September 12, 2001
Erik Koelink
(Delft University of Technology )
Special functions, the quantum dynamical
YangBaxter 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 finitedimensional
irreducible representations related to
special functions, as well as the decomposition of
tensor products of irreducible representations.
The last subject leads to ClebschGordan coefficients
(or 3jsymbols) and Racah coefficients (or 6jsymbols).
From this we can obtain a solution of the
quantum dynamical YangBaxter equation in terms
of Racah coefficients, which can be traced back
to an unpublished paper (1940) by E.P. Wigner
(19021995). In the second part we discuss
the quantum dynamical YangBaxter 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, 1112 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: modeltheoretic investigations.'')
October 10, 2001
Erik Balder
(Utrecht University)
New results in and applications of topological measure theory:
Weakstrong 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 socalled "CohenLenstra 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 CohenLenstra 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 (18541929). 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 (RISCLinz) 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 EulerLagrange 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 portHamiltonian
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,
electromechanical 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 subsystems.
It will be indicated that the class of portHamiltonian systems is closed under
powerconserving interconnection, and how this may be exploited for control and
design. Finally we shall discuss the extension of this Hamiltonian description
of actuated lumpedparameter systems to distributed parameter systems with
energy flow through the boundary of the spatial domain. Key concept is the
notion of the (infinitedimensional) StokesDirac 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 socalled 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
timeconsuming, much interest is directed at the development of efficient
numerical methods on highresolution 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 thirdorder upwind scheme
for the constant state
interpolation. This spatial discretization scheme is robust and
secondorder 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 latitudinallongitudinal (latlon) grid
combined with an explicit time integration method to solve the resulting
semidiscrete system. This prejudice has to do with the pole problem, which
includes all problems related to the nonexistence 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 poleproblem:
(1) a combined latlon reduced grid with two stereocaps in the polar
area and (2) a linearlyimplicit 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 latlon
grid.
Both remedies are investigated by numerical tests on a wellestablished
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 latlon grid. It is even far more
efficient than the solution of the semidiscrete 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 oneparameter
automorphism group (describing the time evolution of the observables).
Thermodynamic equilibrium states are usually characterized by
the KMScondition, 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)
timeevolution. A state that exhibits thermodynamic equilibrium with
respect to any timeevolution 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 ChebyshevType Inequalities
Consider a training sample X_{1},...,X_{n} 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 wellknown 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 extremevalue 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)=z^{3}g_{2}zg_{3} is a squarefree polynomial in z and
n and B are constants.
The general question we try to answer is for which n, g_{2},
g_{3} 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 g_{2}, g_{3} 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 welldocumented or discussed in peerreviewed 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 phasetransition
in the BakSneppen evolution model The BakSneppen 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 selforganising behaviour. In this lecture, I will first
describe the underlying paradigma of selforganised criticality, and explain how the BakSneppen
model fits in this framework. I will explain why physicists believe that this model shows
selforganisational 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)= (zp)g(z) ? The answer is yes; one can simply divide out
a factor (zp) in the power series of f.
We now step into the world of several complex variables : let G be a
domain in C^{n}. We wonder if we can find bounded holomorphic functions
f_{1},...,f_{n} such that
f(z) = (z_{1}p_{1})f_{1}(z) + ^{...} + (z_{n}p_{n})f_{n}(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
"halftwist": to the data of a Hodge structure with the action of a
CMfield, he associates a Hodge structure of lower weight, its halftwist
(under certain hypotheses).
In this work (joint with Van Geemen), we consider the case where the Hodge
structure is the primitive cohomology of an ndimensional hypersurface of
degree d, cover of P^{n} totally ramified along an (n1)dimensional
hypersurface of degree d. In this case the CMfield is a field of roots of
unity. We determine when the halftwist exists, and, when it does, embed
it into the cohomology of a "nice" algebraic variety. We then prove a
Torelli theorem for the halftwist when d=3. Finally, we prove the
existence of a "KugaSatake" 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, socalled 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.
Wellknown 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 nonextreme 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 areintegrable 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 timeinvariant 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 BaumConnes 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 socalled BaumConnes 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
BaumConnes 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 nonnegative. 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., qrecursive 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 mathsentrances 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 multispike solutions to a system of
reactiondiffusion equations.
The GiererMeinhardt equations are a system of reactiondiffusion
equations of activatorinhibitor 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 13 by
1. a rational chiral CQFT with a finite symmetry group G,
2. a modular crossed Gcategory, 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 k1. 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(N^{1/3}log 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 CramérRao 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 rightangled 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 (nonlogical)
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
scorebased algorithm. One drawback of a scorebased 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 multidimensional 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 twodimensional
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
pseudoelasatic 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 threedimensional
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 socalled "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.
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
logreturns 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 intraday data of the U.S. S&P 500
stock index are used to test the hypothesis that
the logreturn process may be a timechanged
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 timechange and the
volatility (the variability of the price process). We
observe that volatility is very unstable on small timeintervals.
However, on a larger timescale 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 nonselfadjoint 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
NevanlinnaPick 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 Hinfinity
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 wellknown fact that the symmetric random walk on the integer
lattice Z_{d} 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 Z_{d} 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 nonzero
drift.
In applications like queueing, one often has to deal with slightly more
general processes called `facehomogeneous random walks': there is a
finite partition of the state space Z_{d}, 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, nonnegative) harmonic
function f_{D}. The value f_{D(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 numbertheoretic results.
September 3, 2003
HaeWon 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 lambdastable 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 padic 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 socalled 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 socalled 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 minusinfinity 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 1skeleton (edge graph) of this polytope is defined as the graph
with vertices the vertices of the polytope and edges its 1dimensional
faces. In 1957 W.M. Hirsch stated his famous conjecture
(cf. [Dantzig, 1963]) saying that any ddimensional polytope with
n facets has diameter at most nd. So far the best bound for any
polytope is O( n^{log 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+n2). 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+n2)
steps along the edges.
February 11, 2004 Daniel Alpay
(BenGurion 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 nonstationary 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
h_{x}, rather than on a more evident function denoted as β_{x} (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 twopart code consisting of
one part model description and one part datatomodel code (essentially
the celebrated MDL code), subject to a given
modelcomplexity constraint, as well as minimizing the onepart code
consisting of just the datatomodel 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 twopart code, or the onepart 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.
References
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 (NashMoser), 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 extralogical content too.
For Brouwer, the
ultimate basis for all mathematics is the urintuition of `the move of
time', that is, the experience of the fact that two notcoinciding
mental events are connected by a time continuum. Departing from this
urintuition, 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 reactionadvectiondiffusion 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 (19161987) 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
ALGOLVerschwörung. 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.
Conflict
Van Wijngaarden and his team readily accepted this agenda
for a universal as to the mathematics and machineindependent
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 ALGOLgroup 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 AtiyahSinger index theorem
The 2004 Abel prize in mathematics has recently been awarded to
M.F. Atiyah and I.M. Singer in recognition of their socalled
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 AtiyahSinger index theorem and its significance, starting
from elementary linear algebra.
May 19, 2004
Misja
Nuyens
(Universiteit van Amsterdam)
Queues, heavy tails and the ForegroundBackground 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 socalled heavy tails. This means that very large values show up
relatively often. For servicetime 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 ForegroundBackground discipline, which will
be introduced in the talk. The queue with this discipline works
effectively, even for heavytailed 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 mathematicalbird'seye 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 driftdiffusion 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 decisionmaking 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 biophysicallybased 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 driftdiffusion 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 AstonJones (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 CP^{4}.
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 compactopen topology. We present a complete
solution to the topological
classification problem for H(M,D) as follows. If
M is a onedimensional 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 l^{2} 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 TemperleyLieb algebra. The components of this eigenvector have been
observed to be equal to specific counts of the socalled 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 padic 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 padic volume. Starting with the definition of density I
will show how the padic 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 longrange and shortrange 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 rareevent 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 largebuffer asymptotics (that were mainly
derived during the 1990s). Relatively new are the socalled many sources
asymptotics: relying on Schilder's theorem, I've found elegant explicit results,
not only for singlenode 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 spatiotemporal structures
in RayleighBenard 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 pairwise deltafunction 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 UMDspaces
It is well known that the theory of stochastic integrals can be extended to
Hilbert spacevalued processes in a very satisfactory way. The reason for
this is that the Itô isometry is an L^{2}isometry. At the same time this
indicates that serious obstructions may be expected for setting up a theory
of stochastic integration for Banach spacevalued 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 operatortheoretic framework. The rôle of L^{2}spaces is then
replaced by the operatorideal of space of radonifying operators,
which in the Hilbert space case coindices with the operator ideal of
HilbertSchmidt 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
GromovWitten 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 socalled DonaldsonThomas theory
are vector bundles, or more generally sheaves.
In the case of toric threefolds 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 canjust as classical bitsbe 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 S_{l} and certain representations
of the Lie algebra sl_{n}. 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
sl_{n} 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
SchurWeyl type of equivalence involving Cherednik algebras and Yangians
of affine type.
May 25, 2005
Charles Dunkl
(University of Virginia)
Nonsymmetric Jack Polynomials and CalogeroMoser Models
This is an overview of the technique of differentialdifference 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 CalogeroMoser 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 Lundadesigns and
Likidesigns that display interesting symmetries and conducts to the
introduction of several types of matrices with attractive visual properties.
References
[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 errorcorrecting 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 halfinteger
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
(BarIlan 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)
Nonsymmetric Wilson polynomials
The Hankel transform is a selfdual Fourier transform with a
Bessel function as a kernel. A "nonsymmetric" version of the Hankel
transform can be studied using a degeneration of a double affine Hecke
algebra. We use a similar approach to study nonsymmetric 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 noncompact type, i.e. the
Lietheoretic generalizations
of the upper half plane H=SL(2,R)/SO(2).
For such a space X we introduce
its natural complexification, the socalled 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
measuretheoretic 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 measuretheory hinges are also given (for
instance, the monotone class theorem and monotone
convergence for quantum integrals).
To demonstrate usefulness, we conclude with a
noncommutative generalization of the RadonNikodym
theorem (which is also valid in nonseparable Hilbert spaces),
that covers (a version of) Gleason's theorem,
Schrödinger's timeevolution, the RadonNikodym 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 semilinear variant of a compactification of
the projective linear group PGL(n) (the socalled 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 OUprocesses
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 OrnsteinUhlenbeck 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)
Flowlevel performance in wireless data networks
Channelaware scheduling algorithms provide an effective mechanism
for improving throughput performance in wireless data networks
by exploiting multiuser diversity. In this talk, we focus on the
flowlevel performance of channelaware scheduling algorithms in
a dynamic setting with random finitesize data transfers.
We present simple conditions for flowlevel stability, and show
that in certain cases the flowlevel performance may be evaluated
by means of a ProcessorSharing model where the service rate varies
with the total number of users. Time permitting, we conclude
with a discussion of capacity issues and flowlevel 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 semisimple symmetric Frobenius algebra
allows to construct a twodimensional 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
(HeinrichHeineUniversitüt Düsseldorf)
Teichmüller curves and triangle groups
Teichmüller curves are algebraic curves isometrically embedded in the
moduli space of curves M_{g}. 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)
GromovWitten Theory, Localization, and Large N Duality
GromovWitten 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 AtiyahBott localization theorem. In this
talk we will present a new approach to localization in GromovWitten
theory based on large N string duality. We will focus on the interplay
between localization, ChernSimons theory and extremal transitions,
emphasizing recent developments for nontoric CalabiYau threefolds.
March 8, 2006
Ton Dieker
(CWI)
Extremes and fluid queues
One of the cornerstones of queueing theory is the singleserver 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 wellknown integral
equations) induce semiflows (or semigroups) 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 variationofconstantsformula. 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
variationofconstants 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 regimeswitching
behavior. Examples include stickslip 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: sewagebacteriumworm. 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 nutrientlimited system of two interacting populations and analyse the local persistence at an internal equilibrium over a parameter interval bounded by a saddlenode bifurcation point at one side and a Hopf bifurcation point at the other side.
Literature:
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.267280.
J. Grasman, M. de Gee and O.A. van Herwaarden,
Breakdown of a chemostat exposed to stochastic noise,
J. Engrg. Math. 53 (2005), 291300.
S.A.L.M. Kooijman, J.Grasman and B.W.Kooi, A new class of nonlinear 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 selfdual 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
(MarnelaVallee)
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 Salgebras 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 AndréQuillen cohomology for commutative algebras.
October 4, 2006
Erdal Emsiz
(UvA)
Affine Weyl groups and integrable systems with deltapotentials
I will talk about affine Weyl groups generalizations of the
onedimensional quantum Bosegas on the ring with pairwise
deltafunction interaction. Outline of the talk:
* Symmetry: By considering Dunkltype
differentialreflection operators associated to the integrable
system, we show that the fundamental object controlling the
algebraic relations between the Dunkltype 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 spinparticles with pairwise deltafunction 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 NonStandard 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 nonparametric, and a
semiBayesian 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
twodimensional projection images of the threedimensional
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 farfield
breakup, and spatiotemporal 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 dynamicalsystems 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 primepair 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 paircorrelation
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)
padic 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 HahnBanach 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 padic number fields, where p is
a prime number. By way of an introduction we treat the 10adic
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 nonHausdorff 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 nonHausdorff 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' noncommutative 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 lowerdimensional) 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
fulldimensional 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
nonuniqueness) 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 finitedimensional systems
(ODEs). We will conclude this talk with a short discussion of
invariant sets for infinite dimensional parabolic semiflows
generated by reactiondiffusion 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 coworkers (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)
19392004 : the development of classifying finite groups by means
of nisoclism 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, 1isoclinsm 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 1isoclinic to each other, but unfortunately all abelian
groups are 1isoclic to each other. It was Philip Hall who started to
study equivalence classes of 1isoclinic groups. In Amsterdam and Leiden
three dissertations dealt with the subject and with the more general notion
of nisoclinsm. In this talk I will say something about the development of
properties of nisoclinsm classes of groups, and mention some numerical
results. If time permits,I shall close the talk by indicating the solution
of the socalled nisoclism embedding subgroup problem, as obtained in 2003
and published in 2004. Today, properties of nisoclisms of groups and of
socalled nisologisms 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
rforms.
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.
References:
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 3connected cover of Spin(n).
By definition, this means that it is a topological group with π_{0}=0,
π_{1}=0, π_{2}=0, π_{3}=0,
and that it is equipped with a group homomorphism
String(n) → Spin(n) which induces isomorphisms on all homotopy groups
π_{i}, 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 Ktheory, 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., unilateral security. Secure computation focuses instead
on multilateral 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 multilateral 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 wellknown theorem of GarciaStichtenoth on
curves with many rational points they showed how to perform secure
computation with improved efficiency. This is the first nontrivial
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
qdifference 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. qdifference
equations are deformations of differential equations, as the
formal limit q → 1 turns qdifference equations into
differential equations. When q=1 the behaviour of the solution
space of qdifference equations changes radically. In particular,
there often exist no nonzero solutions in this case. However,
in such cases we can consider the 'doubled version' of the
qdifference equation. This doubled version is a particular
system of two ordinary difference equations on C canonically
associated to the original qdifference equation on C^{*}. A
solution to the doubled system can now serve as a solution to
the original qdifference equation. Even in cases where we start
with a qdifference 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
E_{7}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
wellknown being Fourier transformation). The analytic difference
operators arise from certain integrable systems, including nonlocal
soliton equations and relativistic Nparticle systems of
CalogeroMoser and Toda type.
December 12, 2007
Rob Stevenson
Adaptive wavelet methods for solving high dimensional PDE's
We present a nonstandard 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 twodimensional grid (using nlimit just intonation), they
form convex or starconvex regions. We found that this holds for
all scales in 5limit 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
patternbased musicanalysis program, which is part of my current
Viciproject.
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 socalled GKZfunctions,
and describe a combinatorial criterion for their algebraicity.
May 14, 2008
Dietrich Notbohm
Combinatorics of simplicial complexes and StanleyReisner algebras
from a topological point of view
Given a finite simplicial complex there are several associated algebraic and
topological invariants, e.g. the StanleyReisner algebra and
DavisJanuszkiewics 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
DavisJanucskiewicz space and depth of the StanleyReisner 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 lowdimensional
condensed matter systems, we study quantum mechanical models
for fermionic particles (`hard squares') on twodimensional
grids. The quantum ground states of a particular supersymmetric
model are in 1to1 correspondence with homology cycles of
the independence complex of the grid. For a generic
twodimensional 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
welltested Standard Model of highenergy 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
socalled 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 computerverified 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 downup 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 18792009
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 socalled 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 socalled 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 GuseinZade
Poincare series of multiindex filtrations and their generalizations.
For a multiindex 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 oneindex 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 zetafunction 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
HeegaardFloer 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 crossfertilization 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 twoplayer 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 zerosum 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 zerosum 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 socalled Ngroups, projective planes,
inverse Galois theory, finite simple groups like those of Ree, Suzuki, FischerGriess, and Thompson, the socalled jfunction,
the moonshine connection, etc.. His work (joint with Walter Feit) on the Burnsideconjecture:
"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 twodimensional 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 particlelike 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
Nonnegative polynomials constant on a hyperplane
We will consider polynomials with nonnegative 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 nonzero 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 CRGeometry, 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 realworld networks share
fascinating features. Many realworld networks are smallworlds, in
the sense that typical distances are much smaller than the size of the
network. Further, many realworld networks are scalefree 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 socalled 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 cellsat 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 semipassivity, which can be understood as an (energy) boundedness of the system behavior. In addition, a result regarding covergency is needed for the noncoupled 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 timedelayed coupled neuronal systems and induce a conjecture regarding network synchronization under timedelayed coupling.
Ref.: E.Steur, I.Tyukin, H.Nijmeijer (2009), 'Semipassivity and synchronization of diffusively coupled neuronal oscillators', Physica D 238, 21192128.
November 11, 2009
Kees Oosterlee
The Heston model with stochastic interest rates and pricing options with
Fouriercosine expansions
In this presentation we give a brief outline of our research on pricing
financial derivatives with numerical techniques.
We favor option pricing by Fouriercosine 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 HullWhite processes. We present approximations in the
HestonHullWhite 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 K_{2} of curves?
If F is any field, then K_{2}(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 Kgroups of curves we concentrate mostly on K_{2} of curves defined over number fields. The Beilinson conjectures then predict a relation between the
regulator of (a part of) K_{2} of such a curve and the value of its
Lfunction 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 KarhunenLoeve 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 Nterm, 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 proteinprotein 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 wellknown prizecollecting
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 noncompact 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 noncompact hypersurfaces M in ( R^2n , \omega) coming from
mechanical Hamiltonians, and prove an analogue of Viterboâ€™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 noncompact 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 socalled inclusion
process is a natural (bosonic) analogue of the wellknown 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 nonequilibrium 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 Spinstructure. 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 1dimensional
object, the Spinstructure of the space has to lift to a
Stringstructure, where the Stringgroup is the universal 3connected
cover of the Spin group. Contrary to the Spingroup, the Stringgroup
cannot be refined to a (finite dimensional) Lie group. Therefore the
question arises what a smooth differential refinement of a
Stringprincipal bundle would be, that encodes the dynamics of these
1dimensional 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 2group 
a differentiable groupstack. This allows to refine a topological
Stringprincipal bundle to a genralization of a differentiable
nonabelian gerbe: a smooth principal 2bundle. 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
Stringprincipal 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,
RiemannHilbert 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
Kortewegde 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 electromagnetic 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) twodimensional fluid flows, a numerical simulation inevitably becomes underresolved. 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 finitereservoir effects. In my talk I will also discuss how this approach may be extended to gridbased 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 Kortewegde 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 nonfunctorial 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 nonfunctorial 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 VovkShafer theory of gametheoretic (rather than measuretheoretic) 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
tangentcontinuous 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 FejesToth (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 JeanDaniel Boissonnat, David CohenSteiner 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 spacetime 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 spacetime. 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 DiaconisSturmfels 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 stabilisesas far as polynomial equations are concernedas 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 GreenTao 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 resourcesharing networks
Resourcesharing 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 toysize examples, but "nearoptimal" 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 binaryvalued 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 binaryvalued 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
binaryvalued hidden Markov processes are zero sets of determinantal
equations which draws a connection to wellstudied 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 toporelational 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 wideranging 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 KummerVandiver conjecture
The KummerVandiver 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 nonexpert 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 FokkerPlanck equation as a gradient flow in the nonlinear 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 nonpositively curved spaces.
An integral part of the classical theory of gradient flows in Hilbert spaces are the so called TrotterKato 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 nonpositively 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 AmbrosioGigliSavaré.
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 NPhard, 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 qAskey 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 Askeyscheme. 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 noninteger 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 EllipticCurve Cryptography
The first part of this talk presents results on attacking
ellipticcurve cryptography, in particular an ongoing effort
to break the Certicom challenge ECC2K130 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 massmarket quadcore 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 sidechannel 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 valuefunction as the smallest stochastic supersolution. We find also an explicit necessary and sufficient condition for optimality of a single dividendband strategy, in terms of a particular GerberShiu function. Joint work with F Avram and Z Palmowski.
February 1, 2012
Bart Vlaar
Nonsymmetric particle creation operators for the quantum nonlinear Schrodinger model
We introduce the quantum nonlinear Schr=9Adinger (QNLS) model which describes a certain quantummechanical manybody 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 quantummechanical 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 socalled Dunkltype operators, which feature in one representation, can be constructed using a second representation. This eigenfunction is nonsymmetric; by symmetrizing it one obtains the QNLS wavefunction. We present an alternative way of constructing this nonsymmetric eigenfunction, namely recursively in a QISMtype 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 gametheoretic 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
NPcomplete.
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 Tduality
"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 2form 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 Tduality 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 InsuranceReinsurance Networks
We Prose a dynamic insurance network model that allows to deal with reinsurance counterparty 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 heavytailed, 2) the contractual obligations, which are assumed to follow popular contracts in the insurance industry (such as stoploss and retrocesion), and 3) the interconnections of the insurancereinsurance 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
Noncommutative 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 infinitedimensional systems theory has become a wellestablished field within mathematics and systems theory. There are basically two approaches to infinitedimensional 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 portHamiltonian systems. For portHamiltonian systems described by ordinary differential equations this approach is very successful. PortHamiltonian 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 infinitedimensional portHamiltonian systems we derive easy verifiable conditions for wellposedness.
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 recentlydefined 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 PaleyWiener theorems
The main motivation for the study of entire functions of
exponential type is the fact that they are FourierLaplace 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 PaleyWiener theorem, which
enable us to locate the convex hull of the support of a function or a
distribution through estimates of its FourierLaplace 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 PaleyWiener 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 samplepath technique and contrast it with the
direct but often rather longwinded 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 BriÃ«t
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 NPhard 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 atmosphereocean 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, datadriven approach to
the problem, in which unresolved processes are represented by a network of Markov processes that are
conditioned on resolvedscale model variables. These Markov processes are estimated from data of e.g. highly
resolved, limitedarea simulations, hence they mimick, or emulate, in a statisticaldynamical way the
feedback from smallscale processes as simulated by a highresolution model.
March 6th 2013
Arthemy Kiselev
On the geometry of the BatalinVilkovisky Laplacian
We approach the selfregularization in a functional definition of the
BatalinVilkovisky Laplacian, which is a necessary ingredient in the
quantization of gaugeinvariant 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
byproduct we clarify the derivation of EulerLagrange 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 BVLaplacian do not hold; such identities involve higherorder
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 BVLaplacian 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
ErdosRenyi 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 ErdosRenyi 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)
