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PREVIOUS LECTURES:   2009   2008   2007   2006   ≤2005


     Lectures in 2009:

December 17
room 174
Prof. Rob de Jeu (Vrije Universiteit Amsterdam): Algebraic K-theory and arithmetic.

Abstract. The Riemann zeta-function, which encodes information about the integers and the prime numbers, has been studied extensively. Its values at 2,4,... are well-known, but much less is known about its values at 3,5,... . This difference can be explained to an extent by the different behaviour of certain groups (algebraic K-groups) of the rationals.

In this talk, we discuss some basic examples of such K-groups, and some links between them and arithmetic.

October 22
room 312
Prof. Tanja Lange (Technische Universiteit Eindhoven): Coppersmith's factorization factory.

Abstract. RSA, named after its inventors Rivest, Shamir, and Adleman, is the most widely used public-key cryptosystem. RSA bases its hardness on the observation that factoring large integers is much harder than multiplying two big numbers. In particular, there is one public parameter n in RSA which is the product of two large primes p and q. Finding p and q means breaking the system.
Factorization records for such RSA numbers (products of two big primes) show that factorization is fully feasible if p and q have only 256 bits each. This problem is called RSA-512 since n has 512 bits. For academic teams it is still too expensive to break RSA-1024, i.e. RSA where p and q have 512 bits each - but the computation is not infeasible for the current generation of computers. So agencies and sufficiently motivated criminals can have the means to break RSA for these sizes.
However, since RSA-1024 is estimated to cost a year of computation on a machine worth 100 million Euros it is commonly believed that real world attackers will not invest the effort to attack an individual user (or any other public key worth less than 100 million Euros).
In 1993 Coppersmith suggested a "factorization factory" which factors many numbers of the same size in significantly less time than handling each of them individually. This means that there can be financial gain in attacking users having less than 100 million Euros if many RSA keys are attacked simultaneously.
The currently best factorization methods are based on the Number Field Sieve (NFS). Coppersmith's method is a variant of the NFS but requires different optimizations. In particular, Coppersmith's method generates many auxiliary numbers that cannot be sieved. For these numbers the Elliptic Curve Method of factorization (ECM) is optimal. We recently sped up ECM by choosing Edwards curves instead of general elliptic curves, choosing curves with larger torsion and better linking the implementation to the processor it runs on.
In this talk we will review RSA and standard factorization methods and then explain how Coppersmith's method works. Then we will show the impacts of Coppersmith's method and faster ECM on concrete parameter choices.

October 8
room 312
Prof. Peter Jagers (Chalmers University of Technology and University of Gothenborg): Extinction: how often, how soon, and in what way?

Abstract. Branching processes were born out of the observation that extinction (of separate families or subpopulations) is ubiquitous in nature and society. This lead to Galton's and Watson's famous error, as they claimed that all family lines must die out, even in exponentially growing populations. We look back at this discussion, and proceed to exhibiting the time and path to extinction.

September 24
room 312
Dr. Wouter Kager (Vrije Universiteit, Amsterdam): Aggregation based on uniformly layered walks

Abstract. Consider the following basic aggregation model on a graph: Initially, the aggregate consists of one site (the origin). The aggregate expands by repeatedly starting a random walk in the origin and adding the first vertex outside the aggregate which is visited by the walk to the aggregate. In this colloquium I will consider such a growth model based on so-called uniformly layered walks. These are random walks on the graph which, roughly speaking, remain uniformly distributed on layers of the graph. It is to be expected that the geometry of the layers will be reflected in the asymptotic shape in the aggregation model. After a brief review of properties of a special subfamily of uniformly layered walks, I will discuss recent results on the limit shape of the aggregation model based on these walks, and present some challenging problems for future research.

June 25 Prof. Evgeny Verbitskiy (Philips Research/Rijksuniversiteit Groningen): Mahler measure and solvable models of Statistical Mechanics.

Abstract. Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that for an algebraic dynamical system its entropy is the Mahler measure of the defining polynomial. Connection between the lattice models and the algebraic dynamical systems is still rather mysterious. In a recent joint paper with K. Schmidt (Vienna), we give a first example of such correspondence: namely, an explicit equivariant encoding of a solvable model - the so-called Abelian sandpile model, onto an algebraic dynamical system of equal entropy.

Wednesday May 20
Room 312
Prof. Harry Kesten (Cornell University): A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains

May 14

room 312

Kloosterman lecture

Prof. Ted Chinburg (University of Pennsylvania): Two note number theory.

Abstract. This talk will be about number theory connected with certain kinds of music in which just two types of notes are played. The first example is an auditory version of the Droste effect which has been described in the visual arts by Lenstra, de Smit and others. The second example has to do with some forms of classical Indian music, and was discussed by M. Bhargava in Leiden a few years ago. I'll explain how results of Baker, Fel'dman and others about linear forms in logarithms of algebraic numbers lead to new results about this kind of music.

May 7 Dr. Ronald van Luijk (Universiteit Leiden): A plethora of Heron triangles.

Abstract. A Heron triangle is a triangle with integral sides and integral area. We will see that there exist arbitrarily many Heron triangles with all the same area and the same perimeter. The proof uses the arithmetic of an elliptic K3 surface. We will also see some very basic open problems about the arithmetic of K3 surfaces.

April 2 Prof. Mai Gehrke (Radboud Universiteit Nijmegen): Duality theory as a Rosetta stone.

Abstract. Modal logic was first developed by philosophers in order to sort out the relationship between possibility, implication and truth, but has since been recognized to have many applications in computer science, linguistics, and information science in general. For this reason a large body of specialized semantic tools for modal logics has been developed. Using extended Stone duality as a Rosetta stone one can see how to translate many of these tools to obtain new results in other areas. This will be illustrated with several examples including the search for semantics for substructural logics and the use of semigroup invariants in automata theory. The talk is pitched at a general mathematical audience and will not require prior knowledge of logic or duality theory.

March 26 Prof. Christian Skau (NTNU, Trondheim): Rendez-vous with Abel's and Ruffini's proof of the unsolvability of the general quintic.

Abstract. A result that has fascinated generation upon generation of mathematicians is the theorem that the general quintic can not be solved algebraically ("by radicals"). Today this result follows as a corollary of Galois theory - this beautiful subject - which requires,however, for the student to absorb and understand many abstract new concepts. Therefore, when at the end of a course in Galois theory the unsolvability of the quintic is given as a corollary, it is often experienced by the students not as a climactic ending - as it should be - simply because of exhaustion at learning this extensive theory.

Abel's original proof - with a certain input from Ruffini - is on the other hand much more direct and easily available. Besides,the proof is strikingly elegant and deserves not to get forgotten. In the talk I will sketch Abel's proof,and also stress the new ideas he thereby introduced into algebra, preparing the way for Galois' work.

March 12 Dr. Martijn de Vries (Technische Universiteit Delft): Unique expansions of real numbers in non-integer bases.

Abstract. Following a seminal paper of A. Rényi, many works were devoted to probabilistic, measure theoretical and number theoretical aspects of representations for real numbers in non-integer bases. In this talk we consider for each fixed real number q>1 the topological structure of the set Uq consisting of those real numbers x for which exactly one sequence (ci) of integers belonging to [0,q) satisfies the equality ∑i≥1 ciq-i=x. If time permits we will also give a characterization of the sets Uq, arising from a dynamical system that was recently introduced by K. Dajani and C. Kraaikamp.
The talk is based on joint work with V. Komornik.

February 12 Prof. Jing Yu (National Tsing-Hua University, Taiwan): On a Galois Theory of Several Variables.

Abstract. Half a century ago, Grothendieck had the idea of developing a Galois theory of several variables. This is supposed to be a theory of finitely generated field extensions, instead of finite extensions. One is given a finite set of numbers generating such an extension, and the aim is to find and explain all the algebraic relations among these transcendental generating numbers.
Recently, such a program has been carried out successfully in the positive characteristic world, by Anderson, Brownawell, C.-Y. Chang, Papanikolas, and myself. I shall sketch this theory, with its applications to various transcendental arithmetic invariants, and special values in positive characteristic.


     Lectures in 2008:

December 11 Prof. Sebastian van Strien (University of Warwick): On some questions of Fatou and Milnor on iterations of polynomial maps.

Abstract. This talk is about iterations of polynomials acting on the complex plane and their associated Julia, Fatou and Mandelbrot sets. I will give a survey of some recent results in this area.

November 27
On November 27 there is a special double General Colloquium session, to celebrate the appointments of Peter Grunwald and Vladas Sidoravicius as part-time Professors at our Institute per November 1 in the frame of an exchange program with the CWI.

Prof. Peter Grunwald (CWI/Universiteit Leiden): Learning when all Models Are Wrong.

Abstract. Statistical analysis of data often results in a model that is *wrong yet useful*: it is wrong in that it is a gross simplification of the process actually underlying the data; it is useful in that predictions about future data taken on the basis of the model are quite successful. For example, we often assume highly dependent variables to be independent (e.g. in speech recognition); we assume nonlinear relationships to be linear (e.g. in econometrics), and so on. Yet most existing statistical methods were designed under the assumption that one of the candidate models is actually "true". By assuming at the outset that this is not the case, we can design algorithms that are provably more robust, and that provably learn faster. Here I will give an overview of some of the remarkable properties of such algorithms:

- "when ignorance is bliss": sometimes, it is a good idea to ignore some of the data;
- there exist prediction methods which, magically, (in some sense) perform well *no matter what data are observed*;
- Bayesian methods perform remarkably well if the model is wrong, if one is solely interested in prediction - yet they can fail dramatically if one is also interested in estimation (identifying which model is "closest to being true"). We explain how this is related to convexity of probability models.

November 27
Prof. Vladas Sidoravicius (CWI/Universiteit Leiden): Markov chains with unbounded memories.

Abstract. The central topic of the talk is long time behaviour and phase transition for Markov chains with unbounded memories (infinite connections). After general introduction I will speak about recent results regarding multiplicity of limiting measures.

November 20 Dr. Steven Wepster (Universiteit Utrecht): On longitude and Tobias Mayer's lunar tables.

Abstract. During the early modern period, finding the longitude was regarded as difficult a task for a navigator as squaring a circle for a mathematician. This changed in mid-eighteenth century, when two methods of longitude finding became available. One of these methods depended on accurate knowledge of the motion of the moon, which in turn was embodied in the lunar tables of the German astronomer and mathematician Tobias Mayer (1723-1762).

After a brief history of the longitude problem, I will show that Mayer's tables depend on a hybrid of kinematics, dynamics, and relatively large-scale model fitting.

November 13 Dr. Karma Dajani (Universiteit Utrecht): Beta-expansions revisited.

Abstract. We give an overview of some of the old and new results describing the ergodic, combinatorial and arithmetic properties of algorithms generating expansions to non-integer base.

October 23 Dr. Jochen Heinloth (Universiteit van Amsterdam): Counting points on classifying spaces of bundles.

Abstract. The trick to study the geometry of varieties described by polynomial equations by counting numbers of solutions of the equations over finite fields has been used for a long time. I would like to explain some examples of spaces which are usually considered as infinite dimensional for which the same trick can be applied. In particular this can be done for classifying-spaces of bundles on Riemann surfaces. As a corollary we obtain a geometric computation of an arithmetically defined invariant, the so called Tamagawa number of a group.

October 9 Prof. Christian Maes (Katholieke Universiteit Leuven): Large deviation theory in a non-commutative setting.

Abstract. The theory of large deviations (2007 Abel Prize for S.R.S. Varadhan) deals with probabilities of rare events. It also relates to analysis in the asymptotic evaluation of certain integrals. In statistical mechanics it is fundamental to the construction of the equilibrium ensembles. For quantum systems, or more generally when the classical phase space is replaced with a non-commutative algebra, similar questions can be asked. We discuss these questions and we give some partial results.
[Joint work with Wojciech De Roeck and with Karel Netocny]

September 18 Dr. Bas Spitters (Radboud Universiteit Nijmegen): A computer-verified implementation of Riemann integration - an introduction to computer mathematics.

Abstract. (Joint work with Russell O'Connor).
The use of floating point real numbers is fast, but may cause incorrect answers due to overflows. These errors 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.

July 10 Prof. Anton Wakolbinger (Johann Wolfgang Goethe Universität, Frankfurt am Main): How often does the ratchet click?

Abstract. In an asexually reproducing population where (slightly) deleterious mutations accumulate along the individual lineages and the individual selection disadvantage is assumed to be proportional to the number of accumulated mutations, the current best class will eventually disappear from the population, a phenomenon known as Muller's ratchet. A question which is simple to ask but hard to answer is: 'How fast is the best type lost'? (or 'How many times does the ratchet click?') We highlight the underlying mathematical problem, review various diffusion approximations and discuss rigorous results in the case of a simplified model. This is joint work with Alison Etheridge and Peter Pfaffelhuber.

June 5 Prof. Shigeki Akiyama (Niigata University, Japan): On the pentagonal rotation sequence.

Abstract. Let (an) be the sequence of integers defined by the recurrence

     0 ≤ an + ωan+1 + an+2 < 1

and by the initial values a0,a1Z where ω is the golden ratio. There are several ways to prove that the sequence is periodic for all initial values. In this talk, we prove this by using the self-inducing structure of a piecewise isometry emerging from the discretized pentagonal rotation. One can also define analogues for Sturmian sequences and β-expansions in this system.

May 22 Dr. Nelly Litvak (Universiteit Twente): Power law behavior of the Google PageRank distribution.

Abstract. PageRank is a popularity measure designed by Google to rank Web pages according to their importance. It has been noticed in empirical studies that PageRank and in-degree in the Web graph follow similar power law distributions. This work is an attempt to explain this phenomenon. We model the relation between PageRank and other Web parameters through a stochastic equation inspired by the original definition of PageRank. Further, we use the theory of regular variation to prove that in our model, PageRank and in-degree follow power laws with the same exponent. The difference between these two power laws is in a multiplicative constant, which depends mainly on the settings of the PageRank algorithm. Our theoretical results are in good agreement with experimental data.

This a joint work with Yana Volkovich (UTwente) and Debora Donato (Yahoo! Research)

April 24 Prof. Maarten Jansen (Katholieke Universiteit Leuven): Multiscale analysis and estimation for data on irregular structures.

Abstract. Wavelets have proven to be a powerful tool in nonlinear approximation (data compression) and nonlinear estimation (data smoothing). The nonlinearity is essential in applications with data that are not smooth but piecewise smooth. The key motivation behind the nonlinear estimation is the fact that a wavelet transform is a multiscale (or multiresolution) analysis of the data, leading to a sparse representation. Data are well approximated by reconstruction from a few, large coefficients in this representation. This talk starts with a summary of the most essential properties and results. Next, we introduce the concepts of lifting and second generation wavelets. Lifting is both a technique for implementing wavelet transforms and a philosophy for the design of new wavelet transforms, the second generation wavelets. Whereas the `first generation wavelets' are limited to applications with equidistant observations in an n-dimensional Euclidean space, second generation wavelets (or general multiresolution analyses) can be defined on a wide variety of structures, including networks, large molecules, and so on. Giving up the equidistancy leads to new theoretical issues with respect to convergence, numerical stability and smoothness of the approximation or estimation.
We conclude with a discussion on adaptive and nonlinear lifting schemes and a few examples.

March 27 Dr. Vladas Sidoravicius (CWI, Amsterdam): Stochastic Structure of Critical Systems.

Abstract. What is in common between the Clairvoyant Demon scheduling problem, the Riemann hypothesis and quasi-isometries of large objects? All these questions can be represented and studied as critical or near-critical percolative systems. Can we handle it by stochastic methods?

March 13 Dr. Hans Maassen (Radboud Universiteit Nijmegen): Quantum measurement, purification of states and protected subspaces.

Abstract. Quantum systems under repeated or continuous observation can be considered as Markov chains on the space of quantum states. The gain of information by the observer is reflected by a tendency towards pure states on the part of the system. In certain subspaces of the Hilbert space the quantum system may be shielded off from observation. We study such spaces and relate them to the problem of protecting information against decoherence in some future quantum computing device.

February 14 Prof. Eric Opdam (Universiteit van Amsterdam): The spectrum of an affine Hecke algebra.

Abstract. Hecke algebras arise in a surprisingly wide variety of situations in algebra, geometry, number theory, and mathematical physics. An affine Hecke algebra has a natural harmonic analysis attached to it, depending on a set of continuous parameters. In recent joint work with Maarten Solleveld the spectra of the affine Hecke algebras were completely determined. We will discuss some basic aspects of these results.

January 24 Dr. Wieb Bosma (Radboud Universiteit Nijmegen): Wiskundige Verpoozingen.

Abstract. Onder de titel `Wiskundige Verpoozingen' schreef P.H. Schoute vanaf 1882 enkele jaren een rubriek in `Eigen Haard'. Mijn nieuwsgierigheid werd gewekt door twee verwijzingen hiernaar in de wiskundige literatuur. In deze voordracht zal ik aan de hand van een flink aantal plaatjes en citaten verslag doen van mijn speurtocht naar Eigen Haard, naar P.H. Schoute, en naar de context waarin deze soms verrassend moderne rubriek verscheen.

De voordracht is zeer geschikt voor studenten.


     Lectures in 2007:

December 6 Prof. Barry Koren (CWI, Amsterdam): Compressible two-fluid flow: model, method and results

Abstract. Multi-fluid flows are found in many applications: flows of air and fuel droplets in combustion chambers, flows of air and exhaust gases at engine outlets, gas and petrolea flows in pipes of oil rigs, water-air flows around ship hulls, etc. To gain better insight in the behavior of multi-fluid flows, especially two-fluid flows, numerical simulations are needed.
We assume that the fluids do not mix, but remain separated by a sharp interface. With this assumption a model is developed for unsteady, compressible two-fluid flow, with pressures and velocities that are equal on both sides of the two-fluid interface. The model describes the behavior of a numerical mixture of the two-fluids (not a physical mixture). This type of interface modeling is called interface capturing. Numerically, the interface becomes a transition layer between both fluids.
The model consists of five equations: the mass, momentum and energy equation for the mixture (the standard Euler equations), the mass equation for one of the two fluids and an energy equation for the same fluid. In the latter, a novel model for the energy exchange between both fluids is introduced. The energy-exchange model forms a source term.
The spatial discretization of the model uses a monotone, higher-order accurate finite-volume approximation, the temporal discretization a three-stage Runge-Kutta scheme. For the flux evaluation a Riemann solver is constructed. The source term is evaluated using the wave pattern found with the Riemann solver.
The two-fluid model is validated on several shock-tube problems and on two standard shock-bubble interaction problems.


November 15 Dr. Ben Kane (Radboud Universiteit Nijmegen): The triangular theorem of 8.

Abstract. The famous ``Eureka" theorem of Gauss states that every integer n can be written in the form T(x)+T(y)+T(z) for integers x, y, and z, where T(x):=x(x+1)/2 is the x-th triangular number. In this talk, we will investigate the more general question which sums of triangular numbers, that is, expressions of the form a1T(x1) +...+ arT(xr) where r, a1,..., ar are given positive integers, will indeed also generate all integers. A recent theorem of Bhargava shows that a positive definite quadratic form of a certain type will generate every positive integer if and only if it generates the integers 1,2,3,5,6,7,10,14 and 15. We shall find that an equally simple result holds for sums of triangular numbers. Indeed every positive integer is represented by a1T(x1) +...+ arT(xr) with integers x1,...,xr if and only if the integers 1,2,4,5 and 8 can be represented in this way.

November 1 Prof. Bas Edixhoven (Universiteit Leiden): How to count vectors with integral coordinates and given length in Rn.

Abstract. The question will be addressed how fast one can compute the number of ways in which an integer m can be written as a sum of n squares. At the moment the answer is not know. I will explain why I think that if n is even and m is given with its factorisation into primes, this counting can be done in time polynomial in n.log(m). The proposed method uses a generalisation of the main results of joint work with J-M. Couveignes, R. de Jong and F. Merkl on the complexity of the computation of coefficients of modular forms.

October 18 Dr. Gunther Cornelissen (Universiteit Utrecht): Listening to Riemann surfaces.

Abstract. A Dedekind zeta function doesn't always encode the isomorphism class of a number field. The dynamical Laplace operator zeta function doesn't always encode the isometry type of a manifold (e.g., a Riemann surface): you cannot hear the shape of a Riemannian manifold. We look at such problems using tools from noncommutative geometry: to a compact Riemann surface of genus at least two, I will associate a finite-dimensional noncommutative Riemannian manifold (a.k.a. spectral triple) that encodes the isomorphism class of the Riemann surface up to complex conjugation. The encoding lies in the zeta functions of the spectral triple: the spectra of various operators in the spectral triple reconstruct the Riemann surface, via application of an ergodic rigidity theorem a la Mostow. Joint work with Matilde Marcolli.

Paper available at http://www.arxiv.org/abs/0708.0500

October 4 Prof. Rob van den Berg (CWI, Amsterdam): Ponds and power laws.

Abstract. Invasion percolation is a random spatial growth model with very simple rules but surprisingly rich and complex behaviour. It was introduced around 1980 by reserachers related to the oil industry but soon drew attention from many others, including theoretical physicists and mathematicians. After defining the model, I will concentrate on an object called a 'pond', and explain that this object has indeed a natural 'hydrologic' interpretation. Although there is no special tuning of a parameter in this model, it turns out that these ponds are, in a sense which will be explained, critical. Such 'self-organized critical behaviour' seems to be quite common in nature, but this is one of the very few 'natural' models where it can be rigorously proved.

The talk is based on joint work with Yuval Peres (Berkeley and Microsoft), Vladas Sidoravicius (Rio de Janeiro; now CWI) and Eulalia Vares (Rio de Janeiro), and on joint work with Antal Jarai (Ottawa) and Balint Vagvolgyi (VU).

September 13 Dr. Ronald de Wolf (CWI, Amsterdam): Fourier analysis, hypercontractive inequalities, and quantum computing.

Abstract. Fourier analysis of real-valued functions on the Boolean hypercube has been an extremely versatile tool in theoretical computer science in the last decades. Applications include the analysis of the behavior of Boolean functions with noisy inputs, machine learning theory, design of probabilistically checkable proofs, threshold phenomena in random graphs, etc. The Bonami-Beckner hypercontractive inequality is an important result in this context. In this talk I will briefly introduce Fourier analysis of real-valued functions on the Boolean hypercube, and then prove a generalization of the Bonami-Beckner inequality for *matrix-valued* functions. Time permitting, I will also describe an application of this new inequality to a problem in quantum information theory.

The talk is based upon joint work with Avi Ben-Aroya and Oded Regev, paper available at http://arxiv.org/abs/0705.3806.

May 24

room 312

Kloosterman Lecture

Prof. Michael Bennett (University of British Columbia, Vancouver): Open Diophantine Problems.

Abstract. This talk will focus on a number of open problems from the field of Diophantine equations, and related areas. I will attempt to indicate where these questions arise, why they turn out to be so difficult, and whether modern methods can provide, if not their complete resolution, at least a certain amount of insight.

Prof. Michael Bennett was the Kloosterman Professor 2007. His research focuses on proving results for Diophantine equations by combining various theoretical and computational techniques. Prof. Bennett was visiting our institute during April and May 2007.

May 3 Dr. Cristian Giardina (Technische Universiteit Eindhoven): Hamiltonian and stochastic models for heat conduction.

Abstract. Non-equilibrium statistical mechanics aims at describing the macroscopic properties of systems which are in contacts with two thermal baths starting from simple microscopic models made of interacting particle systems. We will give an introduction to this largely open problem by considering models in one spatial dimension, i.e., chains of particles with nearest neighbor interaction. We will present both Hamiltonian and stochastic dynamics. The role of conservation law (energy, momentum,etc.) will be discussed.

April 5 Dr. Joost Batenburg (Universiteit Antwerpen): Japanese Puzzles for Experts.

Abstract. Japanese puzzles, also known as "nonograms", are a form of logical drawing. Initially, the puzzle consists of a grid of small, empty squares, along with certain logical descriptions for every row and column in the grid. The puzzler gradually fills the grid by colouring the squares, using either black or white, based on the row and column descriptions. Although Japanese puzzles have never reached the level of popularity of the more recent Sudoku puzzles, they are still highly ranked on the favourite puzzle list of many people in The Netherlands and around the world.
All of the Japanese puzzles in regular puzzle magazines can be solved by repeatedly applying a relatively small set of logical rules. All such puzzles have a unique solution. However, the general Japanese puzzle problem is NP-hard. It is possible to construct puzzles that cannot be solved using simple rules and that have many different solutions.
In this talk, I will present an approach to solving Japanese puzzles which is far more powerful compared to the simple logical rules used by most human puzzlers. The approach is based completely on logical reasoning and can be used to find, with proof, all solutions of a puzzle.

This is joint work with Walter Kosters (LIACS, Leiden).

March 8 Prof. Richard Gill (Universiteit Leiden): Perfect passion at a distance (how to win at Polish poker with quantum dice).

Abstract. I explain quantum nonlocality experiments and discuss how to optimize them. Statistical tools from missing data maximum likelihood are crucial. New results are given on Bell, GHZ, CGLMP, CH and Hardy ladder inequalities. Open problems - there are indeed many! - are discussed.
Prior knowledge of quantum theory or indeed physics is not needed to follow the talk; indeed its lack could be an advantage ;-)
It will be difficult to resist discussion of the metaphysical implications of Bell's inequality.

Slides for a previous version of this talk, and reference to an overview paper:
[the latter to appear in IMS Lecture Notes - Monographs series; volume on "Asymptotics: particles, processes and inverse problems"]


     Lectures in 2006:

December 7 Dr. Daan Crommelin (CWI): Stochastic modeling for atmospheric flows.

Abstract. When modeling or analyzing atmospheric flow, a major question is how to deal with the wide range of spatio-temporal scales that are active in the atmosphere. Stochastic methods have become increasingly important for dealing with this problem, and are used for topics such as analyzing atmospheric low-frequency variability, development of reduced models, the study of atmosphere-ocean interaction and improvement of parameterization schemes in models for weather and climate prediction. I will discuss several approaches that use stochastic methods for studying atmospheric dynamics; among them are inverse modeling for stochastic differential equations (SDEs), elimination of fast variables in stochastic systems, and the use of Hidden Markov Models.

November 16 Dr. Cor Kraaikamp (Technische Universiteit Delft): On multi-dimensional subtractive algorithms.

Abstract. Define the mapping Sd on the set of ordered d-tuples x of positive reals as follows: keep the smallest number, subtract it from the others and reorder the result in a non-decreasing way. Meester and Nowicky proved that, for d=3 and almost all x, the n-th iteration (Sd)n(x) tends to a vector different from the null-vector, as n tends to infinity. Later, Meester and K. showed that this result holds for d≥3. In this lecture, the proof of this results is outlined, and some applications and generalizations will be given.

November 2 Prof. Eduard Looijenga (Universiteit Utrecht): Invariants of the geometric and of the automorphic kind.

Abstract. One of the highlights of 19th century mathematics is the identification of the invariants of cubic forms in three variables with the classical modular forms in one variable (as algebras). This correspondence comes about by means of what we might now call the period mapping for polarized elliptic curves. After a review of this classical fact, we discuss some of its higher dimensional generalizations, among which are the recently settled cases of cubic forms in four and five variables. The last case leads us to consider a natural class of automorphic forms with poles.

September 28 Prof. Anton Bovier (Weierstraß Institut für angewandte Analysis and Stochastik, Berlin): Metastability: a potential theoretic approach.

Abstract. Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems. In this talk, I describe recent attempts towards a model-independent approach to metastability in the context of reversible Markov processes. I will present an outline of a general theory, based on careful use of potential theoretic ideas and indicate a number of concrete examples where this theory was used very successfully. I will also indicate some challenges for future work.

September 21 Dr. Frank Redig (Universiteit Leiden): Sleepy walkers and abelian sandpiles.

Abstract. I will give a short introduction to the abelian sandpile model and discuss recent results on its infinite volume limit. Next, I'll discuss applications to a model of interacting (sleepy, and sometimes activated) random walkers in which we can show a phase transition as a function of the initial density of walkers. This is an example of rigorous connection between self-organized and ordinary criticality, conjectured before by physicists.

June 15

lecture hall no. 1

Prof. Jean-Pierre Serre (Collège de France): Bounds for the orders of the finite subgroups of G(k).

Abstract. A well-known theorem of Minkowski gives a sharp multiplicative upper bound for the order of a finite subgroup of GL(n,Q). We shall see how this result can be extended to other ground fields and to other reductive groups.

June 8 Prof. Alexander Schrijver (Universiteit van Amsterdam/Centrum voor Wiskunde en Informatica): Tensors, invariants, and combinatorics.

Abstract. We give a characterization of those tensor algebras that are invariant rings of a subgroup of the unitary group. The theorem has as consequences several "First Fundamental Theorems" (in the sense of Weyl) in invariant theory.

Moreover, the theorem gives a bridge between invariant theory and combinatorics. It implies some known theorems on self-dual codes, and it gives new characterizations of graph parameters coming from mathematical physics, related to recent work with Michael Freedman and Laszlo Lovász and of Balázs Szegedy.

In the talk we give an introduction to and explanation of these results.

June 1 Prof. Jun Tomiyama (Tokyo Metropolitan University): The interplay between topological dynamics and the theory of C*-algebras.

Abstract. Contrary to a long history of the interplay between measurable dynamics (ergodic theory) and the theory of von Neumann algebras (factors), its counterpart of the interplay between topological dynamics and the theory of C*-algebras is far from mature yet, although there are many results available on the side of the C*-algebras as part of the general theory of transformation group C*-algebras.

In this talk, taking the simplest case of a dynamical systems where a single homeomorphism acts on a compact (not neccessarily metrizable) space, we discuss first how a noncommutative C*-algebra is naturally associated to such a dynamical system, and then show some aspects of the present state of knowledge of the interplay surrounding the simplicity of this associated C*-algebra.

May 11 Prof. Jürgen Klüners (Universität Kassel, Germany): On polynomial factorization.

Abstract. It is well known that the factorization of polynomials over the integers is in polynomial time. Unfortunately this algorithm was not useful in practice. Recently, Mark van Hoeij found a new factorization algorithm which works very well in practice. We present the ideas of his algorithm and extend this algorithm to an algorithm for factoring polynomials in F[t][x], where F is a finite field. Surprisingly the algorithm is much simpler and more efficient in this setting.

We prove (in the rational and the bivariate case) that the new algorithm runs theoretically in polynomial time. We will explain why the expected running times should be heuristically much better than the given worst case estimates.

April 20 Prof. Joost Hulshof (Vrije Universiteit Amsterdam): The hole-filling problem for the porous medium equation.

Abstract. The porous medium equation is a nonlinear degenerate version of the heat equation. It appears in many physical applications such as flows in porous media and thin film viscous flow. Compacly supported solutions of this equation have expanding supports. Holes in the support are filled in finite time. I will discuss radially symmetric hole-filling solutions and their stability properties under radially and non-radially symmetric perturbations.

April 6 Prof. Aad van der Vaart (Vrije Universiteit, Amsterdam): Estimating a function using a Gaussian prior on a Banach space.

Abstract: After a general introduction to nonparametric statistical estimation we discuss recent work [joint with Harry van Zanten] on Bayesian estimation using Gaussian prior distributions. As a concrete example consider estimating a probability density p using a random sample X1,..., Xn from this density (i.e. the probability that Xi falls in a set B is the area under p above B). A Bayesian approach would be to model the density x→p(x) "a-priori" as proportional to the function x→eWx for W a Gaussian process, e.g. Brownian motion, indexed by the set in which the observations take their values. The Bayesian machine (Bayes, 1764) then mechanically produces a "posterior distribution", which is a random measure on the set of probability densities, can be used to infer the "true" value of p, and anno 2006 is computable. We investigate the conditions under which this Bayesian approach gives equally good results as other methods. A benchmark is whether it works well if the unknown p is known to belong to a given regularity class, such as the functions in a Holder or Sobolev space of a given regularity. This depends of course on the Gaussian process used. It turns out to be neatly expressible in the reproducing Hilbert space of the process.

March 16 Prof. Gerard van der Geer (Universiteit van Amsterdam): The Schottky Problem.

Abstract: An algebraic curve determines an abelian variety, the Jacobian of the curve. For example, for a Riemann surface the Jacobian is a complex torus associated to the periods of integrals over the Riemann surface. Not every abelian variety is the Jacobian of a curve and the Schottky problem, due to Riemann, aks for a characterization of the Jacobians among all abelian varieties. Various answers have been proposed. We shall discuss the problem, its history and some of the proposed answers to this problem.

March 2 Dr. Onno van Gaans (Universiteit Leiden): Invariant measures for infinite dimensional stochastic differential equations.

Abstract: 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.