GENERAL COLLOQUIUM

PREVIOUS LECTURES:   ≥2006   2005   2004   2003   2002   2001   2000   1999   1998   1997   1996

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Lectures in 2005:
(the lectures from February-June were organized in cooperation with TU Delft)

Lectures in 2004:

Lectures in 2003:

Lectures in 2002:

 December 12 Dr. A.J. Lenstra (Universiteit van Amsterdam, EURANDOM Eindhoven): On Information bounds. Abstract: In empirical sciences, the question at hand is often not a question regarding the available data themselves, but a question regarding the (at least partially) random mechanism that, one thinks, produced these data. In such situations statistical procedures are invoked to extract an answer from the data. Information bounds are bounds for the precision with which this can be done; as such, they provide optimality criteria for statistical procedures. The best known classical information bound is the Cramér-Rao inequality. It has a history of eighty years, but the customary proofs do little to reconcile us to its truth. We present a viewpoint from which it is as obvious as the observation that in a right-angled triangle the hypotenuse is the longest side. Another famous bound, the van Trees inequality, then follows from Pythagoras' theorem. WednesdayDecember 4 Gorlaeus, C1 4:00-5:00 pm In cooperation with the Seminar on Recreational Mathematics organized by Chris Zaal Prof.dr. H.W. Lenstra (Universiteit Leiden, University of California at Berkeley, USA): Escher and the Droste effect. Abstract: Many newspapers reported recently that a team of Leiden mathematicians worked for two years on completing Escher's lithograph Print Gallery' from 1956. However, not much publicity was given to the techniques from algebraic topology and complex function theory by which this purpose was achieved. In the present lecture, both the mathematical methodology and the pictures produced by it will receive ample attention. November 14 Prof.dr. T.P. Hill (Georgia Tech, Atlanta USA): The Significant-digit Phenomenon, or Benford's Law. Abstract: A century-old empirical observation now called Benford's Law says that the significant digits of many real datasets are logarithmically distributed, rather than uniformly distributed, as might be expected. This talk will briefly survey some of the colorful history of the problem, including recent proofs and applications to fraud detection, and will then describe some very new discoveries that Benford sequences are typical in many deterministic sequences such as 1-dimensional dynamical systems and differential equations. For example, the orbit of iterates of almost every rational function obeys Benford's Law. A number of open Benford-related problems in dynamical systems, probability, number theory, and differential equations will be mentioned. October 31 Prof.dr. S.M. Verduyn Lunel (Universiteit Leiden): Calculating Hausdorff dimensions of invariant sets using spectral theory. Abstract: The dimension of an invariant set of a dynamical system is one of the most important characteristics. In this talk we present a new approach to compute the Hausdorff dimension of conformally self-similar invariant sets. The approach is based on a direct spectral analysis of the transfer operator associated with the dynamical system. In the case that the maps defining the dynamical system are analytic, our method yields a sequence of successive approximations that converge to the Hausdorff dimension of the invariant set at a super-exponential rate. This allows us to estimate the dimension very precisely. We illustrate our approach with examples from dynamical systems and from number theory via Diophantine approximations. October 10 Dr. F. Wiedijk (NIII, Katholieke Universiteit Nijmegen): Formalizing mathematics in the computer. Abstract: I will present "formal mathematics". This field studies how to verify the correctness of mathematical theories with the computer. I will discuss the reliability and the feasibility of this endeavor. At the end of the talk I will show a small formalization in the Mizar proof system. April 25 Dr. M. de Jeu (Universiteit van Amsterdam): Subspaces with equal closure Abstract: We take a unifying and radically new approach towards polynomial and trigonometric approximation in an arbitrary number of variables and in arbitrary topological vector spaces. The key idea is to show in considerable generality that a module, which is generated over the polynomials or trigonometric functions by some subset of a topological vector space, necessarily has the same closure as the module which is generated by this same subset, but now over the compactly supported smooth functions. The particular properties of the ambient topological vector space or of the generating subset are to a large degree irrelevant. This a priori translation - which goes in fact beyond modules - simplifies matters considerably and allows one, by what is now essentially a straightforward check of a few properties, to replace many classical results by more general and stronger statements of a hitherto unknown type. The method ultimately rests on harmonic analysis and can be formulated for Lie groups in general. The underlying conceptual framework then shows that many classical approximation theorems are "actually" theorems on the unitary dual of n-dimensional real space. Polynomials then correspond to the universal enveloping algebra and trigonometric functions correspond to the group algebra.

Lectures in 2001:

 November 29 Prof. dr. P. Groeneboom (Delft University of Technology, Vrije Universiteit Amsterdam): Ulam's problem and Hammersley's process Abstract: Let Ln be the length of the longest subsequence of a random permutation of the numbers 1,...,n for the uniform distribution on the set of permutations. Hammersley's interacting particle process, implicit in [4], has been used in Aldous and Diaconis [1] to provide a "soft" hydrodynamical argument for proving that limn→∞ n-1/2ELn =2. I will show that the latter result is in fact an immediate consequence of properties of a random 2-dimensional signed measure, associated with Hammersley's process. References: [1] Aldous, D. and Diaconis, P. (1995). Hammersley's interacting particle process and longest increasing subsequences. Probab. Th. Relat. Fields 103, 199-213. [2] Deift, P. (2000). Integrable systems and combinatorial theory. Notices of the AMS 47, 631-640. [3] Groeneboom, P. (2001). Ulam's problem and Hammersley's process. Annals of Probability 29, 683-690. [4] Hammersley, J.M. (1972). A few seedlings of research. In: Proc. 6th Berkeley Symp. Math. Statist. and Probability, vol. 1, 345-394. November 15 Prof. dr. F. Beukers (Universiteit Utrecht): Spectral problems in Number Theory Abstract: Consider an ordinary linear differential equation whose coefficients are polynomials in Z[x]. Consider a Taylor series solution of this equation with rational coefficients. The question for which differential equations these coefficients are arithmetically well-behaved is one that has arisen in algebraic geometry, p-adic differential equations and recently in studies on mirror-symmetry. In this lecture we deal with a particular case of differential equations. October 18 Prof. dr. R.J. Schoof (Universitá di Roma "Tor Vergata"): Class numbers of cyclotomic fields Abstract: No good algorithms are known to compute class numbers of cyclotomic fields with large conductor. In this lecture we describe an experimental approach to this problem. September 27 Dr. M. Fiocco (Universiteit Leiden): Statistics for the contact process Abstract: A d-dimensional contact process is a simplified model for the spread of an infection on the lattice Zd. At any given time t > =0, certain sites x in Zd are infected while the remaining ones are healthy. Infected sites recover at constant rate 1, while healthy sites are infected at a rate proportional to the number of infected neighboring sites. The model is parametrized by the proportionality constant $\lambda$. If $\lambda$ is sufficiently small, infection dies out (subcritical process), whereas if $\lambda$ is sufficiently large infection tends to be permanent (supercritical process). In this talk we present an estimation problem for the parameter $\lambda$ of the supercritical contact process starting with a single infected site at the origin. Based on an observation of this process at a single time t, we obtain an estimator for the parameter $\lambda$ which is consistent and asymptotically normal as t tends to infinity. May 17 Prof. dr. R.D. Gill (Univ. Utrecht): Teleportation into Quantum Statistics Abstract: Using the example of the Delft Qubit, or Schrodinger SQUID (Mooij et al, Science 1999), I sketch the basis ingredients of quantum measurement theory: states, evolutions, entanglement, and measurement. I illustrate these by the example of quantum teleportation. In order to prove that experimenters really did do, what they claim, one must repeatedly measure and estimate quantum states. Then the question arises, what is the best experimental design, and an important issue is whether joint measurement of many quantum systems together is advantageous compared to separate measurements on separate systems. I give some recent results on this, connecting to the recently discovered phenomenon (in the fields of quantum information, quantum computation) of "non-locality without entanglement". An extended abstract is available at http://www.math.uu.nl/people/gill/Preprints/abstract_gill.pdf. April 26 Dr. E. Belitser (Univ. Leiden): Exact asymptotics in nonparametric estimation Abstract: In nonparametric estimation, contrary to parametric, the quantity to be estimated is no longer a finite dimensional vector, but an infinite dimensional object - density, regression function, distribution function, image, to name a few. We discuss the basic nonparametric minimax estimation problem and consider several models (regression, white noise model, random censorship model, blurred signal) where it has been possible to derive the exact asymptotic behaviour of the minimax risk. March 29 Prof. dr. G.J. Heckman (KU Nijmegen): Hypergeometric functions and some moduli spaces Abstract: In a number of exceptional (but not irrelevant) cases a moduli space of algebro-geometric objects of some kind can be identified via a period map with a ball quotient. During the lecture three examples will be discussed: 1. 4 points on a line 2. 12 points on a line 3. pencils of cubics in the plane The first example is classical 19th century mathematics, the second example is due to Deligne and Mostow (1986), and the last example is recent joint work with E. Looijenga. February 15 Dr. W.H. Hundsdorfer (CWI, Amsterdam): Implicit and Explicit Time Stepping for Convection Problems Abstract: Numerical discretization for convection-diffusion problems with dominating convection remains one of the major problems in numerical analysis. In this talk we shall mainly consider time discretization for such problems by implicit and explicit methods and by suitable combinations. A fully implicit treatment of convection terms is often not very efficient. This is mainly due to the fact that monotonicity properties of implicit time stepping methods are in general comparable to those of their explicit counterparts (unlike stability properties). On the other hand, in spatial regions where the solution is smooth and the convective velocity is large, implicit methods seem preferable. We shall consider combined methods that are implicit only locally in space. Test results are presented for some simple two-phase flow problems.

Lectures in 2000:

 October 19 Dr. N. Bruin (Clay Mathematics Institute, MSRI, Berkeley): Solving generalised Fermat equations Abstract: In this talk we will explain a method that can be used to bound the number of rational solutions of a bivariate polynomial or, more precisely, the rational points on an algebraic curve of genus at least 2. We use this method to solve some instances of the generalised Fermat-equation. The generalised Fermat-equation is defined to be xr +ys =zt where r,s,t are integers >1. A solution in integers x,y,z is considered to be trivial if xyz=0 and primitive if gcd(x,y,z)=1. While we expect no primitive non-trivial solutions for big enough r,s,t, there are some surprisingly big solutions, such as 438 +962223 =300439072. In fact, using the method mentioned above, we can prove that for {r,s,t}={2,3,8}, this is the biggest primitive solution and we can list the others. This lecture should appeal to anyone with a mathematical background. September 14 Prof. S. Angenent (University of Wisconsin, Madison): The hole filling problem in the Porous Medium Equation Abstract: We describe the singularities of solutions to the porous medium equation which fill a concavity. We will discuss similarities with a number of other problems such as Mean Curvature Flow and the Hele-Shaw problem. May 18 Prof. J. Gonzalo (Universidad Autónoma de Madrid): Isoperimetric problems Abstract: The statement of the isoperimetric problem is this: "enclose a specified amount of volume using the least possible boundary area." Already for Euclidean space it is a nontrivial problem, where the solution is the round ball. The problem is even more interesting in Riemannian manifolds, where the shape (and the topology) of the solution changes as we change its volume. It is deeply connected with questions in Partial Differential Equations and Geometric Measure Theory. I will review the subject and explain my results concerning the change of shape in product spaces. May 4 Prof. dr. R. Tijdeman (Universiteit Leiden): Sturmian sequences and words Abstract: Sturmian sequences were introduced by Morse and Hedlund around 1940 in the context of Symbolic Dynamics. They are closely connected with the Beatty sequences which occur in Number Theory, also known as cutting sequences. The corresponding words play a role in Theoretical Computer Science because of their low complexity, in Operations Research (because of balancedness) and in Quasi-crystallography (because of being almost periodic. They form the class of nonperiodic sequences which are nearest to periodic ones. The survey provides the basic properties of Sturmian sequences, some variants and some applications. April 6 Dr. M. Zieve (Universiteit Leiden): Nice equations for nice groups Abstract: Every finite group occurs as the Galois group of a Galois extension of K(t), where K is an algebraically closed field of positive characteristic. More precisely, the Abhyankar Conjecture (proved by Raynaud and Harbater) describes the Galois groups of Galois extensions of K(t) which are only ramified at a prescribed set of points. However, little is known about Galois extensions of k(t), where k is a finite field. I will present work of Abhyankar and Elkies which yields (by two completely different methods) explicit polynomials in K(t)[x] having certain classical groups as Galois groups, and the implications of their results for the Inverse Galois Problem/Abhyankar Conjecture over k(t). March 9 Prof. dr. S.A. van de Geer (Universiteit Leiden): Entropy and estimation Abstract: Consider the problem of estimating an object, a curve or an image, in statistical terms: a parameter. If a priori little is known about the parameter, it will in general be hard to estimate it. We will quantify this by relating the speed of estimation to the entropy of parameter space. The idea is best explained for the regression problem. Suppose we recorded a signal with additive Gaussian noise, at n discrete time instants. To recover the signal, let us suppose it is smooth, say it is of bounded variation. The rate of convergence of the least squares estimator is then O(n-1/3). This follows from the calculation of the entropy of the set of all functions with total variation bounded by some constant. A somewhat less general description of the complexity of parameter space is in terms of a smoothness index. The general theory then says: suppose we know that the parameter is in a class with smoothness index s (bounded variation is s=1), then the rate of convergence is O(n-s/(2s+1)). For example, k times differentiable functions of d variables have smoothness index s=k/d. We will present some nice entropy results: the entropy of convex spaces with "few" extreme points (bounded variation being a special case), and the entropy of Besov spaces. We also consider penalized estimation, with the penalty being e.g. the total variation or a Besov norm. By taking a soft thresholding penalty, we obtain estimators that are (almost) adaptive, i.e. (apart from some logarithmic factors) they attain the rate O(n-s/(2s+1)), without knowing a priori the smoothness index s. Finally, as an illustration that the theory can be extended to other nonparametric estimation problems, we briefly consider the least absolute deviations estimator. February 17 Prof. dr. E.G.F. Thomas (Rijksuniversiteit Groningen): Path distributions Abstract: An attempt at a rigorous theory of path integrals in which summable distributions on the space Rn, regarded as a finite dimensional space of paths', forms the starting point. January 20 Prof. dr. T. Koornwinder (Univ. Amsterdam, Korteweg-de Vries instituut): On the work of 1998 Fields medal winner Richard Borcherds Abstract: The work of Richard Borcherds brings together a number of beautiful and significant mathematical structures: automorphic forms, the Leech lattice, the monster group, generalized Kac-Moody algebras, vertex algebras. The theory of the last two concepts was developed by him. In this work ideas from quantum field theory and string theory were helpful, and conversely Borcherds' results further stimulate these theories. By combining the various tools Borcherds succeeded to prove the so-called moonshine conjectures of Conway and Norton. These state that the monster group has an infinite dimensional graded representation such that the traces of elements of the monster on this representation are given by certain Hauptmoduls for some genus zero subgroups of SL(2,Z). The lecture will explain part of this work.

Lectures in 1999:

 September 16 May 6 October 14 Prof. dr. R. van der Hout (Akzo-Nobel, Universiteit Leiden): Singularities and nonuniqueness in cylindrical flow of nematic liquid crystals Abstract: This is joint work with E. Vilucchi (Roma II). A nematic liquid crystal is a simple fluid, equipped with a director field, which is a unit vector field, representing the local direction of chain molecules. Associated with this field is an energy density, and the problem of determining the steady director field in a given cylindrical flow may be formulated as a variational problem. Minimizing sequences for the total energy are in principle unbounded in an appropriate Hilberts space. In the case without flow, this unboundedness gives rise to smooth solutions (harmonic mappings in this case) with, dependent on the boundary condition, a discrete finite-energy singularity on the axis of the cylinder. The level of energy, stored in the singularity, determines the homotopy type of the solution. For a given boundary condition, there are at most two "admissible" homotopy classes, and in each class the solution is unique. In the case where flow is present, the questions whether such singularities do indeed occur, and whether solutions in a given homotopy class are unique, have been left open up till quite recently. We shall prove that the first question can be answered affirmatively, and the second question not: we give an example of three solutions in the same homotopy class. Dr. V. Berthé (Univ. Luminy, France): Sequences with low complexity function Abstract: One can classically measure the disorder'' of a sequence with finitely many values, by introducing its complexity function: this function counts the number of factors of given length. The aim of this lecture is to exhibit, through the study of sequences of low complexity function, connections between Combinatorics on words, Ergodic Theory, Arithmetic, Diophantine approximation, and geometry of tilings. We will first recall some classical properties of low complexity sequences. We will focus on a remarkable family of sequences, the so-called Sturmian sequences: these are the sequences with minimal complexity function among non-periodic sequences. We will then see how to extend these results to two-dimensional sequences. Prof. dr. H.W. Lenstra (Univ. Leiden, Univ. California, Berkeley): Exceptional polynomials Abstract: The subject of this lecture was originally suggested by the study of permutation polynomials over finite fields. We will discuss some recent results and open problems. Further, we will go into the connection with the theory of finite groups.

Lectures in 1998:

 December 10 November 12 Prof. dr. R. Dijkgraaf (Universiteit van Amsterdam): On the work of Maxim Kontsevich Abstract: This year Maxim Kontsevich was awarded the Fields Medal. I will review his work, that ranges from moduli of algebraic curves, knots and three-folds, enumerative geometry, to deformation quantization. In all of this the influence of ideas from theoretical physics has been crucial. Dr. B. Gaujal (Inria, France): Regular sequences in high dimension Abstract: In the first part of this talk we concentrate on binary sequences. We define the notion of regular binary sequences and give several characterizations of such sequences. In the second part, we present several extensions to higher dimensions and present the Fraenkel Conjecture. We also define an order of regularity that formalizes the fact that one sequence is more regular than another. Applications to optimal control problems will be mentioned in the last part of the talk. October 22 Dr. J. Melissen (Hogeschool 's-Hertogenbosch): Kepler's conjecture: close packings of round things Abstract: Last August Thomas Hales announced his proof of Kepler's conjecture. This very old conjecture (part of Hilbert's 18th problem) states that spheres in three dimensions cannot be packed more densely than the obviously densest packing. In spite of the obviousness of the result, the fact that the conjecture has rested unproven for almost 400 years just gives some indication of the magnitude of the problem. Indeed, it took about 250 pages of proof in combination with several Gigabytes of computer code and data to complete this Herculean quest. In this lecture I will put Kepler's conjecture in a historical perspective and I will treat a selection from the plethora of related interesting results from discrete geometry. Prof. D.W. Masser (Univ. Basel): Equalities and inequalities for elliptic functions Abstract: The equalities of the title are number-theoretic in nature. In 1975 Brownawell and I proved (independently) that if p(z) is a Weierstrass elliptic function with complex multiplication then under a suitable normalization there are two essentially distinct linear relations with algebraic coefficients between the periods and quasi-periods. One of these coefficients seems to be slightly unpredictable; for example, if the period quotient of p(z) is 2×(-2)½ this coefficient is equal to -6(-2)½(343 + 225×2½)/4991 when the invariants are normalized to be equal. I discuss how this and other examples were recently calculated. The inequalities of the title are analytic in nature. In 1990 Wüstholz and I proved the existence of an absolute constant C such that |p'(z)|

Lectures in 1997:

 April 17 March 20 May 1 Prof. dr. T.A. Springer (Universiteit Utrecht): Some results on complex reflection groups Abstract: A complex reflection group is a finite group of invertible linear maps of a complex vector space Cn which is generated by reflections, i.e. maps with n-1 eigenvalues 1. In the talk I shall discuss some old and new results about eigenvalues of elements of such groups. The new results are joint work with G. Lehrer. Dr. B. de Pagter (Technische Universiteit Delft): Unconditional Decompositions and their Applications Abstract: The notion of unconditional decomposition (and random unconditional decomposition) has turned out to be of importance for a number of results in analysis and operator theory. In the first part of this talk the main concepts concerning unconditional convergence and decompositions in infinite dimensional spaces will be introduced and illustrated with some classical examples. In the second part some applications will be given to certain problems in operator theory, in particular concerning Schur multipliers and commutator estimates (for matrices and operators in infinite dimensional spaces). Prof. dr. C.A.J. Klaassen (Universiteit van Amsterdam): Bonus-Malus in Quality Inspection (Not all that glitters is gold) Abstract: Our Bonus-Malus system is completely different from classical approaches to quality inspection and based on the concept of credit. The credit of a producer is defined as the total number of items accepted since the last rejection. In this Bonus-Malus system the sample size for a lot will depend on the lot size and on the credit of the producer. We will sketch the simple, basic ideas underlying this approach, which resulted from a consultancy project for the Waarborg Platina, Goud en Zilver N.V. at Gouda. Dr. R.W. van der Waall (Universiteit van Amsterdam): Modular Frobenius Groups and Homogeneous Character Induction Abstract: The representation theory of finite groups was initiated by Georg Frobenius a century ago. His so-called Frobenius groups satisfy nice properties in this respect, when working over characteristic-zero-representation theory. To give an idea, let G be a finite group with proper non-trivial subgroup H. Assume that the intersection of any two conjugates of H is trivial. Frobenius proves by means of representation theory, that (G minus the set-theoretic union of the conjugates of H) together with the unit element constitutes a non-trivial normal subgroup of G. An elementary proof of this fact is unknown. About ten to five years ago Modular Frobenius Groups were described in analogy to the classical Frobenius groups, and their investigation is still going on. It is our aim to present a survey of results thereoff to a general mathematical audience, starting around 1895 and ending in 1996. Prof. dr. A. Blokhuis (Vrije Universiteit, TU Eindhoven): Polynomials in Combinatorics and Finite Geometry Abstract: It is illustrated how elementary properties of polynomials can be used to attack extremal problems in finite and Euclidean geometry and in combinatorics.

Lectures in 1996:

 October 10 May 9 November 7 Prof. dr. M. van der Put (Rijksuniversiteit Groningen): Differential equations in characteristic 0 and characteristic p Abstract: An ordinary linear differential equation with coefficients in the field Q(x) can have Liouvillian solutions, i.e. solutions which can be expressed in rational functions by means of exponentials, algebraic equations and logarithms. It is not so easy to find those expressions. One can reduce the equation modulo a suitable rational prime p. The reduced equation is much easier to solve and this leads to a guess for the possible solutions in characteristic 0. On the background of this is a conjecture of Grothendieck. We will give some examples which will lead to some problems in elementary number theory. Prof. dr. J.J. Duistermaat (Universiteit Utrecht): Counting eigenvalues of higher order Sturm-Liouville problems Abstract: Some time ago, Boris Levit asked me about the eigenvalues of the operator d2m/dx2m on a real interval, with Neumann-type boundary conditions. It turned out that for this special case an asymptotic expansion of the eigenvalues can be given with exponentially decreasing error term. For very general Sturm-Liouville problems for higher order operators one can give asymptotic expansions of the eigenvalues which suffice to give explicit short intervals far away on the real axis, each of which contains exactly one eigenvalue (or two, for a special subclass of boundary conditions). However, the asymptotics does not give any information on the number of eignvalues which precede these far away intervals. In this talk we propose to use topological intersection theory for curves in a Grassmann manifold in order to determine this number. This may even be useful for the numerical determination of the small eigenvalues. Dr. J. Brinkhuis (Erasmus Universiteit Rotterdam): On linear optimization Abstract: In practice many optimization problems are solved by linear programming. The well-known standard way to solve these combinatorial problems is by using a combinatorial algorithm, called the simplex-method. A new algorithm is being developed which comes down to the calculation of a limit. This has become widely known through a publication of Karmarkar in 1984. Aim of this lecture is to give a geometrical treatment of the basic principles of this new algorithm. Dr. B.M.M. de Weger (Erasmus Universiteit Rotterdam, Rijksuniversiteit Leiden): Equal binomial coefficients Abstract: In the Pascal Triangle, many numbers occur several times. To a large extent this is due to trivialities, but non-trivial solutions to C(n,k)=C(m,l) (where C(a,b)=a(a-1)...(a-b+1)/b!) do exist. We will formulate a precise conjecture, and present partial results. If k and l are fixed, the equation becomes a polynomial equation, which in some cases can be solved completely. The only case known until recently was k=2, l=3. Now we can add to this the cases k=2, l=4 and k=3, l=4. The case k=2, l=4 is far from easy, and requires considerable efforts, both theoretical and computational. In contrast the case k=3, l=4 is, surprisingly, a lot easier. We will try to explain for a non-specialist audience some of the ideas that go into these proofs. Prof. dr. F. Takens (Rijksuniversiteit Groningen): Chaotic dynamics in variations of the Hénon attractor Abstract: We consider dynamical systems which are given by a map φ: P→ P - one thinks of P as the space of possible states, and of φ as the transformation transforming the present state to the next state (one unit of time later). In our examples P will be finite dimensional (mostly R2), φ differentiable and all evolutions bounded in the future in the sense that {φn(p)} n>0 is bounded (or has a compact closure) for each p∈P. Such an evolution {φn(p)} n>0 is called chaotic if it is not asymptotically stationary or (quasi)-periodic - often other, more complicated, definitions are used but that need not concern us here. We say that a dynamical system exhibits chaos if there is a set of initial values with positive (Lebesgue) measure, such that the evolutions starting in that set are all chaotic. It is not easy to give dynamical systems, say with P equal to Rn given by simple explicit maps for which one can prove that they exhibit chaos (in a `persistent way'). On the other hand there are many examples of systems which exhibit chaos, but for which we have no (simple explicit) equation and there are also dynamical systems with simple explicit maps for which numerical simulation suggests that they exhibit chaos, but for which we have no proof. One of the first (hard) examples where one has proved a simple map to exhibit chaos was the Hénon example in dimension 2: for a set of parameter values (a,b) with positive measure, (x,y)→(1 - ax2 + y, bx) exhibits chaos. In the lecture I want to explain how this example, and variations of it, can be used to show that many other explicit examples also exhibit chaos. Dr. R.W.J. Meester (Universiteit Utrecht): Ergodic theory and dynamical systems in probability Abstract: It can be very useful to study stationary stochastic (spatial) processes from the viewpoint of measure-preserving dynamical systems and ergodic theory. I will explain the connections between probability and ergodic theory, starting from scratch. This will be illustrated with one or two detailed examples arising from random walks and percolation theory. Prof. dr. S.J. van Strien (Universiteit van Amsterdam): Topology, geometry and robustness of Julia sets Abstract: For a long time it was conjectured that the Julia set of a polynomial could not have positive Lebesgue measure. Recently a counter-example to this conjecture was given. In this talk I want to discuss this and related results.