focus on the algorithms instead. For maximal certainty, one uses a computer to check the proof of correctness of the implementation of this algorithm. We illustrate this process by implementing the Riemann integral in constructive mathematics based on type theory. The implementation and its correctness proof were driven by an algebraic/categorical treatment of the Riemann integral which is of independent interest.
This work builds on O'Connor's implementation of exact real arithmetic. A demo session will be included. (Joint work with Russell O'Connor)

November 26, 2008  Jonas Hartwig 
Generalized Weyl algebras
Generalized Weyl algebras were introduced by Bavula in 1992. They are defined by generators and relations involving an arbitrary commutative ring and one or several automorphisms of this ring. Many important and interesting algebras can be presented as generalized Weyl algebras, including the (quantum) Weyl algebra, the (quantum) enveloping algebra of the Lie algebra sl(2), so called down-up algebras and many more. This allows one to study all of these algebras from a unifying point of view. I will give an overview of generalized Weyl algebras and their modules, and then focus on some particularly interesting examples: a Hopf algebra analogous to the quantum enveloping algebra of sl(2), and the so called fuzzy torus, a special case of the fuzzy Riemann surfaces recently introduced by Arnlind et al. in 2006.

December 10, 2008  Robin de Vilder 
The causes and effects of the credit crisis
In this talk the background of the credit crunch will be outlined. For a broad audience the effects that caused the asset price bubble to inflate and deflate will be discussed. It will be argued that excess financial leverage together with seemingly investment diversification and lax monetary policy has led to the recent collapse. It will also be explained how risk models were fooled by the system. The role of hedge funds in the simultaneous break down of uncorrelated markets will also be discussed.
(download sheets)

Janary 21, 2009  Lawrence Zalcman 
Picard Theorems 1879-2009
A selective survey of the history of the famous theorems of Emile Picard (and related results) from 1879 to the day before yesterday, with an emphasis on some surprising recent developments.

February 4, 2009  Said El Marzguioui 
Fine aspects of pluripotential theory
Pluripotential theory can be briefly described as the study of plurisubharmonic functions, and was mainly developed in the eighties of the last century. However, the topological properties of plurisubharmonic functions have never met a wide interest among researchers. Very recently, it turned out that there are phenomena in pluripotential theory that could not be fully explained without understanding the so-called plurifine topology. This topology is associated to plurisubharmonic functions in a very natural way.
Since pluripotential theory is the higher dimensional counterpart of classical potential theory in the complex plane, I will begin this talk by a brief introduction to potential theory. This is necessary to introduce the so-called classical fine topology and the theory of finely subharmonic functions associated to it.
Then I will turn to discuss plurisubharmonic functions. Here I will focus on the study of the sets were these functions can take the value minus infinity. Results about these sets are obtained via "plurifine" objects.

February 18, 2009  Sabir Gusein-Zade  
Poincare series of multi-index filtrations and their generalizations.
For a multi-index filtration on a vector space (e.g. on the ring of germs of functions on a variety) one can define a notion of the Poincare series, generalizing the usual one for a one-index filtration (the generating series for the dimensions of the consecutive factors). (The definition is not very straightforward.) The Poincare series can be written as a certain integral with respect to the Euler characteristic. It appears that the Poincare series of a natural filtration associated to a plane curve singularity coincides with the classical monodromy zeta-function of the corresponding equation. There is a generalizations of this notion for an equivariant situation. Another generalization is obtained by substituting the usual Euler characteristic by a generalized one. This leads to the Poincare series depending on an additional variable. E.Gorsky has found that, for irreducible plane curve singularities, the generalized Poincare series is strongly connected with the generating series of the Heegaard-Floer homologies of the corresponding algebraic knot (defined by P.Ozsvath and Z.Szabo).

March 4, 2009  Kostas Skenderis  
Holography and Mathematics.
About 15 years ago the idea of "holography" emerged from black hole physics. This idea found a concrete realization in string theory in the form of gauge/gravity dualities and grew to become the dominant research direction in theoretical high energy physics. This field has strong interface with conformal geometry and hyperbolic manifolds and has led to continuous and growing cross-fertilization between physics and mathematics. The aim of this talk is to explain what holography is and describe the connections to mathematics.

March 18, 2009  Harry van Zanten  
Prior and Posterior Stochastic Differential Equations
In Bayesian nonparametrics, a prior distribution on an unknown function is often defined as the law of a certain stochastic process. If the statistical model is dominated, this opens up the possibility of using change of measure theory for stochastic processes to characterize and study the posterior distribution. In this paper we develop this approach for nonparametric, univariate regression. As prior on the unknown regression function we use the law of the solution of a linear stochastic differential equation (SDE). Using change of measure theory for continuous semimartingales we derive an explicit SDE characterization for the posterior. We use this dynamical characterization of the posterior to obtain fast algorithms for sampling from the posterior and computation of the posterior mean. Moreover, we obtain exact uniform Bayesian confidence bands.

April 1, 2009  T.A. Springer  
On the work of Jacques Tits
In 2008 the Abel Prize in mathematics was awarded jointly to John Thompson and Jacques Tits, for their contributions to group theory. In the talk I will try to give some idea of Tits's contributions and of their geometric aspects.

April 15, 2009  Sebastiaan van Strien 4pm room P.017 !  
Games, Fictitious play and Chaos
In this talk I will discuss two-player finite games. In these games, players may try to 'learn' the Nash equilibrium of a game, by repeatedly playing the best response to the other player. So the game evolves as a dynamical system (called fictitious play). This approach was suggested in the early 1950's, but later it was shown that convergence to the Nash equilibrium does not necessarily hold, except in zero-sum games. In this talk, I will show that (i) the dynamics of these games is surprisingly rich, and chaotic switching between strategies can occur, (ii) the way orbits fit together geometrically is different from what one would expect and that (iii) in zero-sum games, the dynamical system is closely connected to that of Hamiltonian systems (from classical mechanics). This talk will be aimed at a general audience, and will not require any background in game theory or dynamical systems.

April 29, 2009  Manfred Lehn  
Holomorphic symplectic varieties

May 13, 2009  Rob van der Waall  
On the life and work of John Griggs Thompson
The Abel Price in mathematics for the year 2008 was bestowed to John Thompson and Jaques Tits. In this talk I will give an overview of the Life and Work of John Thompson. His mathematical interests run over finite group theory, self dual even codes, the so-called N-groups, projective planes, inverse Galois theory, finite simple groups like those of Ree, Suzuki, Fischer-Griess, and Thompson, the so-called j-function, the moonshine connection, etc.. His work (joint with Walter Feit) on the Burnside-conjecture: "Every finite group of odd order is solvable" , published in 1963 (255 pages long!) made him already famous. In order to keep the talk as simple as possible, in order that it will be interesting also for a general mathematical orientated audience, I will confine myself mainly to finite group theory as far as the mathematics in the talk is concerned. A minimum of definitions and notions in finite group theory will be provided, needed to understand the talk. As to a good impression of more details on the work and life of John Thompson, the reader may consult the contribution I did publish (in Dutch) in the journal Nieuw Archief voor Wiskunde, page 250, vijfde serie, deel 9, nummer 4, december 2008.

May 27, 2009  Sander Bais  
The physics of quantum groups and their breaking
In recent years there has been a growing interest in two-dimensional media that exhibit topological order, because of their conceivable applications in quantum information technologies. Such media are described by topological field theories, i.e. by particle-like excitations that have only topological interactions which are characterized by the braid group and some underlying quantum symmetry. A relevant notion is that of the topological entanglement entropy.
We give some examples of such systems and will also discuss the breaking of quantum groups due to the formation of a Bose condensate. This allows for an interesting description of topological interfaces that occur between different topological phases. Though it is a typical physics subject it touches upon quite a few topics in mathematics: topology, braided tensor categories,conformal symmetry, affine lie algebra's, quasi triangular Hopf algebra's etc.

September 2, 2009  Han Peters 
Non-negative polynomials constant on a hyperplane
We will consider polynomials with non-negative coefficients that are constant on the hyperplane where the sum of the variables is 1. Such polynomials arise naturally when one studies proper holomorphic mappings from balls to balls in different dimensions. It turns out that there is a subtle relationship between the number of variables, the number of non-zero coefficients, and the degree of such polynomials. This relationship is well understood for polynomials in two variables, we will consider the case of 3 and more variables.
I will mention briefly how the problem arises in CR-Geometry, but most of the talk will only deal with elementary mathematics and should be accessible for a general mathematics audience. (Joint work with Jiri Lebl)

September 16, 2009  Stefan Kolb 
Weyl group combinatorics and quantum groups
Much of the representation theory of semisimple Lie algebras is governed by their Weyl group. In the simplest example of the special linear Lie algebra sl(n), the Weyl group is nothing but the symmetric group in n Elements. Quantum groups, which arose in certain integrable models of statistical physics around 25 years ago, provide deformations of enveloping algebras of Lie algebras. The reign of the Weyl group survives this quantization in many respects and even captures new effects related to the quantization. Lie subalgebras of semisimple Lie algebras should translate to the quantum world as so called 'coideal subalgebras'. While many classes of coideal subalgebras are known, there is so far no general classification. In this talk I will discuss a classification result for coideal subalgebras in terms of Weyl group combinatorics. The talk is based on joint work with Istvan Heckenberger.

September 30, 2009  Remco van der Hofstad 
Critical behavior in inhomogeneous random graphs
Empirical findings have shown that many real-world networks share fascinating features. Many real-world networks are small-worlds, in the sense that typical distances are much smaller than the size of the network. Further, many real-world networks are scale-free in the sense that there is a high variability in the number of connections of the elements of the networks. Therefore, such networks are highly inhomogeneous.
Spurred by these empirical findings, models have been proposed for such networks. In this talk, we shall discuss a particular class of random graphs, in which edges are present independently but with unequal edge occupation probabilities that are moderated by appropriate vertex weights. We characterize when these models have a so-called giant component, meaning that a positive proportion of the vertices is connected to one another. This characterization has important consequences for the robustness of such networks to (deliberate and random) attacks. Alternatively, when thinking of the edges as allowing water to flow through them, when the model has a giant component, then choosing a source and sink uniformly at random, with positive probability, water will flow from the source to the sink.
We discuss what happens precisely at criticality, a problem having strong connections to statistical mechanics. Simply put, we study how large the maximal regions are that become wet after letting water drop on a uniform vertex. Our results show that, for inhomogeneous random graphs with highly variable vertex degrees, the critical behavior admits a transition when the third moment of the degrees turns from finite to infinite. Similar phase transitions have been shown to occur for typical distances in such random graphs when the variance of the degrees turns from finite to infinite.

October 14, 2009  Eduard Looijenga 
Some highlights of the work of Mikhail Gromov, Abel laureate of 2009

October 28, 2009  Henk Nijmeijer 
The electronic brain: does it synchronize?
The talk consists of two parts. In the first part we review shortly some basic mathematical properties that are valid for all existing models of neuronal cells-at least as regarding their electrical activity. We show that under fairly general conditions a network of neuronal cells will exhibit synchronization provided their coupling structure is sufficiently strong. This result, though often accepted, can be proven using the concept of semi-passivity, which can be understood as an (energy) boundedness of the system behavior. In addition, a result regarding covergency is needed for the non-coupled cell equations.
In the second part, experimental results regarding an electronic realization of a network of neuronal Hindmarsh Rose systems are presented. Some of these results illustrate the aforementioned theory, whereas additional experiments deal with (partial) synchronization of time-delayed coupled neuronal systems and induce a conjecture regarding network synchronization under time-delayed coupling.
Ref.: E.Steur, I.Tyukin, H.Nijmeijer (2009), 'Semi-passivity and synchronization of diffusively coupled neuronal oscillators', Physica D 238, 2119-2128.

November 11, 2009  Kees Oosterlee 
The Heston model with stochastic interest rates and pricing options with Fourier-cosine expansions
In this presentation we give a brief outline of our research on pricing financial derivatives with numerical techniques. We favor option pricing by Fourier-cosine expansion techniques. The pricing method, called the COS method, is explained in detail.
Furthermore we discuss the Heston model with stochastic interest rates driven by a Hull-White processes. We present approximations in the Heston-Hull-White hybrid model, so that a characteristic function can be derived and derivative pricing can be efficiently done by the Fourier Cosine expansion technique. We furthermore discuss the effect of the approximations in the hybrid model on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities.

November 25, 2009  Rob de Jeu 
What is known about K2 of curves?
If F is any field, then K2(F) can be defined using generators and relations. We first discuss this group for the field of rationals numbers, and its connection with quadratic reciprocity. After making some general remarks on the K-groups of curves we concentrate mostly on K2 of curves defined over number fields. The Beilinson conjectures then predict a relation between the regulator of (a part of) K2 of such a curve and the value of its L-function at 2. We discuss this conjecture, and various results that corroborate it.

December 3, 2009  Christoph Schwab 
Finite Element Methods for PDEs with Stochastic Coefficients
We present a deterministic FEM for the solution of elliptic problems with stochastic coefficients which are given as spatially inhomogeneous random fields. Neither ergodic nor stationary input is assumed.
The method is based on a Fast Multipole Method and a Krylov Eigensolver to compute the Karhunen-Loeve expansion of the input data.
A spectral Galerkin approximation of `Polynomial Chaos' type in the sense of N. Wiener of the joint probability densities of the random solution of the SPDE. Numerical analysis of the random solution's regularity and of the complexity of the method are given, based on ideas and techniques from best N-term, nonlinear approximation. Sufficient conditions on the random field input for a convergence rate which is superior to that of Monte Carlo Methods are given.
Several types of probability measures and their corresponding polynomial systems from the Askey Scheme (as suggested by W. Schoutens) will be discussed. Recent Numerical Experiments with an implementation of the adaptive algorithm with an elliptic PDE with polynomials expansions in up 1500 independent random variables will be presented.
Joint work w. R. Andreev, M. Bieri, and C.J. Gittelson of ETH and with A. Cohen (Paris VI) and R. DeVore (Texas A&M).

December 09, 2009  Gunnar Klau 
Combinatorial optimization and algorithmics for disease classification
In this talk I will highlight mathematical and algorithmic aspects of two projects that deal with the classification of clinical data. The first part will be about identification of functional modules in protein-protein interaction networks. Besides shedding light on molecular disease mechanisms, these modules might help to better classify clinical data of different tumor subtypes. I will present an exact integer linear programming solution for this problem, which is based on its connection to the well-known prize-collecting Steiner tree problem from Operations Research. In the second part of my talk, I will speak about an algorithmic approach for transitivity editing. This problem appears in hierarchical disease classification and consists of adding and removing a minimum number of directed edges in a given graph so that the resulting graph is transitive.

February 03, 2010  Rob van der Vorst 
Closed characteristics on non-compact manifolds
Viterbo demonstrated that any (2n - 1)-dimensional compact hypersur- face M in ( R^2n , \omega) of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (R^2n ,\omega). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces M in ( R^2n , \omega) coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H_i( M ), i = n, . . . , 2n - 1, and the existence of closed characteristics in the non-compact case (including the compact case).

February 17, 2010  Frank Redig 
Duality and bosonic particle systems
Duality is a powerful tool in the study of interacting particle systems and models of population genetics. Recently, we proved that a system of interacting diffusions used as a model of heat conduction is dual to a system of particles hopping randomly on a lattice and attracting each other. This process, the so-called inclusion process is a natural (bosonic) analogue of the well-known exclusion process (which is fermionic). We show that the attractive interaction leads to interesting phenomena such as clustering and condensation. We also show how duality can be used to give exact expressions of several non-equilibrium correlation functions. The talk is based on joint work with C. Giardina, J. Kurchan and K. Vafayi.

March 03, 2010  Urs Schreiber 
Differential geometry in an ∞-topos
The familiar theory of smooth Spin(n)-principal bundles with connnection has a motivation from physics: for the quantum mechanics of a spinning point particle to make sense, the space it propagates in has to have a Spin-structure. Then the dynamics of the particle is encoded in a smooth differential refinement of the corresponding topological Spin(n)-principal bundle to a smooth bundle with connection.
It has been known since work by Killingback and Witten that when this is generalized to the quantum mechanics of a spinning 1-dimensional object, the Spin-structure of the space has to lift to a String-structure, where the String-group is the universal 3-connected cover of the Spin group. Contrary to the Spin-group, the String-group cannot be refined to a (finite dimensional) Lie group. Therefore the question arises what a smooth differential refinement of a String-principal bundle would be, that encodes the dynamics of these 1-dimensional objects.
It turns out that this has a nice answer not in ordinary smooth differential geometry, but in "higher" or "derived" differential geometry: String(n) naturally has the structure of a smooth 2-group -- a differentiable group-stack. This allows to refine a topological String-principal bundle to a genralization of a differentiable nonabelian gerbe: a smooth principal 2-bundle. In the talk I want to indicate how the theory of smooth principal bundles with connection finds a natural generalization in such higher differential geometry, and in particular provides a good notion of connections on smooth String-principal bundles.

March 17, 2010  Eric Cator 
The Hammersley interacting particle process
In recent years, there has been a surge of results on the interface of probability, mathematical physics and random matrix theory, using a wide variety of techniques, such as Younq tableaux, Riemann-Hilbert problems, steepest descent methods and determinantal processes. Central to these results is a relatively simple interacting particle process, originally invented by Hammersley, and later generalized by Aldous and Diaconis. In this talk we will introduce the particle system, mention some of the connections with other fields, and then describe the probabilistic techniques we have developed to give more intuitive proofs of some of the results in the literature, and to show new properties of the process. We were also able to extend these proofs to more general particle systems, for which similar results were already conjectured by mathematical physicists.

March 31, 2010  Vivi Rottschäfer 
Formation of singularities in natural systems
In this talk we study the formation of singularities in natural systems. Singularities arise when nonlinear effects dominate the dispersive ones, up to the formation of the singularity. We focus on projects that are motivated by concrete applications coming, for example, from optics or the aggregation of bacteria.
As a model problem, we study singular solutions of the generalised Korteweg-de Vries equation (KdV). The stability of solitary waves of the KdV has already been studied extensively. These solitons can become unstable and become infinite in finite time, in other words blow up. We analyse the structure of these blowup solutions. After introducing a dynamical rescaling the solutions are found as bounded solutions to an ODE. We study this ODE using asymptotic methods to construct the solutions. Through the asymptotic analysis, we determine the parameter range over which these solutions may exist.

April 14, 2010  Tamás Hausel 
Arithmetic harmonic analysis on character varieties
In this talk I will give motivations from geometry to study the character tables - as introduced by Frobenius in 1896 - of finite Chevalley groups; such as SL_2(F_q) studied by Schur in 1907 and GL_2(F_q) studied by Jordan in 1907. I will explain how these can be used to gain cohomological information on the representation varieties of the fundamental group of a Riemann surface to SL_2(C) and GL_2(C) respectively. I will conclude with a calculation which proves agreement of certain Hodge numbers of SL_2 and PGL_2 character varieties; which exploits the difference in the character tables of Schur and Jordan. This agreement of Hodge numbers is referred to as topological mirror symmetry and has roots in string theory and reflects a generalization of the basic electro-magnetic duality of Maxwell's equations.

April 28, 2010  Jason Frank 
A thermostat model for unresolved dynamics
Due to the downscale cascade of vorticity in (quasi-) two-dimensional fluid flows, a numerical simulation inevitably becomes under-resolved. Any finite discretization includes a closure model, either explicit or implied. Using a simple point vortex model as proof of concept, we propose a statistically consistent closure based on the idea of canonical statistical mechanics, which models the exchange between a system of particles and a large reservoir. We construct a thermostat device that simulates the exchange of vorticity with a large reservoir of point vortices, but using just a single additional degree of freedom. With this approach we are able to reproduce numerical results of Bühler (2002), who studied the equilibrium statistics of a set of 4 strong vortices coupled with a set of 96 weak vortices. For an accurate comparison, the usual canonical ensemble must be modified to account for finite-reservoir effects. In my talk I will also discuss how this approach may be extended to grid-based models.

May 19, 2010  Marjan Sjerps 
Forensic Statistics: recent developments and brand new plans
'Forensic statistics' is the field of statistics and probability theory, applied to forensic science and criminal law. It is concerned with the interpretation of forensic evidence. A main topic is the derivation of evidential force, which is expressed as a Likelihood Ratio. Key questions are: how can (forensic) researchers determine the evidential force of their observations? How can they report this to the police or the court? What is the evidential force of a combination of several pieces of evidence? In forensic statistics, mathematical models are used to examine these questions. The result is a mixture of new applications for statistical techniques and the development of new theories, as well as fundamental research. The latter involves questions about the essence of statistical evidence and dealing with probability in the courtroom.
In my presentation I will give a taste of the field using an imaginary legal case. Furthermore, I will outline my plans for the future at the Korteweg-de Vries institute.

May 26, 2010  Alessandra Palmigiano 
Dualities for noncommutative spaces
Quantales are very simple ordered algebras which can be thought of as pointfree noncommutative topologies. In recent years, their connections have been studied with fundamental notions in noncommutative geometry such as groupoids and C*-algebras. In particular, the class of quantales corresponding to certain very well behaved groupoids (the etale groupoids) has been identified by means of a non-functorial duality. However, there are very interesting examples of groupoids that do not belong to this class. For instance, groupoids that arise from group actions on topological spaces. In a joint work with Riccardo Re, the non-functorial duality has been extended to these latter groupoids as well. In the talk, I will introduce this line of research, sketch the main ideas of the duality, and discuss some examples.

September 1, 2010  Roland van der Veen 
Knot invariants: from the Jones polynomial to hyperbolic geometry
In this talk I will introduce two rather different ways to study knots: One is to compute the Jones polynomial. This is a knot invariant that comes from representation theory and statistical physics, but I will give an elementary combinatorial description. The second way to look at knots is to study metrics on the space around the knot. Complete hyperbolic metrics turn out to be unique and I will sketch how to find them. Finally I will describe a conjecture which relates the above two views of knot theory in an unexpected way.

September 8, 2010  Peter Grünwald 
Statistics without Stochastics
Consider a set of experts that sequentially predict the future given the past and given some side information. For example, each expert may be a weather(wo)man who, at each day, predicts the probability that it will rain the next day.
We describe a method for combining the experts' predictions that performs well *on every possible sequence of data*. In marked contrast, classical statistical methods only work well under stochastic assumptions ("the data are drawn from some distribution P") that are often violated in practice.
Nonstochastic prediction schemes can be used as a basis for robust, nonstochastic versions of more standard statistical problems such as parameter estimation, curve fitting and model selection. The resulting theory is closely related to Bayesian statistics, but avoids some of its conceptual problems, essentially by replacing "prior distributions" by "luckiness functions".
This talk summarizes insights from Dawid's Prequential Analysis,the Vovk-Shafer theory of game-theoretic (rather than measure-theoretic) probability, Rissanen and Barron's work on the Minimum Description Length Principle, as well as adding some ideas of my own.

September 22, 2010  Gert Vegter 
Geometric Approximation
We consider geometric approximation of smooth objects, like curves, surfaces, manifolds, by `simpler' shapes, like polygons or polyhedra, piecewise quadric surfaces. Topological correctness, good approximation of geometric invariants -- like area, normals, and curvature -- and optimal complexity are key issues, both from a theoretical and from a practical point of view.
After a general discussion of these issues, illustrated for piecewise linear approximation of surfaces, we focus on approximation of smooth curves by tangent-continuous splines whose elements are line segments, circular or conic arcs (in 2D), or helical arcs (in 3D). The complexity of such splines is expressed in terms of the maximal Hausdorff distance between the curve and the approximating spline, and in terms of intrinsic differential features, in particular curvature or affine curvature (depending on the type of spline).
History. The first result in this direction was obtained by Fejes-Toth (1948), who derived the minimal complexity of a polygon inscribing a given convex curve in the plane to within a given Hausdorff distance. Schneider (1981) generalized this result by determining the complexity of an optimal polytope inscribing a convex hypersurface in arbitrary dimensions - a result that was later rederived by Gruber (1993) under weaker assumptions on the differentiability of the hypersurface, and generalized by Clarkson (2006) to the context of general, not necessarily convex hypersurfaces.
This is joint work with my PhD students Nico Kruithof, Simon Plantinga and Sunayana Ghosh, and with Jean-Daniel Boissonnat, David Cohen-Steiner and Sylvain Petitjean.

October 20, 2010  Eric Verlinde 
Emergence of Gravity
Results from string theory and insights obtained from black hole physics give strong indications that gravity is an emergent phenomenon. I will explain the central concepts and ideas from a physics standpoint and as much as possible from a mathematical perspective. When a quantum or classical Hamiltonian system with a fast dynamics is driven by a slowly evolving system it leads to a reaction force on the slow system. The magnitude of this force can be determined with the help of an adiabatic invariant, namely the volume of phase space. I will argue that this basic mechanism lies at the origin of gravity. To arrive at the conventional space-time description of physics one has to separate the underlying dynamical system in to a slow system describing the motion of material objects through space, and a fast system whose degrees of freedom are usually ignored. Gravity is then caused by changes in the amount of phase space of these fast degrees of freedom due to the displacement of the material objects in space-time. I will present a heuristic derivation of Newton's law of gravity based on these principles.

November 3, 2010  Jan Draisma 
Finiteness results in statistics using algebra
I will give three interrelated examples of how polynomial algebra can be used to settle finiteness questions arising from statistics. I will assume no prior familiarity with any of these, and I will emphasise the fundamental algebraic tools that go into the proofs.
The first example is the by now classical Diaconis-Sturmfels algorithm for sampling from contingency tables with prescribed marginals, where algebra proves the existence of a finite Markov basis. The second concerns recent work by Hillar and Sullivant, where such Markov bases are shown to stabilise as some of the sizes of the contingency tables tend to infinity. The third is a proof that Gaussian factor analysis with a fixed number of (latent) factors stabilises---as far as polynomial equations are concerned---as the number of observed variables tends to infinity.

November 17, 2010  Jan Pieter van der Schaar 
Cosmological inflation - Theory and Observations
The stunning developments in observational cosmology over the last decade, which I will briefly review, have affirmed the fascinating potential of cosmology to probe physics at extremely high energy scales. In this colloquium I will introduce and review the paradigm of cosmological inflation. I will highlight its general properties and predictions, compare those to the most recent cosmological observations, and discuss the exciting potential of observations in the near future to discriminate between different models of inflation. I will then explain why a complete understanding of inflation requires a string theoretical framework and discuss some attempts to use inflation as a cosmological microscope to probe string scale physics.

December 1, 2010  Jan Brandts 
Numerical Analysis meets Geometry: Acute and Nonobtuse Simplicial Partitions.
The requirement that a numerical approximation of the solution of a PDE satisfies similar maximum- and comparison principles as the solution itself can lead to unusual constraints on the geometry of simplicial partitions of the domain on which the PDE is defined.
Of particular interest are partitions into simplices without obtuse dihedral angles, and those with only acute dihedral angles.
In this presentation we outline the origin of the constraints, and give details on the geometrical questions - and (some of) their answers.

February 23, 2011  Tanja Eisner 
Arithmetic progressions via ergodic theory.
We sketch the development from van der Waerden's theorem on arithmetic progressions to the recent Green-Tao theorem and show how methods from ergodic theory have been decisive in this field.

March 2, 2011  Christoph Thiele 
Carleson's theorem, variations and applications.
A famous theorem of Carleson states that the Fourier series of a square integrable function on an interval converges almost everywhere. This theorem relates to at least three other topics in analysis: boundedness of generalized eigenfunctions of Schroedinger operators in one dimenson, weighted ergodic averages as in Bourgain's Return Times theorem, and the Hilbert transform along vectorfields. In the talk we discuss these connections.

March 9, 2011  Sindo Nunez Queija 
Resource allocation in resource-sharing networks
Resource-sharing networks are a useful stochastic modeling approach for document transmission in the Internet. Mathematically, the network can be represented as a graph, of which the nodes represent the resources (routers, links) and the edges indicate whether they are connected. The transfer of a document (a "flow") simultaneously requires capacity from all resources on its path. Flows are initiated according to stochastic processes, which for tractability are assumed to be Poissonian. Each flow is endorsed with a random size, and the flow is terminated as soon as the document transmission is completed.
In this talk we will discuss resource allocation mechanisms and explain the inefficiency of standard uncoordinated scheduling strategies leading to long transmission delays. "Optimal" strategies can only be numerically determined for toy-size examples, but "near-optimal" behavior (asymptotically optimal under certain scaling conditions) can be achieved with quite simple strategies. (based on joint work with M. Verloop and S. Borst)

March 23, 2011  Alexander Schönhuth 
Complete identification of binary-valued hidden Markov processes
The complete identification problem is to decide whether a stochastic process is a hidden Markov process and if yes to infer a corresponding parametrization. So far only partial answers to either the decision or the inference part have been given all of which depend on further assumptions on the processes. Here we present a full, general solution for binary-valued hidden Markov processes. Our approach is rooted in algebraic statistics hence geometric in nature. We demonstrate that the algebraic varieties which describe the probability distributions associated with binary-valued hidden Markov processes are zero sets of determinantal equations which draws a connection to well-studied objects from algebra. As a consequence, our solution provides immediate algorithmic access where tests come in form of elementary (linear) algebraic routines.
Along the way I will provide a gentle introduction to algebraic statistics which does not require other than elementary knowledge.

April 6, 2011  Neil Walton 
Insensitive, maximum stable allocations converge to proportional fairness
We describe a queueing model where service is allocated as a function of queue sizes. We discuss allocation policies that are insensitive to service requirements, policies that have a maximal stability region and policies which maximize a certain proportionally fair utility function. We discuss the historical and practical significance of such properties and then illude to different formal arguments connecting these.

April 20, 2011  Mai Gehrke 
Profinite algebras as dual spaces
Profinite algebras, such as profinite rings, groups, and monoids are often used in algebra and have, more recently, also found applications in computer science. Duality theory on the other hand plays a central role in the study of logics where it serves as the main mechanism in relating syntactic and semantic approaches.
In recent work with Grigorieff and Pin we have shown that the category of profinite abstract algebras in any signature may be seen as a subcategory of all topo-relational dual spaces of a corresponding type. This technical result has allowed us to generalise a powerful method in automata theory.
The talk will provide an introduction to the concepts involved and a glimpse at the automata and complexity theoretic applications

April 27, 2011  Joop Kolk 
Hans Duistermaat: the Man and his Mathematics
Hans Duistermaat was Professor of Pure and Applied Mathematics at Utrecht University and the first mathematician to be appointed as an Academy (KNAW) Professor, from 2005 through 2009. Last year he passed away rather unexpectedly. Duistermaat was a geometric analyst. His interests were wide-ranging and he has been been influential in different parts of mathematics: ordinary and partial differential equations; differential, symplectic and algebraic geometry; Lie theory; classical mechanics, etc.
In this nontechnical talk I intend to give some impression of the development of Duistermaat as a mathematician. In particular, I will illustrate his approach to mathematics by presenting an extremely concise and transparent proof of the Fourier Inversion Theorem given by Hans. A video fragment, photographs, autographs, graphics, etc. are part of this talk.

May 4, 2011  Lenny Taelman 
Believing in the Kummer-Vandiver conjecture
The Kummer-Vandiver conjecture is a 150 year old open problem in number theory that was born out of attempts to prove Fermat's Last Theorem. If true, it has profound consequences in algebraic number theory as well as in algebraic topology. It is however far from universally believed, and its status is quite controversial. In this introductory talk, which is aimed at a non-expert audience, I will explain the history and statement of the conjecture, and try to give some arguments both for and against the conjecture.

May 11, 2011  Thomas Ward 
Group automorphisms from a dynamical point of view
This will be a survey of the problems and phenomena that arise in attempting to describe the space of all compact group automorphisms modulo natural dynamical equivalences. This natural question turns out to involve diverse problems, and when the same question is asked for more general group actions entirely new rigidity phenomena arise.

September 7, 2011  Wil Schilders 
Model Order Reduction: mathematical methods and applications
Model Order Reduction (MOR) is a flourishing field in numerical mathematics that aims at reducing complex models while retaining dominant features, as well as relevant properties. It originates from the systems and control discipline, the most popular technique being truncated balanced realization that is based on the solution of systems of Lyapunov equations. Since the 1990's, however, numerical mathematicians became interested in the field, especially after the breakthrough work of Feldmann and Freund on using Lanczos methods to generate low order models. This has led to a wealth of developments, to date still mainly for linear models, but also for the nonlinear and parameterized case. There is an intimate relation with numerical linear algebra, most notably the solution of large linear systems and the determination of selected eigenvalues.
In this presentation, we will discuss the most important developments in Model Order Reduction to date from a numerical point of view. Lanczos and Arnoldi type methods, the dominant and sensitive pole algorithms, efficient solution of large Lyapunov systems will be touched upon. In addition, a number of applications in industry will be shown, MOR being of vital importance for challenging simulations.

September 21, 2011  Igor Stojkovic 
Gradient Flows, Product formulas, and Maximal Monotone Operators in Metric Spaces
The optimal transportation theory has been one of the fastest expanding branches of mathematics of the past decade. One of the striking results in this direction is the interpretation of the Fokker-Planck equation as a gradient flow in the non-linear space of probability measures on Rd (Felix Otto et al. 1998). In the first part this talk I will give an overview of the key concepts of the theory of gradient flows in metric spaces, and also introduce two particular classes of spaces which are relevant in this context: the Wasserstein spaces of probability spaces, and the non-positively curved spaces. An integral part of the classical theory of gradient flows in Hilbert spaces are the so called Trotter-Kato product formulas. Furthermore, a natural extension of gradient flows are the well studied maximal monotone operators. It turned out that product formulas for gradient flows in non-positively curved spaces can be proved. Moreover one can introduce maximal monotone operators on Wasserstein spaces and give their systematic treatment, thereby extending the theory of Ambrosio-Gigli-Savaré. The presented results are a part of my recently defended PhD thesis (April 2011, Leiden University).

October 5, 2011  Karen Aardal 
Uncapacitated facility location: A problem we "almost" understand
The Uncapacitated Facility Location Problem (UFLP) is one of the classical discrete optimization problems. UFLP is NP-hard, but it has some practical and theoretical features that make it "almost easy". To mention some examples: for reasonable objective functions, we observe that the solutions to the linear relaxation are typically integral; probabilistic results indicate that the duality gap is small; in terms of approximability the lower and upper bounds on approximability are almost equal. I will illustrate these observations and pose some open problems.

October 12, 2011  Lars Diening 
A decomposition technique for John domains
We develop a method to decompose functions with mean value zero on a (possibly unbounded) John domains into a countable sum of functions with mean value zero and support in balls. John domains may have a very bad boundary, for example the famous Koch's snowflake is a John domain. This method enables us to generalize results known for balls to such bad domains in an almost trivial way. As application we present the solvability of the divergence equation div u = f, the negative norm theorem, Korn's inequality, and Poincaré's inequality.

October 19, 2011  Fokko van de Bult 
A Mendeleev table for classical orthogonal polynomials: Obtaining the q-Askey scheme using elliptic hypergeometric functions
Orthogonal polynomials are polynomials which are orthogonal under a bilinear form which is of the form $\langle f,g\rangle = \int fg d\mu$ for some measure $\mu$. There are several examples which can be described explicitly (using hypergeometric functions), and which occur naturally in several different applications. For example you may have heard of the Chebyshev or the Legendre polynomials. In the 1980s these different example were placed in a single scheme, which has become known as the Askey-scheme. This scheme contains all these ``classical'' examples of orthogonal polynomials, together with the limit transitions between them. In this talk I will discuss how we can obtain the analogous $q$-Askey scheme (which is the scheme for orthogonal polynomials which can be expressed as basic hypergeometric series) by considering limits of the elliptic hypergeometric biorthogonal functions invented (in the 2000s) by Spiridonov and Zhedanov. As a result we can find a very pretty geometric description of the $q$-Askey scheme, and we can extend that scheme to include many families of biorthogonal rational functions. This is joint work with Eric Rains.

November 2, 2011  Karma Dajani 
Two special invariant ergodic measures for random beta transformations
It is well known that if beta is a non-integer greater than 1, then almost every point has uncountably many expansions in base beta. In this talk, we will introduce a transformation, the so called random beta transformation, whose iterations produce all possible expansions in base beta. We exhibit two natural ergodic invariant measures for this transformation, give their properties and prove that these measures are mutually singular.

November 16, 2011  Tanja Lange 
Advances in Elliptic-Curve Cryptography
The first part of this talk presents results on attacking elliptic-curve cryptography, in particular an ongoing effort to break the Certicom challenge ECC2K-130 and a detailed study on the correct use of the negation map in the Pollard- rho method. The second part presents a signature scheme which on a 390 USD mass-market quad-core 2.4GHz Intel Westmere (Xeon E5620) CPU can create 108000 signatures per second and verify 71000 signatures per second on an elliptic curve at a 2128 security level. Public keys are 32 bytes, and signatures are 64 bytes. These performance figures include strong defenses against software side-channel attacks: there is no data flow from secret keys to array indices, and there is no data flow from secret keys to branch conditions.

November 30, 2011  Wieb Bosma 
Some intriguing aspects of continued fractions
Continued fractions provide an alternative representation of real numbers (instead of decimal or binary expansions). This representation gives the best rational approximations but also has serious disadvantages, for arithmetic for example, for instance because the partial quotients (alternative digits) can be arbitrarily large.
In this elementary talk some aspects of continued fractions will be highlighted: connections with the theory of finite automata (the simplest model of computing) and with the distinction between algebraic and transcendental numbers. With elementary means some results for continued fractions with bounded partial quotients will be derived. Among these, a surprising recent result about continued fractions for complex numbers: the existence of algebraic numbers of arbitrary even degree with bounded complex partial quotients.

December 14, 2011  Martijn Pistorius 
Optimal dividend distribution in the presence of a penalty
In this talk we consider an optimal dividend problem for an insurance company which risk process evolves as a spectrally negative Levy process (in the absence of dividend payments). We assume that the management of the company controls timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividends received until the moment of ruin and a penalty payment at the moment of ruin which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. We explicitly solve the corresponding optimal control problem. The solution rests on the characterization of the value-function as the smallest stochastic super-solution. We find also an explicit necessary and sufficient condition for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. Joint work with F Avram and Z Palmowski.

February 1, 2012  Bart Vlaar 
Non-symmetric particle creation operators for the quantum nonlinear Schrodinger model
We introduce the quantum nonlinear Schr=9Adinger (QNLS) model which describes a certain quantum-mechanical many-body system. It is of mathematical interest because it has a rich underlying theory; on the other hand it has been physically constructed in labs (at the UvA, among others). We briefly review the two main methods used to study this model. In the 1980s, the quantum inverse scattering method (QISM) developed by the Faddeev school was applied to the QNLS model yielding recursive relations for the quantum-mechanical wavefunctions solving the QNLS model. Furthermore, in the 1990s another approach proved fruitful, in which representations are studied of the degenerate affine Hecke algebra of type A, which is a certain deformation of the group algebra of the symmetric group. A common eigenfunction of the so-called Dunkl-type operators, which feature in one representation, can be constructed using a second representation. This eigenfunction is non-symmetric; by symmetrizing it one obtains the QNLS wavefunction. We present an alternative way of constructing this non-symmetric eigenfunction, namely recursively in a QISM-type fashion, thereby providing a link between the two solution methods.

February 15th, 2012  Krzysztof Apt 
Choosing Products in Social Networks
Social networks have become a huge interdisciplinary research area with important links to sociology, economics, epidemiology, computer science, and mathematics. We introduce a new threshold model of social networks, in which the nodes influenced by their neighbours can adopt one out of several alternatives (products). We study various algorithmic questions concerning these networks, for example the problem of computing the minimum (resp. maximum) possible spread of a product. Also, using game-theoretic concepts, we analyze the consequences of adopting products by the agents who form the network. In particular, we prove that determining an existence of a (pure) Nash equilibrium is NP-complete. We explain how these results can be used to analyze consequences of the addition of new products to a social network. We show that in some cases such an addition can permanently destroy market stability. Based on joint works with Vangelis Markakis and Sunil Simon.

Febuary 29th, 2012  Gil Cavalcanti 
Generalized geometry and T-duality
"Generalized geometry" is a term which refers to geometric structures on the direct sum of tangent and cotangent bundles of a manifold. They were introduced by Courant and Weinstein in 1990 as a way to unify the geometry of a closed 2-form and of a Poisson bivector. They received renewed interest with the introduction of Generalized complex structures by Hitchin in 2003. In this talk I will review the setup and some of the most basic geometric structures that appear in the context of generalized geometry. Then I will explain how T-duality can be interpreted in this context.

March 14th, 2012  Bert Zwart 
An encounter with Erlang, Gauss, Poisson and Ramanujan
Erlang's formulae describe the probability of blocking, delay or abandonment in three basic qeueing models. Despite (or thanks to) their simplicity, these formulae are among the most celebrated results in Applied Probability. For large systems (for example, call centers with many agents), these formulae become less insightful, and a large body of research is devoted to developing asymptotic approximations of blocking probabilities. This talk is devoted to assessing the quality of such approximations. In passing, we provide new Gaussian approximations of Poisson distributions.

March 28th, 2012  Verbitskiy 
Dimers, sandpiles and algebraic dynamics
In this talk I will address the link between solvable models of statistical mechanics and algebraic dynamical systems. The main reason to believe in the existence of a strong link is the remarkable coincidence of entropies of many celebrated solvable lattice models (dimer matchings, domino tilings, spanning trees, etc) and entropies of certain algebraic dynamical systems. Even though the question about the existence of such a link was raised almost two decades ago, this problem remained largely inaccessible. The development of the theory of symbolic covers of algebraic dynamical systems has only recently provided a suitable framework. I will describe in greater detail the link between the solvable sandpile models and their algebraic counterparts. The talk is based on a series of joint papers with D. Lind (Seattle) and K. Schmidt (Vienna).

April 11th, 2012  Jose Blanchet 
Modeling and Efficient Rare Event Simulation of Systemic Risk in Insurance-Reinsurance Networks
We Prose a dynamic insurance network model that allows to deal with reinsurance counter-party default risks with a particular aim of capturing cascading effects at the time of defaults. We capture these effects by finding an equilibrium allocation of settlements which can be found as the unique optimal solution of a linear programming problem. This equilibrium allocation recognizes 1) the correlation among the risk factors, which are assumed to be heavy-tailed, 2) the contractual obligations, which are assumed to follow popular contracts in the insurance industry (such as stop-loss and retro-cesion), and 3) the interconnections of the insurance-reinsurance network. We are able to obtain an asymptotic description of the most likely ways in which the default of a specific group of insurers can occur, by means of solving a multidimensional Knapsack integer programming problem. Finally, we propose a class of provably strongly efficient estimators for computing the expected los! s of the network conditioning the failure of a specific set of companies. Strong efficiency means that the complexity of computing large deviations probability or conditional expectation remains bounded as the event of interest becomes more and more rare.

April 25th, 2012  Hans Maassen 
Quantum information, probability, and statistics
Non-commutative or "quantum" probability theory considers random phenomena from the point of view of operator algebras. In this way techniques from probability and statistics can, by generalization beyond commutative algebras, be applied to quantum mechanical systems. In order to illustrate the method, we discuss several topics: limits to the copying of information in connection to the Heisenberg principle, entanglement of quantum systems, in particular under symmetry, and the use of Young diagrams as statistical estimators.

May 9th, 2012  Harrie Willems 
Late Style - Yuri Manin Looking Back on a Life in Mathematics
A film by Agnes Handwerk and Harrie Willems.
This biographical documentary follows Yuri Ivanovich Manin¿s stellar career in the "golden years" of Moscow mathematics during the 1960s and 1970s. He was one of the key players in the development of algebraic geometry, at a time when a constellation of brilliant minds - of which Manin's was but one - were conducting outstanding mathematical research. This happened under the structures of a closed society that put severe restrictions on academics (despite diplomatic détente) right up to the fall of the Iron Curtain in 1989. Yuri Manin's brilliance, and his unswerving integrity, helped him to evade the pitfalls of Sovietera academia.
His full cooperation with the documentary allows the world a fascinating glimpse into an era of scientific enquiry that is as celebrated as it is underreported. It is about the exceptional life of a mathematician in unusual times, whose passion for his subject, as well as his breadth of thinking, allowed him to forge his own freedom. Internationally recognized for his contribution to mathematics, Manin's many prizes include the Lenin Prize in 1967, the Brouwer Gold Medal in 1984 and the Georg Cantor Medal in 2002. The length of the documentary is 57 minutes.

May 23th, 2012  Hans Zwart 
Linear port Hamiltonian Systems
The field of infinite-dimensional systems theory has become a well-established field within mathematics and systems theory. There are basically two approaches to infinite-dimensional linear systems theory: an abstract functional analytical approach and a PDE approach. Many physical systems can be formulated using a Hamiltonian framework. This class contains ordinary as well as partial differential equations. Each system in this class has a Hamiltonian, generally given by the energy function. In the study of Hamiltonian systems it is usually assumed that the system does not interact with its environment. However, for the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take this interaction with the environment into account. This led to the class of port-Hamiltonian systems. For port-Hamiltonian systems described by ordinary differential equations this approach is very successful. Port-Hamiltonian systems described by partial differential equation is a subject of current research.
In this presentation, we combine the abstract functional analytical approach with the more physical approach based on Hamiltonians. For a class of linear infinite-dimensional port-Hamiltonian systems we derive easy verifiable conditions for well-posedness.

September 5th 2012  Sameer Murthy 
Mock theta functions and their appearance in physics
Automorphic forms of various types have arisen repeatedly in diverse areas of physics. In this talk, I shall discuss two related instances of this type, which involve a recently-defined class of functions called mock theta functions. The first instance is connected to quantum black holes in string theory, and the second is connected to the representations of the largest Mathieu group M24. I will explain these connections and discuss some properties of these functions.

September 19th 2012  Ragnar Sigurdsson 
Growth estimates of entire functions and Paley-Wiener theorems
The main motivation for the study of entire functions of exponential type is the fact that they are Fourier-Laplace transforms of functions with compact support, distributions with compact support, and analytic functionals. In the first half of the lecture I will discuss a few variants of the classical Paley-Wiener theorem, which enable us to locate the convex hull of the support of a function or a distribution through estimates of its Fourier-Laplace transform. In the second half I will discuss the following problem: Assume that we have an entire function of several complex variables and that it is of exponential type when restricted to a family of complex lines through the origin. Which growth estimates does it then satisfy in the whole space? The solution of the problem involves a few facts on convex and plurisubharmonic functions and we will see that it enables us to relax conditions in the Paley-Wiener theorems.

October 3rd 2012  Koen Turck 
Poisson's equation for Markov chains and its use in perturbations
I will talk about two recent research topics of mine in which Poisson's equation plays a crucial role. I will start out by introducing Poisson's equation in a Markov chain context, and point out how it relates to the (to some) more familiar notion of Poisson's equation in PDE theory. Determining solutions for Poisson's equation for Markov chains, as in physics, often proceeds via Green's functions. In the first part of the talk, I will derive explicit expressions for the transforms of these Green's functions. It appears this can be done for a surprisingly large class of Markov chains. The second part of the talk centres around perturbation of Markov chains. I will illustrate, by means of a concrete example, a powerful but often hard to formalise sample-path technique and contrast it with the direct but often rather long-winded approach. I conclude the talk by showing how a formulation in terms of Poisson's equation provides the best of both worlds.

October 17th 2012  Mehdi Tavakol 
Intersection theory on moduli spaces of curves.
I will talk about moduli spaces of curves and their invariants. The main focus is on the study of algebraic cycles on these spaces. A distinguished collection of cycles, which are called tautological classes, and some of their properties will be discussed.

November 14th 2012  David Anderson 
Stochastic models of biochemical systems.
I will give an introduction to the most common stochastic models used in the study of population processes, and, in particular, biochemistry. I will develop from first principle arguments the relevant mathematical equations governing the systems and will then discuss some computational challenges presented by these models. In particular, I will discuss recent developments in numerically computing sensitivities (derivatives of expectations with respect to system parameters).

November 28th 2012  Bob Rink 
Coupled cell networks: semigroups, Lie algebras and normal forms
Dynamical systems with a network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web. In this talk I will explain what it means for a coupled cell network to possess the "semigroup(oid) property". Networks with this property form a Lie algebra and we recently developed a method to compute their local normal forms near a dynamical equilibrium. This helped us understand and predict certain seemingly anomalous bifurcations in network systems.

February 6th 2013  Jop Briët 
Grothendieck's inequality for quantum entanglement and combinatorial optimization
Grothendieck's inequality is a fundamental result to the theory of Banach spaces. But many years after Grothendieck published this result, people realized that it also has important applications in theoretical computer science and physics. In particular, the inequality can be used in quantum information theory to study the mysterious phenomenon of quantum entanglement and in combinatorial optimization to prove performance guarantees of efficient approximation algorithms for NP-hard problems. The aim of this talk is to give an overview of these applications and explain why Grothendieck's inequality and recent generalizations of it are powerful mathematical tools to study them.

February 20th 2013  Daan Crommelin 
Stochastic representation of unresolved scales in atmosphere models
In atmosphere-ocean science, the representation (or parameterization) of subgrid scale processes in numerical models is a notoriously difficult problem. In recent years, researchers have turned to stochastic methods in order to improve these parameterizations. I will present a stochastic, data-driven approach to the problem, in which unresolved processes are represented by a network of Markov processes that are conditioned on resolved-scale model variables. These Markov processes are estimated from data of e.g. highly resolved, limited-area simulations, hence they mimick, or emulate, in a statistical-dynamical way the feedback from small-scale processes as simulated by a high-resolution model.

March 6th 2013  Arthemy Kiselev 
On the geometry of the Batalin-Vilkovisky Laplacian
We approach the self-regularization in a functional definition of the Batalin-Vilkovisky Laplacian, which is a necessary ingredient in the quantization of gauge-invariant field theories. We analyse the geometry of variation of functionals: namely, we study the interrelation of bundles in the course of integration by parts, the implications of the locality postulate, and a rigorous construction of iterated functional derivatives. In particular, we show that the conventional formula for calculation of a functional's variation is a consequence of the true geometric definition -- but not a definition itself. Thus, as a by-product we clarify the derivation of Euler-Lagrange equations. The core of known difficulties, which are manifest from elementary counterexamples, is that the standard geometry of functionals is insufficient to grasp the full geometry of the calculus of variations. Indeed, several important identities combining the Schouten bracket and the BV-Laplacian do not hold; such identities involve higher-order variational derivatives but those need to be proclaimed permutable whenever one inspects the response of a functional to a shift of its argument along different test sections. To circumvent the obstructions, we enlarge the spaces of functionals in such a way that there is enough room to store the information about the test shifts. Our approach resolves the problem of intrinsic regularization of the BV-Laplacian and Schouten bracket; the newly defined structures match in all standard ways. The talk is based on the recent work arXiv:1302.4388 [math.DG] joint with S.Ringers.

March 20th 2013  Roland van der Veen 
Knots, representations and some physics
The aim of this talk is to give a brief survey of knot theory with a focus on the so called HOMFLY polynomial. The HOMFLY polynomial is not only useful for telling knots apart but also has a rich structure of its own related to quantum field theory and string theory. We will report on recent progress toward proving a conjecture of C. Vafa on the existence of certain recursions for the HOMFLY polynomial.

April 3rd 2013  Monique Laurent 
Positive semidefinite matrix completion and geometric graph realizations
We consider the problem of completing a partially specified matrix to a positive semidefinite matrix, with special focus on questions related to the smallest possible rank of such completion. We present complexity results and structural characterizations of the graph of specified entries for the existence of small rank completions. We also discuss links to Euclidean graph realizations in distance geometry and to some topological graph parameters of Colin de Verdi\`ere type. In these various topics, the geometry of semidefinite programming provides a unifying setting.

April 17th 2013  Raf Bocklandt 
A noncommutative glance through the mirror
Mirror symmetry is a strange relationship between two types of geometry: symplectic and algebraic. We will explain how mirror symmetry works using some concrete examples and illustrate how one can use techniques from noncommutative geometry to shed light on this phenomenon.

May 29th 2013  Tobias Mueller 
Logic and random graphs
Random graphs have been studied for over half a century as useful mathematical models for networks and as an attractive bit of mathematics for its own sake. Almost from the very beginning of random graph theory there has been interest in studying the behaviour of graph properties that can be expressed as sentences in some logic, on random graphs. We say that a graph property is first order expressible if it can be written as a logic sentence using the universal and existential quantifiers with variables ranging over the nodes of the graph, the usual connectives AND, OR, NOT, parentheses and the relations = and ~, where x ~ y means that x and y share an edge. For example, the property that G contains a triangle can be written as Exists x,y,z : (x ~ y) AND (x ~ z) AND (y ~ z). First order expressible properties have been studied extensively on the oldest and most commonly studied model of random graphs, the Erdos-Renyi model, and by now we have a fairly full description of the behaviour of first order expressible properties on this model. I will describe a number of striking results that have been obtained for the Erdos-Renyi model with surprising links to number theory, before describing some of my own work on different models of random graphs, including random planar graphs and the Gilbert model. (based on joint works with: P. Heinig, S. Haber, M. Noy, A. Taraz)

September 19th 2013  Hitoshi Murakami 
A survey of the volume conjecture and its generalization.
The volume conjecture states that the colored Jones polynomial of a knot would give the volume of the knot complement. It can be generalized to a relation of the colored Jones polynomial to the volume, the Chern-Simons invariant and the Reidemeister torsion associated with a representation of the fundamental group of the knot complement to SL(2;C). I will give a survey of these conjectures.

October 3 2013  Harry van Zanten 
The mathematics of nonparametric Bayesian statistics
In this talk I will try to explain what nonparametric Bayesian statistics is, why it is such a fashionable topic, and what some of the interesting mathematical problems in this area are. I will make a serious attempt to keep the story accessible for a general math audience. If time permits I will talk about recent solutions to some concrete problems as well and give an idea of the math involved.

October 24 2013  Benno van den Berg 
Nonstandard analysis and proof theory
The aim of this talk is to illustrate some themes from proof theory using systems for nonstandard analysis. Proof theory was originally invented by Hilbert in order to settle his foundational disputes with Brouwer. Hilbert's idea was to formalise all of mathematics, included the infinitary (ideal) parts that had been put into doubt, and then prove it consistent using a finitary proof, which Brouwer and others would also be compelled to accept. Of course, this programme collapsed, when Godel proved his famous incompleteness theorems, showing that infinitary notions have real strength and their consistency cannot be established using finitary means. Nowadays, proof theorists have less grand goals and usually see it as their task to understand and measure the precise strength of infinitary notions in mathematics, sometimes with surprising results. In this talk, I will focus on nonstandard analysis as an example of the use of "ideal" objects (such as infinite numbers and infinitesimals) to establish "real" mathematical results and use it to illustrate some proof-theoretic themes. (This will be based on joint work with Eyvind Briseid and Pavol Safarik.)

November 21 2013  Sander Dahmen 
Diophantine problems and the modular method
Since the proof of FLT, many Diophantine problems have been solved using deep results about elliptic curves, modular forms, and associated Galois representations. The purpose of this talk is to discuss some of these results and explain how they can be applied to explicitly solve certain Diophantine problems.

December 5 2013  Angela Ortega 
Prym-Tyurin varieties
The classical Prym varieties introduced by Mumford in the 70's opened new perspectives in the study of the moduli space of abelian varieties: Prym varieties are different from Jacobians, but they arise from coverings between curves, which allows one to describe the geometry of its theta divisor in terms of the curves involved. Later, Tyurin produced principally polarized abelian varieties constructed from Fano 3-folds. He was the first to notice the common feature among these examples: they are subvarieties of Jacobians JC defined via an endomorphism of JC satisfying a quadratic equation. In the first part of the talk I will present the basic constructions of Prym and Prym-Tyurin varieties. I will explain (1) the special case of the only Prym-Tyurin variety of exponent different from 2 associated to a covering of curves, discovered recently in a joint work with H. Lange, and (2) how the general Jacobian of a curve of odd genus can be realized as a Prym-Tyurin variety of exponent equal to the Catalan number. I will close by indicating how Prym-Tyurin varieties can be used to give a structure theorem for polarized abelian varieties of dimension 6.

Feburary 5 2014  Miranda Cheng 
Mock Modular Forms, Niemeier Lattices, and Umbral Moonshine
The term "moonshine", first introduced in the context of Monstrous Moonshine, describes an unexpected relation between modular functions and the representation theory of finite groups. I will give an overview of the recent developments in the area, focussing on the "umbral moonshine" phenomenon which (conjecturally) relates mock modular forms and various finite groups arising naturally from the study of the Niemeier lattices, and its possible relation to K3 surfaces and string theory. This talk is mostly based on joint work with John Duncan and Jeff Harvey.

Feburary 19 2014  Guus Regts 
Partition functions of edge- and vertex-coloring models: characterizations and connections to invariant theory and Hilbert spaces
Edge- and vertex-coloring models are generalizations of the Ising-Potts model of statistical mechanics. The partition functions of these models were introduced as graph parameters by de la Harpe and Jones (P. de la Harpe, V.F.R. Jones, Graph invariants related tostatistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57, 207–227, 1993). In this talk I will introduce these parameters and discus how to characterize them using invariant theory of the orthogonal/symmetric group and Hilbert's Nullstellensatz. If time permits, I will moreover discuss connections to tensor algebras and the recent they of graph limits. This talk is partly based on joint work with Dion Gijswijt, TU Delft; Jan Draisma, TU Eindhoven; László Lovász, Eötvös Loránd University in Budapest and Lex Schrijver, CWI and UvA

March 5 2014  Fabian Ziltener 
Coisotropic Submanifolds of Symplectic Manifolds, Leafwise Fixed Points, and a Discontinuous Capacity
Consider a symplectic manifold $(M,\omega)$, a coisotropic submanifold $N$ of $M$, and a selfmap $\phi$ of $M$. A leafwise fixed point of $\phi$ is a point in $N$ that under $\phi$ is mapped to the isotropic leaf through itself. Symplectic manifolds naturally generalize phase space of classical mechanics. In this setting coisotropic submanifolds arise as energy level sets. Let $\phi$ be the time-one flow of a time-dependent perturbation of a given Hamiltonian function on $M$. Then a leafwise fixed point of $\phi$ is a point on a given energy level set whose trajectory is changed only by a phase shift, under the perturbation. I will discuss lower bounds on the number of leafwise fixed points of $\phi$. As an application one obtains a symplectic capacity by considering the minimal actions of regular closed coisotropic submanifolds of a given symplectic manifold. A variant of this capacity is discontinuous. This talk is partly based on joint work with Jan Swoboda and Kai Zehmisch.

March 19 2014  Rob van der Mei 
Towards a unifying theory on polling models: new results and challenges
Polling systems are multi-queue systems in which a single server visits the queues in some order to serve the customers waiting at the queues, typically incurring some amount of switch-over time to proceed from one queue to the next. Polling models find a wide variety of applications in areas like computer-communication systems, logistics, flexible manufacturing systems, production systems and maintenance systems. In this talk, I will discuss a number of interesting fundamental properties of polling models, and discuss recent progress in an attempt to develop a unifying theory on polling models under heavy-load circumstances.

April 2 2014  Teun Koetsier and Jan van Mill  
Irmgard Gawehn and the topological classification of manifolds. An attempt at rehabilitation.
The talk is about the life and work of Irmgard Gawehn, a female German topologist. In the 1920s she wrote a brilliant Ph D thesis on the topological characterization of the 2-sphere. From 1928-1930 she was L. E. J. Brouwer's assistant at the University of Amsterdam. In the late 20th century Hans Freudenthal claimed that Gawehn had not written her thesis herself. We will discuss the significance of her topological work and we will critically examine Freudenthal's claim.

April 23 2014  Lex Schrijver  
Several graph invariants can be described as `partition functions' (in the sense of de la Harpe and Jones). In the talk we give an introduction to this and we present characterizations of such partition functions among all graph invariants. We show how similar methods describe knot invariants and give rise to varieties parametrizing all partition functions. We relate this to the Vassiliev knot invariants, and show that, in the real case, its Lie algebra weight systems are precisely those weight systems that are "reflection positive". We also give a one-to-one relation between complex Lie algebra weight systems and reductive metric Lie algebras. The talk will be introductory and does not assume any specific knowledge on graphs, knots, or algebras.

May 28 2014  Mingmin Shen  
Differential equations from an algebraic point of view
In analysis one constructs new functions by specifying a differential equation they satisfy while in algebra one constructs new functions (or “numbers”) by specifying an algebraic equation they satisfy. It is very interesting to ask when a solution to a differential equation actually satisfies an algebraic equation. The Grothendieck-Katz conjecture gives a criterion to that by testing the differential equation in positive characteristic. I will explain the basic ideas by examples. I will also explain why this is interesting in algebraic geometry.