General Mathematics Colloquium

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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