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Lectures in 2013 -- 2015: (See below for 2006 -- 2012)
7 January 2016
Dror Bar-Natan (Toronto)
The Kashiwara-Vergne problem and topology

Abstract. I will describe a general machine, a close cousin of Taylor's theorem, whose inputs are topics in topology and whose outputs are problems in algebra. There are many inputs the machine can take, and many outputs it produces, but I will concentrate on just one input/output pair. When fed with a certain class of knotted 2-dimensional objects in 4-dimensional space, it outputs the Kashiwara-Vergne Problem (1978, solved Alekseev-Meinrenken 2006, elucidated Alekseev-Torossian 2008-2011), a problem about convolutions on Lie groups and Lie algebras.
More material, such as a handout.

3 December 2015
Marta Fiocco (MI and LUMC)
Marginal Structural Models to Analyse Causal Effects of Time-varying Treatments: an application to Osteosarcoma data


19 November 2015
Milton Jara
Profile cut-off for randomly perturbed dynamical systems

Abstract. Consider an ordinary differential equation with a fixed point that is a global attractor. Without loss of generality, assume that the fixed point is the origin. Under general conditions, at times goes by any solution of this equation approaches the fixed point exponentially fast. Now add a small random perturbation to this equation. It is well known that, again under very general conditions, as times goes by the solution of this stochastic equation converges to an equilibrium distribution that is well approximated by a Gaussian random variable of variance proportional to the strength of the perturbation. General theory of stochastic processes allows to show that this convergence, for each fixed perturbation, is again exponentially fast. We show that the convergence is actually abrupt: in a time windows of small size compared to the natural time scale of of the process, the distance to equilibrium drops from its maximal possible value to near zero, and only after this time window the convergence is exponentially fast. This is what is known as the cut-off phenomenon in the context of Markov chain of increasing complexity. Under a proper time scaling, we are able to prove convergence of the distance to equilibrium to a universal function, a fact known as profile cut-off.
Joint work with Gerardo Barrera.

12 November 2015
14:00 -- 16:30
De Sitterzaal
Public Symposium: Moduli Spaces and Arithmetic Geometry Celebrating FO80 at Lorentz Center

Description from the workshop web page: ``The Lorentz Workshop 'Moduli Spaces and Arithmetic Geometry' is organized to celebrate the 80th birthday of the mathematician Frans Oort, who is truly a paterfamilias of the Algebraic Geometry community in the Netherlands and beyond.

As part of this workshop, the program for the afternoon of Thursday 12 November 2015 consists of two lectures aimed at a broader audience of mathematicians. We are much honored to have Aise Johan de Jong and Don Zagier as speakers this afternoon. The afternoon is concluded with a reception. Everyone interested is cordially invited.''

more information and registration form

9 June 2015
Yakov Sinai (Princeton / Leiden Kloosterman Professor 2015):
The history of the entropy of dynamical system

Abstract. In this talk prof. dr. Yakov Sinai will present some facts connected with entropy of dynamical system discovered by Kolmogorov.The role of Hopf, Rokhlin, Gelfand and others will be also outlined.

30 April 2015
Roland van der Veen (Universiteit van Amsterdam): Knots and quantum modular forms

Abstract. The purpose of this talk is to give a non-technical introduction to mathematical aspects of topological quantum field theory in three dimensions. First I will demonstrate why knots are key to understanding three dimensional topology and how a simple knot invariant leads to a topological quantum field theory. I will then explain what it means to take the classical limit of a quantum theory and what one can expect in our case. Conjecturally one gets a wealth of geometric and algebraic objects including hyperbolic volume and Zagier's quantum modular forms. I will end with a report on partial progress towards proving such conjectures. No knowledge of physics is required to follow the talk.

23 April 2015
Rongfeng Sun (National University of Singapore): Brownian web, Brownian net, and their universality

Abstract. A central theme of modern probability theory is the identification and study of large scale stochastic fluctuations that are universal among many different models. A classic example is the Central Limit Theorem (CLT), which establishes the universality of the Gaussian distribution. A functional CLT, known as Donsker’s Invariance Principle, establishes the universality of the Brownian motion. In this talk, we introduce two such universal limits, the Brownian web and the Brownian net, which are families of Brownian motions with branching and coalescence. We will explain how they arise as universal scaling limits of many one-dimensional interacting particle systems, such as the voter model, planar aggregation models, and random walks in random environments, etc.

16 April 2015
Martijn Caspers (Münster): Noncommutative De Leeuw theorems

Abstract. Classical "De Leeuw" theorems (1965) concern the norms of Fourier multipliers: these are particular linear maps acting on the Lp-spaces of the real numbers. Recent developments, both in the theory of approximation properties and non-commutative harmonic analysis, motivate an investigation of De Leeuw his theorems for arbitrary locally compact groups. In this talk we first explain the classical theory and then consider these "non-commutative" forms of De Leeuw his theorems: restriction, lattice approximation, periodization, et cetera. This is joint work with J. Parcet, M. Perrin and E. Ricard.

14 and 15
April 2015
(all day)
Nederlands Mathematisch Congres
(Gorlaeusgebouw, Leiden; please register)
9 April 2015
Sara van der Geer (ETH Zürich / Leiden Kloosterman Professor 2014):
High-dimensional data, random questions and random answers (abstract)

19 March 2015
Richard Gill (Leiden): Einstein was wrong, probably

Abstract. In this talk I want to explain to you Bell’s inequality and Bell’s theorem. John Bell (1964) literally added a new twist — of 45 degrees, actually — to the famous EPR argument (Einstein, Podolsky, Rosen, 1935), which was supposed to show the incompleteness of quantum mechanics and vanquish Bohr. Bell’s twist apparently showed that either quantum mechanics was wrong, or is shockingly non-local. His findings have been apparently vindicated, in favour of quantum non-locality, by experiment; the most famous being that of Alain Aspect et al. (1982), who tested the so-called Bell-CHSH inequality (Clauser, Horne, Shimony and Holt’s variant).

However that experiment (and all other experiments to date actually!) have slight defects, so-called loopholes. They actually did the wrong experiment, because if they had done the right experiment, they wouldn’t have got the results they were looking for! Only now, 50 years on, are the experimenters on the threshhold of definitively proving quantum non-locality … as far as experiment ever proves anything. In particular, Delft is in the race, and there are even plans to perform the experiment with Alice and Bob (it’s always about Alice and Bob) located in Leiden and Delft.

Enter: probability and statistics. Rutherford famously once said “if you needed statistics, you did the wrong experiment”. Rutherford was wrong too …

References: a survey paper by myself on statistical issues in Bell-CHSH experiments (debated on PubPeer), and slides of two recent talks.

29 January 2015
room 174
Remco van der Hofstad (Eindhoven): The survival probability in high dimensions

Abstract. A branching process is a simple population model where individuals have a random number of children, independently of one another. Branching processes have a phase transition. Indeed, when the mean offspring is at most 1, the branching process dies out almost surely, while for mean offspring larger than one, the branching process survives with positive probability. Branching processes with mean offspring equal to one are called critical.

A classical result by Kolmogorov from 1938 states that the probability \theta_n that a critical branching process with finite-variance offspring survives to time n decays like 2/(n\gamma), where \gamma is the variance of the offspring distribution. A related result by Yaglom from 1945 states that, conditionally on survival to time n, the number of individuals alive scales like n times an exponential random variable.

In this talk, we extend such results to high-dimensional statistical physical models. Models to which our results apply are oriented percolation above 4+1 dimensions, the contact process above 4+1 dimensions, and lattice trees above 8 dimensions. We give general conditions to show that Kolmogorov's and Yaglom's theorem hold. In the case of oriented percolation, this reproves a result with den Hollander and Slade (that was proved using a 100 page lace-expansion argument), at the cost of losing explicit error estimates.

[This is joint work with Mark Holmes, building on work with Gordon Slade, Frank den Hollander and Akira Sakai.]

December 11, 2014,
Anthony Thornton (Twente): Multi-scale modelling of segregating granular flows: From inclined planes to drums; via a volcano

Abstract. Many flows in the natural environment or industry consist of shallow rapidly moving segregating granular flow (e.g. from snow avalanches, landslides, debris flows, pyroclastic flows to flows in rotating drum mixers, kilns and crushers). It is important to be able the degree of segregation in such flows as it is vital for the accurate prediction of the dynamics.

Continuum methods are able to simulate the bulk behaviour of such flows, but have to make averaging approximations reducing the degree of freedom of a huge number of particles to a handful of averaged parameters. Once these averaged parameters have been tuned via experimental or historical data, these models can be surprisingly accurate; but, a model tuned for one flow configuration often has no prediction power for another setup.

On the other hand discrete particle methods are a very powerful computational tool that allows the simulation of individual particles, by solving Newton's laws of motion for each particle. With the recent increase in computational power it is now possible to simulate flows containing a few million particles; however, for 1mm particles this would represent a flow of approximately 1 litre which is many orders of magnitude smaller than the flows found in industry or nature.

This talk will focus on a simplified example of bi-dispersed (by size and density) dry granular flows on inclined planes and in rotating drums. We will investigate this problem via the continuum modelling approach (both numerical and analytical solutions will be presented), particles simulations and conclude by discussing how both can be combined to reveal deeper insight.

December 4, 2014,
Tim van Erven (Leiden): Game-theoretic Online Learning

Abstract. Online learning deals with prediction problems in which the data are not available all at once, but instead become available sequentially. In this talk I will give an introduction to the game-theoretical model for online learning, which covers standard problems like online classification and online linear regression.

The game-theoretic approach is special, because it avoids making probabilistic assumptions about the data, and leads to very robust methods that are guaranteed to work under any circumstances. For some learning problems, however, this robustness comes at a cost: if the data do satisfy certain probabilistic assumptions or are somehow "easy", then we can learn much faster/make much fewer prediction errors. I will discuss recent adaptive methods that automatically learn faster from such easy data, but at the same time retain game-theoretic robustness for any data.

November 20, 2014,
Arnoud den Boer (Twente): Online learning in stochastic optimization problems

Abstract. Online learning refers to optimization problems where the objective function is unknown to the decision maker, but is gradually learned from accumulating data. This type of problems has many applications in online settings, such as dynamic pricing, online advertisement, and recommendation systems. A decision policy that is frequently used in practice is to use, at each decision moment, the decision that seems optimal given the available data. Perhaps surprisingly, this simple policy may lead to detrimental results. In this talk I will explain some of the intuition behind this phenomenon, discuss alternative policies and characterize their performance, and point to challenging open problems.

November 13, 2014,
Daniel J. Bernstein (University of Illinois at Chicago, TU Eindhoven): Hyper-and-elliptic-curve cryptography

Abstract. Elliptic curves have revolutionized real-world cryptography. For example, when modern web browsers talk to modern HTTPS web servers, they encrypt data using "ECDHE" elliptic-curve keys that are thrown away after the connection ends. The situation before elliptic-curve cryptography was that messages were encrypted using RSA keys; those keys were painfully slow to generate, and consequently were reused for a long time; attackers stealing keys from servers were able to retroactively decrypt earlier connections.
Modern research has produced simpler, faster addition algorithms for elliptic curves; even better speeds from certain surfaces; and, most recently, top speeds from viewing certain groups simultaneously as curves and as surfaces. This talk will introduce these fast groups to a general mathematical audience.

October 30, 2014,
Peter Bruin (Leiden): Heights in arithmetic and geometry

Abstract. Let x be a rational number. The height of x, denoted by H(x), is a measure of the ‘arithmetic complexity’ of x. If x = a/b with a, b coprime integers, then H(x) is defined as max{|a|, |b|}. There are various interesting and useful extensions of this notion; for example, the rational numbers can be replaced by any number field, and x can be replaced by a point on an algebraic variety. Using the machinery of heights, one can quantify ‘how many’ solutions a Diophantine equation has. I will explain this in some detail in the case of elliptic curves.

October 16, 2014,
Johannes Schmidt-Hieber (Leiden): Non-parametric Le Cam theory: Overview and recent results

Abstract. Suppose we are interested in estimation of a parameter in a given statistical model. Under various circumstances there exists a simpler model that contains the same information about the unknown parameter. This concept of sufficiency is useful for the theoretical understanding of the estimation problem and leads to a more unified view on statistics. For non-parametric problems, that is, if we consider estimation of complex objects such as functions, the notion of sufficiency becomes too strong. In the late '90s, it was shown for two non-parametric models, that sufficiency holds in an asymptotic sense as introduced earlier by Le Cam. Since then, few other results of this type could be established. In the first part of the talk we review the development of this field. In the second part, we explain a recent result on asymptotic equivalence for regression under dependent noise.

September 25, 2014,
Martin Bright (Leiden): Diophantine equations: geometry and the local-global approach

Abstract. A Diophantine equation is a polynomial equation to which integer or rational solutions are sought. In general, understanding the solutions of a Diophantine equation is a very difficult problem. During the 20th century, two important themes arose in the theory of Diophantine equations. The first was a realisation that the geometry of the set defined by the equation has a strong influence on its arithmetic. The second was the idea that, to understand integer or rational (“global”) solutions to an equation, one should begin by looking at its real or p-adic (“local”) solutions. I will give examples of successes and failures of these approaches, concentrating on curves and surfaces.

May 15, 2014,
Walter van Suijlekom (Nijmegen): Inner perturbations in noncommutative geometry

Abstract. Starting with an algebra, we define a semigroup which extends the group of invertible elements in that algebra. As we will explain, this semigroup describes inner perturbations of noncommutative manifolds, and has applications to gauge theories in physics. We will present some elementary examples of the semigroup associated to matrix algebras, and to (smooth) functions on a manifold. Joint work with Ali Chamseddine and Alain Connes.

May 8, 2014,
Jason Frank (Utrecht): Application of Thermostats to Discretized Fluid Models

Abstract. In meteorology and climate science, fluid models are simulated on time scales very long compared to the characteristic Lyapunov time of chaotic growth, with the goal of generating a data set suitable for statistical analysis. The choice of a numerical discretization scheme for a problem carries with it a certain bias in the statistical data generated in long simulations. In this talk I will discuss recent research on the use of thermostat techniques, commonly used in molecular dynamics, to control the invariant measure of a discretized model, with the goals of correcting bias or effecting a statistically consistent model reduction. These will be illustrated for a point-vortex gas and a Burgers/KdV equation. I will also discuss challenges to extending the approach to the Euler equations.

April 16 and 17, 2014,
No colloquium. NMC in Delft.
April 3, 2014,
Gianne Derks (Surrey): Existence and stability of stationary fronts in inhomogeneous wave equations

Abstract. Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Homogeneous nonlinear wave equations are Hamiltonian partial differential equations with the homogeneity providing an extra symmetry in the form of the spatial translations. Inhomogeneities break the translational symmetry, though the Hamiltonian structure is still present. When the spatial translational symmetry is broken, travelling waves are no longer natural solutions. Instead, the travelling waves tend to interact with the inhomogeneity and get trapped, reflected, or slowed down.

In this talk, we will discuss the existence and stability of stationary fronts in the inhomogeneous wave equations. After a short overview of the theory for homogeneous wave equations, we introduce the inhomogeneous wave equations, illustrate how a phase plane analysis gives the existence of curves of stationary solutions and present a stability criterion for the stationary fronts. To illustrate how these results can be used to find families of stationary fronts and their stability, we consider two models involving the inhomogeneous sine-Gordon equation. One model involves long Josephson junctions with imperfections. The other model describes aspects of the DNA/RNAP interaction in DNA copying.

March 27, 2014,
Sergey Shadrin (Universiteit van Amsterdam): Intersection theory behind the spectral curve

Abstract. I'll make a survey on the possible applications of the method of the so-called spectral curve topological recursion. This is a method developed by physicists in order to replace matrix models they used in various theories by some mathematically rigorous constructions.

Surprisingly, in its rigorous mathematical formulation the spectral curve topological recursion produces in a universal way solutions to a huge number of combinatorial and geometric problems, and in particular it allows to relate all of this problems to geometry of the moduli spaces of curves.
My main goal will be to show a number of simple but still very surprising examples, in order to illustrate how this theory works.

March 13, 2014,
Francesco Caravenna (University of Milano-Bicocca): Scaling and Universality in Probability

Abstract. The notion of "scaling limit" refers to a situation in which a family of discrete models converges (in a suitable sense) to a continuum model. The interest in such a limiting object relies on its "universality", i.e., typically it does not depend on the fine details of the discrete models from which it arises. Scaling limits and universality are key topics in probability theory and in statistical physics. In this talk, I will present a selection of results, both classical and modern, that convey the main ideas and give the flavor of the topic.

February 27, 2014, Snellius 312
Daniele Sepe (Utrecht): Integral affine geometry: from fundamentals to applications and back

Abstract. The study of symmetries lies at the heart of geometry in all its guises; often the motivation to study such symmetries come from 'real life' problems arising in physics or chemistry amongst others. This talks concentrates on integral affine geometry, which studies some fundamental, simple to define, yet still mysterious objects, which can be thought of as being related to crystals. After a basic introduction outlining the driving questions in the area which date back to the '60s, I shall illustrate how integral affine geometry arises in other fields of mathematics and how these applications can (hopefully!) bring new techniques to solve outstanding problems.

December 19, 2013,
Snellius 174(!)
Ieke Moerdijk (Nijmegen): Topology after Kan

Abstract. There are (at least) three Dutch mathematicians who had a lasting influence on topology. Everybody knows how Brouwer's fixed point theorem led to homology, and how Freudenthal's suspension theorem made stable homotopy theory possible. In this lecture, I will sketch some of the main contributions of Daan Kan, explain the notions of Kan complex and of Kan extension, and sketch how they currently affect the daily life of everybody working in algebraic topology, including my own.

November 14, 2013,
Snellius 312
Marco Streng (Leiden): Elliptic Curves: complex multiplication, application, and generalization

Abstract. Some lattices inside the plane are mapped into themselves by a combination of scaling and rotating. This is called complex multiplication, and turns out to have (through analysis, geometry, and algebra) applications in both pure number theory, and daily life.

November 7, 2013,
Frank den Hollander (Leiden): Extremal geometry for a Brownian porous medium

Abstract. pdf

Wed May 8, 2013,
Snellius 174
Xing Chaoping (Nanyang Technological University): Algebraic curves over finite fields and applications

Kloosterman lecture.

Abstract. For a long time, the study of algebraic geometry belonged to the realm of pure mathematicians. After a series of three papers in the period 1977 - 1982, Goppa found fascinating applications of algebraic curves over finite fields, and especially of those with many rational points, to coding theory. This created much stronger interest in the area and attracted new groups of researchers such as coding theorists, cryptographers, theoretical computer scientists and algorithmically inclined mathematicians. In this talk, we first give a brief survey on algebraic curves over finite fields with many rational points and then present three applications.

April 18, 2013
Snellius 412
Jan Bouwe van den Berg (VU Amsterdam): Braids in dynamics

Abstract. Pieces of string or curves in three dimensional space may be knotted or braided. This physical idea can be used as a topological tool to study certain types of dynamical systems. In particular, such an approach leads to forcing theorems in the spirit of the famous "period three implies chaos" for interval maps. We discuss applications to ordinary and partial differential equations. Finally, we illustrate how the topological arguments can be combined with a computer-assisted approach to obtain insight in an evolution equation from the field of pattern formation.

April 11, 2013
Snellius 312
Eric Cator (Nijmegen): A randomly forced Burgers equation on the real line

Abstract. In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used. This is joined work Yuri Bakhtin and Konstantin Khanin.

March 28, 2013
Snellius 312.
Vitaly Bergelson (Ohio State University): Ergodic Theorems Along Polynomials and Combinatorics

Abstract. We will start with discussing various recurrence and convergence results which demonstrate that dynamical systems exhibit surprisingly regular behavior along polynomials times. Besides being of interest in their own right, these polynomial results have strong applications in combinatorics and number theory. We will discuss some of these applications including various polynomial generalizations of Szemeredi's theorem on arithmetic progressions. We will conclude by formulating and discussing some open problems and conjectures. The talk is intended for a general audience.

     Lectures in 2012:

December 6, 2012
Snellius 174.
Frank Vallentin (Delft/CWI): Upper bounds for geometric packing problems

Abstract. How densely can one pack given objects into a given container? Problems of this sort, generally called packing problems, are fundamental problems in geometric optimization. In this talk I present a method based on semidefinite optimization and harmonic analysis which can be used to compute upper bounds for the optimal density. I will show how to apply it to a variety of situations: packing spherical caps on the unit sphere, packing spheres (of potentially different radii) into Euclidean space, packing translates of regular tetrahedra into Euclidean space.

November 22, 2012
Snellius 174.
Aad van der Vaart (Leiden): On Bayesian curve fitting

Abstract. We illustrate the "nonparametric Bayesian paradigm" by some practice and some theory for the problem of finding a curve y=f(x) that "best" fits a given set of (noisy) points (x_i,y_i). After starting from just curve fitting by smoothing splines we add probability to the description of the problem in two steps, and show the mathematical theory that this enables, with reference to Bayes, Laplace, Gauss and Fisher.

November 8, 2012
Snellius 312.
Hermen Jan Hupkes (Leiden): Differential equations with delayed and advanced terms: where, why & how

Abstract. We discuss the modelling motivation for using MFDEs (functional differential equations of mixed type, also known as delay-advanced differential equations) and illustrate their distinguishing mathematical features.

May 24
Snellius B1.
Filmvertoning: Late style - Yuri I. Manin Looking Back on a Life in Mathematics

Studievereniging De Leidsche Flesch zal de onlangs verschenen biografische documentaire "Late Style - Yuri I. Manin Looking Back on a Life in Mathematics" vertonen gemaakt door Agnes Handwerk en Harrie Willems. Late Style vertelt het verhaal over Yuri Manin tijdens de gouden jaren van de wiskunde in Moskou tijdens de jaren zestig en zeventig - een periode die niet alleen in het teken van de wiskunde stond, maar zeker ook beïnvloed werd door de plitieke situatie in Rusland. Voorafgaand aan de filmvertoning zal Frans Oort een introductie geven.

Programma: 15:45 koffie en thee, 16:00 inleiding Frans Oort, 16:30 vertoning "Late Style"

May 10, 2012
Snellius 174.
Sander Dahmen (Utrecht): Solving Diophantine equations: the modular method

Abstract. Since the proof of FLT, many Diophantine problems have been solved using deep results about elliptic curves, modular forms, and associated Galois representations. The purpose of this talk is to discuss some of these results and explain how they can be applied to explicitly solve certain Diophantine equations. We shall focus in particular on so-called generalized superelliptic equations, i.e. exponential Diophantine equations of the form F(x,y)=z^n where F is a binary form over the integers (to be solved in integers x,y,z,n with n>1 and x and y coprime).

April 26, 2012
Snellius 174.
Roeland Merks (CWI & Leiden): Modeling stochastic self-organization of multicellular tissues: on the growth of blood vessels and glands

Abstract. Morphogenesis, the formation of biological shape and pattern during embryonic development, is a topic of intensive experimental investigation, so the participating cell types and molecular signals continue to be characterized in great detail. Yet this only partly tells biologists how molecules and cells interact dynamically to construct a biological tissue. Mathematical and computational modeling are a great help in answering such questions on biological morphogenesis. Cell-based simulation models of blood vessel growth describe the behavior of cells and the signals they produce. They then simulate the collective behavior emerging from these cell-cell interactions. In this way cell-based models help analyze how cells assemble into biological structures, and reveal the microenvironment the cells produce collectively feeds back on individual cell behavior. In this way, our simulation models, based on a Cellular Potts model combined with partial-differential equations, have shown that the elongated shape of cells is key to correct spatiotemporal in silico replication of vascular network growth. The models have also helped identify a new stochastic mechanism for the formation of branched structures in epithelial gland tissues. I will discuss some recent insights into these mechanisms. Then I will discuss our more recent cell-based modeling studies of cell-extracellular matrix interactions during angiogenesis. I will conclude by suggesting some interesting continuum and stochastic mathematical problems that our cell-based simulations suggest.

April 19, 2012
Snellius 174.
Anthony Wickstead (Queen's University Belfast): The Riesz Decomposition Property for some spaces of real-valued functions

Abstract. An ordered space V has the Riesz Separation Property (RSP) if f_1, f_2, h_1, h_2 \in V and f_1, f_2 \leq h_1, h_2 implies there is a g in V with f_1, f_2 \leq g \leq h_1, h_2. Many, but not all, interesting vector spaces of functions have the RSP even though they do not possess the stronger property of being a vector lattice. The talk will survey results on this topic due to H.H. Schaefer, L. Fuchs and A. Nagel & W. Rudin.

March 22, 2012
Snellius 174.
André Henriques (Utrecht): What is an elliptic object?

Abstract. Elliptic cohomology (also called "topological modular forms" of "TMF") is a cohomology theory that was constructed in the 90ties by homotopy theoretical means. Several strong indicators make people believe that there exist geometric objects that represent elliptic cohomology classes. However, despite multiple attempts by many people, nobody has managed to define those elusive "elliptic objects".

February 23, 2012
Snellius 174
John F. Bukowski (Juniata College & Leiden): The Diverse Interests of Christiaan Huygens: Mechanics and Music

Abstract. Christiaan Huygens contributed to the early history of the problem of the hanging chain when he proved at age 17 that the chain did not take the shape of a parabola. We will examine his proof in detail. Huygens was also one of many seventeenth-century mathematicians interested in the tuning of the musical scale. We will see how he used logarithms to divide the octave into a 31-tone scale, and we will compare his tuning to other tunings of the scale.

     Lectures in 2011:

February 24, 2011
Oort building
Abel in Holland

For details and registration click here

March 3, 2011
Jaap Top (Groningen) Schoute's discriminants

Abstract. On Saturday, May 27, 1893 the Groningen geometer P.H. Schoute (1846-1913) presented three string models of algebraic surfaces during the monthly meeting in Amsterdam of the Royal Netherlands Academy of Arts and Sciences (KNAW). In spite of their trip to the Trippenhuis, these models have survived to this day and can still be admired in the mathematics institute of the university of Groningen. In the talk we discuss what these models show, why Schoute designed them, and we explain some of the beautiful mathematical properties of the corresponding surfaces.

April 21, 2011
Kloosterman lecture

Jean-Louis Colliot-Thélène (CNRS and Université Paris-Sud, Orsay)

From sums of squares in fields to motivic cohomology and higher class field theory

Abstract. L. Euler (1770) proved that any positive rational number is a sum of four squares of rational numbers. E. Artin (1927) proved that any rational function in $n$ variables with real coefficients, if positive on ${\bf R}^n$, is a sum of squares of such rational functions (Hilbert's 17th problem), and A. Pfister (1967) proved that it may then be written as a sum of at most $2^n$ squares.

Artin also showed that positive rational functions in $n$ variables with {\it rational} coefficients are sums of squares of such functions. That they may be represented by a bounded number of squares, more precisely $2^{n+2}$, was predicted in 1991. This relied on two hypotheses, both of which are now known, one by work of V. Voevodsky, the other one by work of U. Jannsen.

I shall go through the history of sums of squares in fields. I shall then try to give a glimpse of the various tools employed in the proof of the $2^{n+2}$-result : the algebraic theory of quadratic forms (as started by E. Witt), Milnor K-Theory, Galois cohomology, motivic cohomology and higher class field theory.

Slides [PDF ]

May 12, 2011
Peter Spreij (Amsterdam) Block Hankel confluent Vandermonde matrices

Abstract. Vandermonde matrices are well-known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, q × q say, in which case the i-th column is given by u(zi), where we write u(z) = (1,z,,zq-1). If all the z i (i = 1,,q) are different, the Vandermonde matrix is non-singular, otherwise not. The latter case obviously takes place when all zi are the same, z say, in which case one could speak of a confluent Vandermonde matrix. Non-singularity is obtained if one considers the matrix V (z) whose i-th column is given by the (i - 1)-th derivative u(i-1)(z).

We will consider generalizations of the confluent Vandermonde matrix V (z) by considering matrices obtained by using as building blocks the q × r matrices M(z) = u(z)w(z), with u(z) as above and w(z) = (1,z,,zr-1), together with its derivatives M(k)(z). Specifically, we will look at the matrix whose ij-th block is given by M(i+j)(z). This in general non-square matrix exhibits a block-Hankel structure. We will answer a number of elementary questions for this matrix. What is the rank? What is the null-space? Can the latter be parametrized in a simple way? Does it depend on z? What are left or right inverses? It turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure. The answers depend on the size of the matrix M(z) and the number of derivatives M(k)(z) that is involved. The talk will be completely elementary, no specific knowledge of the theory of matrix polynomials is needed.

October 13, 2011
Snellius 174
Mark Peletier (Eindhoven) Gradient Flows, Optimal Transport, and Fresh Bread

Abstract. In 1997, Jordan, Kinderlehrer, and Otto pioneered a new way of looking at age-old equations for diffusion, thus giving an exact mathematical description of the sense in which `diffusion is driven by entropy'. This sense revolves around the concept of optimal transport.

Introduced by Monge in 1781, this theory focuses on optimal ways to transport given quantities from A to B. Its development took off after Kantorovich improved the formulation in 1942, and in recent years the theory has exploded, with applications in differential geometry, probability theory, functional analysis, analysis on non-smooth spaces, and many more.

In this talk I will revisit the original connection between the diffusive problems on one hand and the theory of optimal transport on the other. I will show how the two are connected, discuss many consequences of this, and describe recent insight into the deeper meaning of this connection.

December 1, 2011
Snellius 174
Bert Zwart (CWI Amsterdam) Fluid and diffusion limits of Bandwidth Sharing Networks

Abstract. Bandwidth sharing networks are popular as a mathematical model of communication networks such as the Internet. In its simplest form, a bandwidth sharing network is a multidimensional Markov chain with transition rates specified by the solution of a family of convex optimization problems, making the analysis of the dynamics of such networks challenging. We consider such networks in a regime where the total number of users and resources grows large, and obtain fluid and diffusion limits. In many cases, the invariant points of these limit processes can be computed in polynomial time. We also highligth some cases in which the accumulation points satisfy a system of functional equations for which we are unable to establish uniqueness.

December 8, 2011
Snellius 174
Tanja Eisner (Amsterdam) Arithmetic progressions via ergodic theory

Abstract. We sketch the development from van der Waerden's theorem on arithmetic progressions to the recent Green-Tao theorem and show how methods from ergodic theory have been decisive in this field.

     Lectures in 2010:

December 16, 2010
room 312
Prof. Michel Dekking (TU Delft): The mathematics of iterated paperfolding

Abstract. We will present some old and some new results in a project which has been going on for 35 years. There are bits of algebra (representation theory), bits of topology (The Jordan curve theorem, Hausdorff metric), bits of graph theory (Euler's 1736 theorem), bits of number theory (Gaussian primes, Löschian numbers), bits of theoretical computer science (automatic sequences), bits of measure theory (Hausdorff dimension), and I might sneak in some probability.

December 2, 2010
room 174
Dr. Bas Lemmens (University of Kent, UK): Hilbert geometries and symmetric cones

Abstract. In a letter to Klein, Hilbert introduced a collection of metric spaces which naturally generalize Klein's model of the hyperbolic plane. These metric spaces are usually called Hilbert geometries and play a role in Hilbert's 4th problem. In this talk I will focus on several open problems concerning their group of isometries, which were posed by P. de la Harpe in the nineteen-eighties. I will try to convince you that the solution must have something to do with symmetric cones (a concept which I will explain in the lecture) and provide some supporting evidence.

November 4, 2010
room 174
Dr. Jan Draisma (TU Eindhoven): Finiteness results in statistics using algebra

Abstract. I will give three interrelated examples of how polynomial algebra can be used to settle finiteness questions arising from statistics. I will assume no prior familiarity with any of these, and I will emphasise the fundamental algebraic tools that go into the proofs.

The first example is the by now classical Diaconis-Sturmfels algorithm for sampling from contingency tables with prescribed marginals, where algebra proves the existence of a finite Markov basis. The second concerns recent work by Hillar and Sullivant, where such Markov bases are shown to stabilise as some of the sizes of the contingency tables tend to infinity. The third is a proof that Gaussian factor analysis with a fixed number of (latent) factors stabilises---as far as polynomial equations are concerned---as the number of observed variables tends to infinity.

October 14, 2010
room 174
Dr. Ben Moonen (Universiteit van Amsterdam): Galois theory and enumerative geometry

Abstract. From Galois theory we know that for a polynomial of degree at least five there is in general no ’formula’ (using only radicals) for its zeros. Grothendieck taught us that Galois theory (algebra) is really the same as the theory of fundamental groups (geometry). In my talk I will discuss some geometric analogues of the problem of solving equations. We consider situations where there is a finite number of special objects (e.g., lines on a cubic surface, flexes of a plane curve, Weierstrass points on a curve, ...) and where we can ask if we can actually give these by formulas using only radicals.

The talk is expressly aimed at non-experts.

September 23, 2010
room 174
Prof.dr. Karen Aardal (TU Delft and CWI): Cutting planes in integer optimization

Abstract. The use of cutting planes to solve integer optimization problems dates back to the computational success in 1954 of Dantzig, Fulkerson, and Johnson, who solved an, at that time, large traveling salesman instance by problem-specific cutting planes, and to the theoretical work of Gomory in 1958. Gomory developed a pure cutting plane algorithm to solve general integer optimization problems. Until very recently, cutting planes have in practice been derived from one single constraint, either from one of the original constraints, or from a surrogate constraint. Recent research focus on the generation of cutting planes from multiple constraints. We discuss some of these new results and the challenge of generating multiple-row constraints that are computationally effective.

June 17, 2010
room 174
Prof.dr. Nicolai Reshetikhin (UC Berkeley / UvA): Statistical mechanics in the thermodynamic limit: deterministic limit shapes and fluctuations

Abstract. The talk will be focused on how deterministic limit shapes emerge in statistical mechanics in the thermodynamic limit. In tiling and dimer models this phenomenon is known as Arctic circle formation. We will also discuss how the partition function and fluctuations depend on boundary conditions in the thermodynamic limit.

May 27, 2010
room 174
Dr.ir. M.C. Veraar (Delft University of Technology): Regularity for stochastic PDEs

Abstract. In this talk we explain how to formulate a partial di erential equation with noise in a functional analytic way. Moreover, we show how one can apply functional analytic tools to derive sharp regularity properties of solutions to such equations. Some of the mathematical tools one needs for this are operator theory, harmonic analysis, probability theory and evolution equations.

April 29, 2010
room 174
Prof. Arjen Doelman (Universiteit Leiden): Pulses in singularly perturbed reaction-diffusion equations

Abstract. Localized structures such as pulses and fronts are the building blocks of the dynamics generated by reaction-diffusion equations (in one spatial dimension). A significant part of our mathematical insights in the dynamics and interactions of (multi-)pulse patterns is based on the analysis of singularly perturbed reaction-diffusion models of Gray-Scott and/or Gierer-Meinhardt (GS/GM) type. We will briefly review these insights and focus on phenomena exhibited by these models that are problem-specific, i.e. that are caused by the special nature of GS/GM-type models. This will be used as a motivation for the introduction of a general class of systems that significantly extends the GS/GM models. The first steps towards understanding the existence, stability and dynamics of pulses in this class of models will be taken in a schematic fashion. The emphasis will be on the differences with known results in the GS/GM context.

     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.