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

 December 15 Prof. Philippe Clément (TU Delft, Univ. Leiden): Maximal regularity for differential equations in Banach spaces and R-boundedness. Abstract: Vector-valued extensions of the classical Lp-multiplier theorems of Marcinkiewicz ('39), Mikhlin ('56) and others have been established by Bourgain ('86) for an important class of Banach spaces (UMD). Operator-valued extensions obtained in the sixties are valid only for operators acting on a space isomorphic to a Hilbert space. Very recently Lutz Weis ('01) proved a "non- Hilbertian" operator-valued version of the Mikhlin theorem by using in an essential way the notion of an R-bounded family of operators. At the same time he was able to solve a longstanding problem concerning maximal Lp-regularity for abstract parabolic equations in Banach spaces. The aim of this talk is to introduce and discuss these results following the approach of Arendt and Bu ('02). December 8 Dr. Patsy Haccou (Institute of Biology, Univ. Leiden): Effects of deleterious mutations on the evolution of reproductive modes. Abstract: The prevalence of sexual reproduction is still one of the great mysteries of evolutionary biology, since, for all else being equal, asexual populations have a twofold fitness advantage over their sexual counterparts. Thus, whenever the two reproductive strategies compete, the elimination of the sexual mode of reproduction is expected, unless there are factors that counterbalance its disadvantages. Nevertheless, most eukaryotes reproduce sexually. Several theories have been developed to explain this. One of the proposed explanations is that under certain conditions sexual reproduction may lower the average load of deleterious mutations, thus increasing the viability of females.   This argument, as well as the lion's share of arguments used in the discussion concerning competition between sexual and asexual modes of reproduction, is, however, based on comparison of long term population growth rates. The general idea is that when an initially small population of asexually reproducing females has a positive chance to invade in a large, stable population of sexually reproducing individuals, it will eventually replace the resident population. On the basis of this reasoning it is for instance predicted that a clonal population, which has the same expected viability as a sexual diplodiploid population, but a twice as large fecundity, will eventually outcompete the sexual one. This argument, however, disregards the fact that, although the probability of such a take-over may be positive, it can still be extremely small, thus making it a very rare event indeed, even on an evolutionary scale. An additional danger of comparing population growth rates is that it does not necessarily give correct predictions concerning invasion probabilities. Finally, focusing on growth rates entails the danger that effects of initial conditions are ignored. The state of the first female that starts to reproduce asexually is determined by the distribution of female states in the resident population. Establishment chances of asexual reproduction turn out to be considerably affected by such initial conditions.   I examine the probability of succesful invasion of a resident sexual population by an asexually reproducing mutant when the viability of a female is determined by the number of deleterious mutations that she carries. November 10 Prof. Jaap Korevaar (Universiteit van Amsterdam): Tauberian Theory: History and Highlights. Abstract: Roughly following the historical development, and noting the motivation provided by number theory, we discuss various aspects of Tauberian theory. Results from the Hardy-Littlewood era (1910-1935) will lead to Wiener theory (developed in 1927-1932). Complex Tauberian theory (1905-2005) will be considered, and a bit of remainder theory (1950-2000) if time permits. Some open questions will be indicated. November 3 Prof. Frank den Hollander (Universiteit Leiden): The incipient infinite cluster for oriented percolation. Abstract: In this talk we consider oriented percolation on Zd×Z+, where (x,n) is connected to (y,n+1) for all x,y∈Zd with |x-y|≤ L and all n∈Z+. Each bond is open with probability p and closed with probability 1-p. We consider the case d> 4, L large and p=pc, the critical percolation threshold. We are interested in the probability θn that the origin (0,0) is connected to the plane at time n, i.e., to Zd×{n}. With the help of the so-called lace expansion technique, we derive a recurrence relation for θn up to quadratic order, from which we deduce that       limn→∞ n2(θn -θn+1)=C  ∈ (0,∞). This limit shows that a key critical exponent for oriented percolation assumes the mean-field value above the upper critical dimension. This fact, in turn, has a number of interesting consequences for the geometry of the incipient infinite cluster, i.e., the cluster that is about to become infinite at criticality. October 20 Dr. Robin de Jong (Universiteit Leiden): Moduli spaces of curves. Abstract: The study of complex curves is one of the central topics in algebraic geometry. We briefly describe the various viewpoints that one can take for this study - one more algebraic, one more geometric and the other more analytic. On the level of topology, complex curves are distinguished by their genus, that is, the number of holes that they have when embedded in real three dimensional space as a compact topological surface. Curves of low genus admit a very concrete description and we will discuss some examples. We will see that all curves of a fixed low genus can be naturally parametrised, and that in fact such natural parameter spaces exist for all genera. What we get is the so-called moduli space of curves. One of the surprising facts is that it itself has the structure of a (higher dimensional) analytic variety. We advocate the idea that it is very interesting to study this variety from the analytic point of view. In particular we ask if one can construct natural non-trivial non-vanishing analytic functions on this space, like for example the classical discriminant modular form in the case of moduli of elliptic curves (that is, curves of genus one). June 16 Delft Dr. Marcel de Jeu (Universiteit Leiden) Non-convex and real Paley-Wiener theorems for the Fourier transform. Abstract: Theorems which give information about the support of a function on basis of the growth rate of an integral transform of that function are commonly known as Paley-Wiener theorems. These theorems are named after the first and prototypical result by Paley and Wiener in 1934, who characterized the Fourier transforms of compactly supported L2-functions on the real line as being precisely the L2-functions on the real line which have an entire extension to the complex plane of exponential type. In the mean time, more theorems in this vein have become available, also in higher dimension, typically establishing a link between the convex hull of the support of a function in real n-space on one side and the growth rate of its transform on complex n-space on the other side. After giving a brief introduction to Fourier analysis, the classical Paley-Wiener theorems and their relevance, we will present some results in this direction which are of a different flavour. These Paley-Wiener type theorems, which are joint work with Nils Byrial Andersen, show not only that the support itself (and not just its convex hull) can be recovered from certain growth rates related to the transform, but also that one does not have to go beyond real n-space to retrieve this information. June 2 Leiden Prof. Roger Cooke (TU Delft): Model Inference for graphical models. Abstract: We develop a theory of model inference for graphical models based on the vine-copula representation of high dimensional distributions. This is usually seen as a problem of discovering conditional independences in multivariate data (as in the work of eg Terry Speed and Joe Whittaker). Vines are nested sets of trees which encode conditional bivariate information. When associated with partial correlations, they are algebraically independent and uniquely determine the correlation matrix. When associated with conditional rank correlations, they determine a sampling algorithm. Further, it is shown that the determinant equals the product of all terms (1-(ρih|K) 2) where ρih|K is the partial correlation associated with an edge in the vine. Each regular vine represents a factorization of the determinant. We seek a factorization which, when logged, dominates all others in the sense of majorization. That is, a factorization whose terms (1-(ρih|K)2) are closest to 0 or 1. This enables us to replace insignificant partial correlations with 0's. Unlike the method of Speed and Whittaker, there is no need introduce 'corrections' to restore positive definiteness. An example illustrates the procedure. May 19, Delft Dr. Jan-Hendrik Evertse (Universiteit Leiden): Recent results on linear recurrence sequences. Abstract: A (complex) linear recurrence sequence U={un}n≥0 is given by a linear recurrence     (1)  un=c1 un-1+...+ ck un-k    (n≥k) with complex coefficients c1,..., ck and complex initial values u0,..., uk-1. We give a survey of some recent results on linear recurrence sequences which have been proved using Diophantine approximation techniques. In particular, we are interested in the set of zeros of a linear recurrence sequence U, that is the set of n such that un=0. According to a classic theorem by Skolem-Mahler-Lech, the number of zeros of U is finite if U is 'non-degenerate.' Their proof was by means of p-adic analysis. In 1999, Schmidt obtained the following striking result: if U is non-degenerate, then its number of zeros is bounded above by a constant C(k) depending only on the length of the recurrence (1). Schmidt's proof was by means of some heavy machinery from Diophantine approximation. We will discuss this and some other results. May 12, Leiden Dr. Ben de Pagter (TU Delft) Representations, Boolean algebras and R-boundedness. Abstract: Relations between Hilbert spaces and general Banach spaces have been somewhat pacified by the introduction and systematic study of so-called R-bounded (=Randomized bounded) collections of operators. R-Bounded sets of bounded linear operators in Banach spaces play an increasingly important role in various branches of functional analysis, operator theory, harmonic analysis and partial differential equations. In some special situations, this notion was actually introduced in the mid-1990 s by E. Berkson and T.A. Gillespie (R-property), but was already implicit in earlier work of J. Bourgain (1983). In this talk we shall discuss the concept of R-boundedness and present some recent applications of this notion to Boolean algebras of projections in Banach spaces and to the representation theory of groups and of spaces of continuous functions on Banach spaces. April 21 Delft Prof. Bas Edixhoven (Universiteit Leiden): How fast can one compute Ramanujan's tau-function? Abstract: Ramanujan's tau-function τ(n) will be defined (very easy). I will explain why I expect that for prime numbers p, one should be able to compute τ(p) in time polynomial in log p, and I will indicate how far we are from having all details written down. I will also explain why this is of interest. April 14Leiden Prof. Arnold Heemink (Technische Universiteit Delft) Filtering algorithms for large scale systems. Abstract: Data assimilation methods are used to combine the results of a large scale numerical model with the measurement information available in order to obtain an optimal reconstruction of the dynamic behavior of the model state. Many data assimilation schemes are based on solving the Euler-Lagrange equations. A recursive algorithm to solve this two-point boundary value problem can be derived and results in the well-known Kalman filtering algorithm. This standard filter however would impose an unacceptable computational burden for large scale systems with a state dimension of more then, say, 100 000. In order to obtain a computationally efficient filter simplifications have to be introduced. Recently many new algorithms have been proposed in literature, all of the square root type: Ensemble Kalman filter (EnKF), Reduced Rank Square Root filter (RRSQRT), SSQRT, RRTKF, SEIK, POENKF, COFFEE, ... . In the presentation we will first formulate the general data assimilation problem and will discuss a number of square root filter algorithms. For a class of algorithms we will present a convergence theorem. The characteristics and performance of the methods will be illustrated with a number of real life data assimilation applications in ground water flow, reservoir engineering, ocean dynamics and air pollution. March 24 Delft Prof. Sjoerd Verduyn Lunel (Universiteit Leiden): Mixed-type functional differential equations, holomorphic factorization and applications. Abstract: In this talk we introduce and motivate the mixed-type functional differential equation     x'(t) = ax(t) + bx(t-1) + cx(t+1) defined on the real axis. Such equations arise naturally in various contexts, for example, in the study of travelling waves in discrete spatial media such as lattices. Since this mixed-type equation is not an initial value problem, it is our goal to decompose solutions of this equation as sums of "forward" solutions and "backward" solutions. We show that the set of all forward solutions defines a semigroup which can be realized by a retarded functional differential equation except for possibly finitely many modes, and similarly for the set of backward solutions as an advanced functional differential equation. Holomorphic factorizations play a crucial role in our results. Finally, we study the boundary value problem on intervals of long but finite length in the spirit of the finite section method. March 17 Leiden Dr.ir. Kees Oosterlee (Technische Universiteit Delft): Evaluation of European and American options with grid stretching and accurate discretization. Abstract: In this talk, we present several numerical issues, that we currently pursue, related to accurate approximation of option prices. Next to the numerical solution of the Black-Scholes equation by means of accurate finite differences and an analytic coordinate transformation, we present results for options under the Variance Gamma Process with a grid transformation. The techniques are evaluated for European and American options. February 17Leiden Dr. Klaas Pieter Hart (Technische Universiteit Delft): Embeddability of the measure algebra Abstract: The purpose of this talk is to compare the Boolean algebra P(N)/fin and the Measure algebra, i.e., the algebra of Borel sets of the real line modulo the ideal of Lebesgue null sets. The first algebra is one of the most widely studied by Boolean algebraists and general topologists alike. The second algebra is well-known among analysts and topologists have used its Stone space in the construction of (counter)examples. I shall discuss just one question: which of the two is embeddable into the other. It is relatively easy to see (but surprising to some) that the first is not embeddable into the second. The Measure algebra sometimes can and sometimes cannot be embedded into P(N)/fin. We shall see what this means and how one goes about proving it. February 10Delft Prof. Sara van de Geer (Leiden): Adaptive learning Abstract: Let (X,Y) be random variables, with X subset of X a feature and Y a label. We study the problem of predicting Y given X. We restrict ourselves to the case where Y takes only two values, say in {±1}. An example is the case where X represents features of a mushroom (size, shape, color) and Y indicates whether it is edible or not. A classifier is a subset G of X. It predicts the label 1 if X is in G and else the label -1. Our aim is to find a classifier that produces small prediction error. The optimal classifier is Bayes rule, which is to predict the most likely label given X. However, the distribution of (X,Y) is generally not at all known. A strategy is learned using a training set (X1,Y1) ,...,(Xn, Yn) of independent copies of (X,Y). In this talk, we will consider support vector machines (SVM's). Roughly speaking, the idea is to find the classifier which minimizes the number of errors in the training data among a certain collection of classifiers G. But this is generally computationally infeasible. As convex relaxation one instead minimizes the hinge loss (1-Y1 f(X1))++...+ (1-Yn f(Xn))+ over a collection F of real-valued functions. Denoting the minimizer by ffn, we use {ffn ≥0} as classifier. We will show that by applying a particular complexity regularization method, closely related to soft thresholding, the SVM procedure can be adaptive. For example, when some variables in X are irrelevant for the prediction, the SVM will behave as if it knew this a priori. Similarly, if the boundary of Bayes classifier is smooth, SVM will mimic this. To prove such behavior, the main effort lies in handling the random part of the problem: the so-called estimation error. Here, we use some nice probabilistic tools: symmetrization, contraction and concentration of measure.

Lectures in 2004:

 December 16 Dr. Vivi Rottschäfer (Universiteit Leiden): Blowup solutions of the Nonlinear Schrödinger equation Abstract: In my talk, I will study the cubic Nonlinear Schrödinger equation (NLS), this equation arises as a model equation in a variety of problems coming from physics, biology and chemistry. Numerical simulations show that there exist solutions of the NLS that become infinite in a finite time, hence these solutions blow up. Blowup solutions for the NLS have been studied extensively via numerical methods and asymptotic analysis including the so-called multi-bump solutions. Building on these results we proved the existence of these multi-bump blowup solutions under certain conditions. I will give an overview of the known results for these solutions starting with the numerics. December 2 Prof. Frans Oort (Universiteit Utrecht): Conjectures in Mathematics Abstract: In my talk I will discuss the stimulating aspects of conjectures in modern mathematics. Examples of questions and conjectures in number theory and algebraic geometry will be discussed in order to illustrate the concepts mentioned. I will present and propose what criteria could be given for a question to be called a conjecture. Most material will be presented in a way accessible for a general mathematical audience. Although part of my lecture is "about mathematics" there will be enough mathematical contents to satisfy those of you who also want to see proofs and facts, beautiful structures, expectations, and problems to work on. November 4 GORLAEUS, C2 4:00-5:00 pm Prof. Willem van Zwet (Universiteit Leiden): Statistics and the law: the case of nurse Lucia de B Abstract: In a hospital in The Hague a number of unexpected cases of death or re-animation of patients occurred. When it was found that in all of these cases a nurse named Lucia de B was present and caring for these patients, she was arrested and tried for multiple murder. The case for the prosecution rested on toxicological evidence as well as a statistical analysis showing that her presence in all cases could not be attributed to chance. On the basis of this Lucia was sentenced to life imprisonment.    The case was appealed by the defendant and considered by the Appellate Court (Gerechtshof) in The Hague. The defense now produced another expert witness who claimed that the statistical evidence presented earlier was unconvincing. In an interview in the newspaper NRC-Handelsblad a third expert went quite a bit farther and claimed that the statistical analysis was completely wrong and when performed correctly, should have lead to a verdict of not guilty. In the resulting confusion the court made it clear that it could not credit any of the statistical arguments anymore and confirmed the life sentence without mentioning the word statistics at all, though clearly still impressed by the unlikely presence of the defendant in all cases.    It seems that the various experts have succeeded only in confirming the well-known distrust of statistical arguments that goes back to Disraeli's dictum that there are lies, damned lies and statistics. In this talk I'll explain that if the ordinary rules for statistical consulting had been followed, it would have been quite clear what statistics could and could not contribute to this case. September 30 Prof. Ronald Cramer (CWI Amsterdam/Universiteit Leiden): Information-theoretically secure cryptography Abstract: The security of cryptographic methods often relies on assumptions about the computational hardness of certain mathematical problems. For example, the RSA cryptosystem relies on the difficulty of factoring large integers. Interestingly, there exist several meaningful tasks that can be performed reliably and privately and yet are not based on computational infeasibility.   Multi-party computation, which involves a network of collaborating processors whose goal it is to evaluate a given function on secret pre-images but without revealing the function value, can be achieved securely under the sole assumption that at most a certain quorum of the processors are under the control of a malicious adversary. As another example, consider a scenario where a sender and a receiver are connected by n independent communication channels. Even if a malicious adversary corrupts a minority of the channels and subsequently eavesdrops on their traffic and modifies it in any way he pleases, it is still possible for the sender to securely pass messages to the receiver. This means that, except with very small probability, the receiver correctly recovers the original messages, whereas the adversary has no Shannon information about these messages.   This talk addresses solutions that (nearly) minimize the required traffic on each channel. In order to securely pass an l-bits message, the data sent on each channel amounts to approximately l + k bits. The error probability is 2-k. This works by an interplay of algebraic error correcting techniques and combinatorial authentication codes. March 25 Dr. Robbert J. Fokkink (Technische Universiteit Delft): Egbert van Kampen Abstract: During his short life the Dutch mathematician Egbert van Kampen produced some results that are still of interest today, but his name is sadly missing from the mathematical history books. In an attempt to set that right, the aim of this talk is to shed some light on the life and work of one of our most prolific mathematicians, touching upon various aspects of nationalistic interest. March 11 Dr. Pieter Moree (Universiteit van Amsterdam, Max Planck Institut für Mathematik, Bonn): The hexagonal versus the square lattice Abstract: Schmutz-Schaller formulated in 1995 a conjecture concerning lattices of dimensions 2 to 8 and proved its analogue in hyperbolic geometry. As a particular case he mentioned that the hexagonal lattice ought to be 'better' than the square lattice. This statement is equivalent with the statement that for every x the number of integers n smaller or equal than x that can be written as a sum of two squares is not less than the number of integers m smaller or equal than x that can be written as a sum of a square and three times a square. Together with Herman te Riele (CWI, Amsterdam) I recently proved this by methods from computational number theory and the asymptotic theory of arithmetic functions. As a by product I disproved some claims on the divisibility of the tau-function Ramanujan made in his unpublished intriguing manuscript on the partition and tau function (two famous functions in number theory). February 19 Dr. Peter Grünwald (CWI, Amsterdam): Two Theories of Information: Shannon & Kolmogorov Abstract: We introduce, compare and contrast the theories of Shannon information and Kolmogorov complexity. We investigate the extent to which these theories have a common purpose and where they are fundamentally different. We discuss the fundamental relations 'entropy = expected Kolmogorov complexity' and 'Shannon mutual information = expected algorithmic information'. We show how 'universal coding/modeling' (a central idea in practical data compression) may be viewed as a middle ground between the two theories, and how it leads to the 'minimum description length principle', a practically useable theory for statistical inference with arbitrarily complex models.   (joint work with P.M.B. Vitányi) January 29 Dr. Wim Couwenberg (Universiteit van Amsterdam): Reflection arrangements and ball quotients. Abstract: In 1986 Deligne and Mostow showed that under certain simple conditions the moduli space of weighted n-tuples of points on P1 is a quotient of a complex ball by a discrete group of automorphisms. The covering map is described in terms of Lauricella hypergeometric functions FD on the geometric quotient (P1)n//SL(2,C). This envelops (for n=4) work by Schwarz on the Gauss hypergeometric function. In this talk I will discuss a further generalisation in terms of weights on arrangements of finite (complex) reflection groups. The Deligne-Mostow results now correspond to arrangements of type An. For certain weights we find a projective variety (just like the geometric quotient) that is again a ball quotient. Presently, only a few of those are identified as moduli spaces.

Lectures in 2003:

 December 4 Dr. Jaap Top (Rijksuniversiteit Groningen): Quartic curves over a finite field. Abstract: This lecture is about the "why" and "how" of a very elementary question: how many (or how few) rational points can a plane curve such as in the title have? October 30 Dr. Jan Hogendijk (Universiteit Utrecht): The determination of π and sin 1o in ca. 16 decimals by Al-Kashi (ca. 1420). Abstract: In the 1410s, the Iranian mathematician Al-Kashi determined π and sin 1o far more accurately than any of his (Greek and Islamic) predecessors. Al-Kashi's pi-determination is extant; his treatise on sin 1o is lost, but survives in mutilated form in a commentary by Qadi-Zadeh al-Rumi (died 1437). These two mathematicians were colleagues with a somehat complicated relationship at the court of Ulugh Beg in Samarkand. I will discuss new information on Al-Kashi's determination of π and sin 1o which can be derived from Arabic manuscripts of the two treatises that have recently become available. Some of the results are joint work with Boris A. Rosenfeld. October 9 Dr. Derk Pik (Universiteit Leiden): Minimal representations of a contractive operator as a product of two bounded linear operators. Abstract: Given a contraction K acting between Hilbert spaces U and Y we present a classification of al pairs of bounded linear operators B: U -> X and C: X -> Y (where X is an auxiliary Hilbert space) such that K = C B. September 18 Prof. Joe P. Buhler (Reed College, Portland, Oregon): A problem in symmetric functions arising from phase determination in crystallography. Abstract: The determination of the structure of a crystal by X-ray diffraction requires that the complex Fourier transform be estimated merely from data about its absolute value, together with natural physical constraints on the crystal structure. In practice, stochastic techniques are used, and these techniques are sometimes successful even when the number of atoms in the unit cell of a crystal is on the order of several thousands. Motivated by a question raised by the crystallographer Herbert Hauptman, we consider an exact algebraic version of the phase determination problem. This gives rise to a question in symmetric functions of two sets of variables whose solution leads to further interesting algorithmic and mathematical problems. These results are joint work with Zinovy Reichstein. May 22 Dr. F. Merkl (Universiteit Leiden): Statistics of Riemann zeta zeros. Abstract: The Riemann hypothesis conjectures that all nontrivial zeros of the Riemann zeta function have real part 1/2. Assuming this hypothesis, one may ask more refined questions about the empirical local distribution of these zeros. There is strong numerical and theoretical evidence - but still no full proof - that the zero distribution, appropriately rescaled, is described by the local eigenvalue distribution of certain random matrices. In the talk, I will review some rigorously known facts on the distribution of Riemann zeta zeros, proven by Montgomery, Hejhal, Rudnick, Sarnak, and others. Furthermore, I will report about recent examinations of the distribution of Riemann zeros on other scales than the "typical" distance between neighboring zeros. April 17 Prof.dr. P. Stevenhagen (Universiteit Leiden): Primes is in P Abstract: In August 2002, the Indian computer scientists Agrawal, Kayal and Saxena proved that primality of an integer can be tested by means of a deterministic algorithm that runs in polynomial time. For several decades, this had been an outstanding problem. We discuss the importance of the result in theory and in practice, and give an impression of the mathematics that goes into it. March 13 Dr. H. Matzinger (Universität Bielefeld): Some Ideas in Scenery Reconstruction Abstract: We present a few ideas used in scenery reconstruction. A scenery is a coloring of the integers, that is a map ξ from Z to a finite set {0,1,...,C-1}. Let {S(k)}=(S(0),S(1),S(2),...) be a recurrent random walk on Z starting at the origin. We will denote by X the color record obtained by observing the scenery ξ along the path of the random walk {S(k)},         X :=  (ξ(S(0)),ξ(S(1)),ξ(S(2)),... ).   The scenery reconstruction problem can be described as follows:  given an unknown scenery ξ, can we "reconstruct" ξ if we can only observe X? Does one path realization of the process {X(k)}=(X(0),X(1),...) a.s. uniquely determine ξ? Not every scenery can be reconstructed but a lot of typical sceneries can. For this we take the scenery to be random itself and show that with the right measure almost every scenery can be reconstructed up to shift and reflection. How to solve such a reconstruction problem depends a lot on the specific properties of the random walk and the distribution of the scenery. February 27 Dr. J. van den Berg (CWI, Amsterdam): Percolation and related phenomena. Abstract: Suppose we have an n x n square grid, and interpret the line segments between adjacent nodes as streets. If each street is randomly blocked with probability p (independent of the other streets), is there, when n is large, still a reasonable probability that we can cross the grid from the left to the right (or, for instance, from the south-west to the north-east corner)? It turns out that the answer is yes if p is above, and no if p is below, a so-called critical value. This is a typical example of a percolation problem. Percolation plays an important role in Statistical Physics, the theory of Interacting Particle Systems and other fields. I will show various examples and briefly discuss some modern developments. January 23 Prof.dr. S.J. Edixhoven (Universiteit Leiden): Counting solutions of systems of equations over finite fields. Abstract: I will first explain what the problem means (i.e., what is a finite field, what kind of equations do we consider), give some examples, and explain how it is related to cryptography. Then I will discuss the currently known algorithms and their complexity and limitations. Finally, I will describe my research plan for the next years concerning the problem of getting rid of at least one limitation: that the characteristic of the field should be small.

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.