Lectures in 2009:
Prof. Rob de Jeu (Vrije Universiteit Amsterdam):
Algebraic K-theory and arithmetic.
The Riemann zeta-function, which encodes information about the integers and
prime numbers, has been studied extensively. Its values at 2,4,... are
well-known, but much less is known about its values at 3,5,... .
difference can be explained to an extent by the different behaviour of
groups (algebraic K-groups) of the rationals.
In this talk, we discuss some basic examples of such K-groups, and some
links between them and arithmetic.
Prof. Tanja Lange (Technische Universiteit Eindhoven):
Coppersmith's factorization factory.
RSA, named after its inventors Rivest, Shamir, and Adleman, is the
most widely used public-key cryptosystem. RSA bases its hardness on
the observation that factoring large integers is much harder than
multiplying two big numbers. In particular, there is one public
parameter n in RSA which is the product of two large primes p and
q. Finding p and q means breaking the system.
Factorization records for such RSA numbers (products of two big
primes) show that factorization is fully feasible if p and q have only
256 bits each. This problem is called RSA-512 since n has 512 bits.
For academic teams it is still too expensive to break RSA-1024,
i.e. RSA where p and q have 512 bits each - but the computation is
not infeasible for the current generation of computers. So agencies
and sufficiently motivated criminals can have the means to break RSA
for these sizes.
However, since RSA-1024 is estimated to cost a year of computation on
a machine worth 100 million Euros it is commonly believed that real
world attackers will not invest the effort to attack an individual
user (or any other public key worth less than 100 million Euros).
In 1993 Coppersmith suggested a "factorization factory" which factors
many numbers of the same size in significantly less time than handling
each of them individually. This means that there can be financial gain
in attacking users having less than 100 million Euros if many RSA keys
are attacked simultaneously.
The currently best factorization methods are based on the Number Field
Sieve (NFS). Coppersmith's method is a variant of the NFS but requires
different optimizations. In particular, Coppersmith's method generates
many auxiliary numbers that cannot be sieved. For these numbers the
Elliptic Curve Method of factorization (ECM) is optimal. We recently
sped up ECM by choosing Edwards curves instead of general elliptic
curves, choosing curves with larger torsion and better linking the
implementation to the processor it runs on.
In this talk we will review RSA and standard factorization methods and
then explain how Coppersmith's method works. Then we will show the
impacts of Coppersmith's method and faster ECM on concrete parameter
Prof. Peter Jagers (Chalmers University of Technology and University of Gothenborg):
Extinction: how often, how soon, and in what way?
Branching processes were born out of the observation that extinction (of
separate families or subpopulations) is ubiquitous in nature and society. This
lead to Galton's and Watson's famous error, as they
claimed that all family lines must die out, even in exponentially growing
populations. We look back at this discussion, and proceed to exhibiting the
time and path to extinction.
Dr. Wouter Kager (Vrije Universiteit, Amsterdam):
Aggregation based on uniformly layered walks
Consider the following basic aggregation model on a graph: Initially,
the aggregate consists of one site (the origin). The aggregate expands
by repeatedly starting a random walk in the origin and adding the first
vertex outside the aggregate which is visited by the walk to the
aggregate. In this colloquium I will consider such a growth model based
on so-called uniformly layered walks. These are random walks on the
graph which, roughly speaking, remain uniformly distributed on layers of
the graph. It is to be expected that the geometry of the layers will be
reflected in the asymptotic shape in the aggregation model. After a
brief review of properties of a special subfamily of uniformly layered
walks, I will discuss recent results on the limit shape of the
aggregation model based on these walks, and present some challenging
problems for future research.
Prof. Evgeny Verbitskiy (Philips Research/Rijksuniversiteit Groningen):
Mahler measure and solvable models of Statistical Mechanics.
Mahler measure of certain multivariate polynomials occurs frequently as the
entropy or the free energy of solvable lattice models (especially dimer models).
It is also known that for an algebraic dynamical system its entropy is the Mahler measure of the defining polynomial.
Connection between the lattice models and the algebraic dynamical systems
is still rather mysterious.
In a recent joint paper with K. Schmidt (Vienna),
we give a first example of such correspondence:
namely, an explicit equivariant encoding of a solvable model -
the so-called Abelian sandpile model,
onto an algebraic dynamical system of equal entropy.
|Wednesday May 20
Prof. Harry Kesten (Cornell University):
A problem in one-dimensional diffusion-limited aggregation (DLA)
and positive recurrence of Markov chains
Prof. Ted Chinburg (University of Pennsylvania):
Two note number theory.
This talk will be about number theory connected with certain kinds of music
in which just two types of notes are played.
The first example is an auditory version of the Droste effect which has
been described in the visual arts by Lenstra, de Smit and others.
The second example has to do with some forms of classical Indian music,
and was discussed by M. Bhargava in Leiden a few years ago.
I'll explain how results of Baker, Fel'dman and others about linear forms
in logarithms of algebraic numbers lead to new results about this kind
Dr. Ronald van Luijk (Universiteit Leiden):
A plethora of Heron triangles.
A Heron triangle is a triangle with integral sides and integral area.
We will see that there exist arbitrarily many Heron triangles with all
the same area and the same perimeter. The proof uses the arithmetic of
an elliptic K3 surface. We will also see some very basic open problems
about the arithmetic of K3 surfaces.
Prof. Mai Gehrke (Radboud Universiteit Nijmegen):
Duality theory as a Rosetta stone.
Modal logic was first developed by philosophers in order to sort
out the relationship between possibility, implication and truth, but has
since been recognized to have many applications in computer science,
linguistics, and information science in general. For this reason a large
body of specialized semantic tools for modal logics has been developed.
Using extended Stone duality as a Rosetta stone one can see how to translate
many of these tools to obtain new results in other areas. This will be
illustrated with several examples including the search for semantics for
substructural logics and the use of semigroup invariants in automata theory.
The talk is pitched at a general mathematical audience and will not require
prior knowledge of logic or duality theory.
Prof. Christian Skau (NTNU, Trondheim):
Rendez-vous with Abel's and Ruffini's proof of the unsolvability of the
A result that has fascinated generation upon generation of mathematicians is
the theorem that the general quintic can not be solved algebraically
Today this result follows as a corollary of Galois
theory - this beautiful subject - which requires,however,
for the student to
absorb and understand many abstract new concepts.
Therefore, when at the end
of a course in Galois theory the unsolvability of the quintic is given as a
it is often experienced by the students not as a climactic
ending - as it should be - simply because of exhaustion at learning this
Abel's original proof - with a certain input from Ruffini -
is on the other
hand much more direct and easily available.
Besides,the proof is strikingly
elegant and deserves not to get forgotten.
In the talk I will sketch Abel's proof,and also stress the new ideas he
thereby introduced into algebra, preparing the way for Galois' work.
Dr. Martijn de Vries (Technische Universiteit Delft):
Unique expansions of real numbers in non-integer bases.
Following a seminal paper of A. Rényi, many works were devoted to
probabilistic, measure theoretical and number theoretical aspects of
representations for real numbers in non-integer bases.
In this talk we consider for each fixed real number q>1 the topological
structure of the set
consisting of those real numbers x for which
exactly one sequence (ci) of integers
belonging to [0,q)
satisfies the equality ∑i≥1
permits we will also give a characterization of the sets
arising from a dynamical system
that was recently introduced by K. Dajani and C. Kraaikamp.
The talk is based on joint work with V. Komornik.
Prof. Jing Yu (National Tsing-Hua University, Taiwan):
On a Galois Theory of Several Variables.
Half a century ago, Grothendieck had the idea of developing a Galois
theory of several variables. This is supposed to be a theory of finitely
generated field extensions, instead of finite extensions. One is given a
finite set of numbers generating such an extension,
and the aim is to find and
explain all the algebraic relations among these transcendental generating numbers.
Recently, such a program has been carried out successfully in the positive
characteristic world, by Anderson, Brownawell, C.-Y. Chang, Papanikolas, and
myself. I shall sketch this theory, with its applications to various
transcendental arithmetic invariants, and special values in positive
Lectures in 2008:
Prof. Sebastian van Strien (University of Warwick):
On some questions of Fatou and Milnor on iterations of polynomial maps.
This talk is about iterations of polynomials acting on the complex plane
and their associated Julia, Fatou and Mandelbrot sets. I will give a survey of
some recent results in this area.
On November 27 there is a special double General Colloquium session,
to celebrate the appointments of
Peter Grunwald and Vladas Sidoravicius as part-time Professors
at our Institute per November 1 in the frame of an
exchange program with the CWI.
Prof. Peter Grunwald (CWI/Universiteit Leiden):
Learning when all Models Are Wrong.
Statistical analysis of data often results in a model that is *wrong yet
useful*: it is wrong in that it is a gross simplification of the process
actually underlying the data; it is useful in that predictions about future
data taken on the basis of the model are quite successful. For example, we
often assume highly dependent variables to be independent (e.g. in speech
recognition); we assume nonlinear relationships to be linear (e.g. in
econometrics), and so on. Yet most existing statistical methods were
designed under the assumption that one of the candidate models is actually
"true". By assuming at the outset that this is not the case, we can design
algorithms that are provably more robust, and that provably learn faster.
Here I will give an overview of some of the remarkable properties of such
- "when ignorance is bliss": sometimes, it is a good idea to ignore some of
- there exist prediction methods which, magically, (in some sense) perform
well *no matter what data are observed*;
- Bayesian methods perform remarkably well if the model is wrong, if one is
solely interested in prediction - yet they can fail dramatically if one is
also interested in estimation (identifying which model is "closest to being
true"). We explain how this is related to convexity of probability models.
Prof. Vladas Sidoravicius (CWI/Universiteit Leiden):
Markov chains with unbounded memories.
The central topic of the talk is long time behaviour and phase transition
for Markov chains with unbounded memories (infinite connections). After
general introduction I will speak about recent results regarding
multiplicity of limiting measures.
Dr. Steven Wepster (Universiteit Utrecht):
On longitude and Tobias Mayer's lunar tables.
During the early modern period, finding the longitude was regarded as
difficult a task for a navigator as squaring a circle for a mathematician.
This changed in mid-eighteenth century, when two methods of longitude
finding became available. One of these methods depended on accurate
knowledge of the motion of the moon, which in turn was embodied in the lunar
tables of the German astronomer and mathematician Tobias Mayer (1723-1762).
After a brief history of the longitude problem, I will show that Mayer's
tables depend on a hybrid of kinematics, dynamics, and relatively
large-scale model fitting.
Dr. Karma Dajani (Universiteit Utrecht):
We give an overview of some of the old and new results describing the
combinatorial and arithmetic
properties of algorithms generating expansions to non-integer base.
Dr. Jochen Heinloth (Universiteit van Amsterdam):
Counting points on classifying spaces of bundles.
The trick to study the geometry of varieties described by polynomial
equations by counting numbers of solutions of the equations over finite
fields has been used for a long time. I would like to explain some examples
of spaces which are usually considered as infinite dimensional for which the
same trick can be applied. In particular this can be done for
classifying-spaces of bundles on Riemann surfaces.
As a corollary we obtain a geometric computation of an arithmetically
defined invariant, the so called Tamagawa number of a group.
Prof. Christian Maes (Katholieke Universiteit Leuven):
Large deviation theory in a non-commutative setting.
The theory of large deviations (2007 Abel Prize for S.R.S.
Varadhan) deals with probabilities of rare events.
It also relates to analysis in the asymptotic evaluation of certain
integrals. In statistical mechanics it is fundamental to the construction
of the equilibrium ensembles. For quantum systems, or more generally when
the classical phase space is replaced with a non-commutative algebra,
similar questions can be asked. We discuss these questions and we give some
[Joint work with Wojciech De Roeck and with Karel Netocny]
Dr. Bas Spitters (Radboud Universiteit Nijmegen):
A computer-verified implementation of Riemann integration -
an introduction to computer mathematics.
(Joint work with Russell O'Connor).
The use of floating point real numbers is fast, but may cause incorrect
answers due to overflows. These errors can be avoided by hand. Better, exact
real arithmetic allows one to move this bookkeeping process entirely to the
computer allowing one to focus on the algorithms instead. For maximal
certainty, one uses a computer to check the proof of correctness of the
implementation of this algorithm. We illustrate this process by implementing
the Riemann integral in constructive mathematics based on type theory.
The implementation and its correctness proof were driven by an
algebraic/categorical treatment of the Riemann integral which is of
This work builds on O'Connor's implementation of exact real arithmetic. A demo
session will be included.
Prof. Anton Wakolbinger (Johann Wolfgang Goethe Universität, Frankfurt am Main):
How often does the ratchet click?
In an asexually reproducing population where (slightly) deleterious
mutations accumulate along the individual lineages and the individual
selection disadvantage is assumed to be proportional to the number of
accumulated mutations, the current best class will eventually disappear from
the population, a phenomenon known as Muller's ratchet. A question which is
simple to ask but hard to answer is: 'How fast is the best type lost'? (or
'How many times does the ratchet click?') We highlight the underlying
mathematical problem, review various diffusion approximations and discuss
rigorous results in the case of a simplified model. This is joint work
with Alison Etheridge and Peter Pfaffelhuber.
Prof. Shigeki Akiyama (Niigata University, Japan):
On the pentagonal rotation sequence.
be the sequence of integers defined by the recurrence
0 ≤ an +
+ an+2 < 1
and by the initial values
∈Z where ω is the
golden ratio. There are several ways to prove that the sequence is
periodic for all initial values. In this talk, we prove this by using
the self-inducing structure of a piecewise isometry emerging from
pentagonal rotation. One can also define analogues for Sturmian sequences
and β-expansions in this system.
Dr. Nelly Litvak (Universiteit Twente):
Power law behavior of the Google PageRank distribution.
PageRank is a popularity measure designed by Google to rank Web pages
according to their importance. It has been noticed in empirical studies
that PageRank and in-degree in the Web graph follow similar power law
distributions. This work is an attempt to explain this phenomenon. We
model the relation between PageRank and other Web parameters through a
stochastic equation inspired by the original definition of PageRank.
Further, we use the theory of regular variation to prove that in our
model, PageRank and in-degree follow power laws with the same exponent.
The difference between these two power laws is in a multiplicative
constant, which depends mainly on the settings of the PageRank
algorithm. Our theoretical results are in good agreement with
This a joint work with Yana Volkovich (UTwente) and Debora Donato
Prof. Maarten Jansen (Katholieke Universiteit Leuven):
Multiscale analysis and estimation for data on irregular structures.
Wavelets have proven to be a powerful tool in nonlinear approximation
(data compression) and nonlinear estimation (data smoothing). The
nonlinearity is essential in applications with data that are not smooth
but piecewise smooth. The key motivation behind the nonlinear
estimation is the fact that a wavelet transform is a multiscale (or
multiresolution) analysis of the data, leading to a sparse
representation. Data are well approximated by reconstruction from a few,
large coefficients in this representation.
This talk starts with a summary of the most essential properties and
results. Next, we introduce the concepts of lifting and second
generation wavelets. Lifting is both a technique for implementing
wavelet transforms and a philosophy for the design of new wavelet
transforms, the second generation wavelets. Whereas the `first
generation wavelets' are limited to applications with equidistant
observations in an n-dimensional Euclidean space, second generation
wavelets (or general multiresolution analyses) can be defined on a wide
variety of structures, including networks, large molecules, and so on.
Giving up the equidistancy leads to new theoretical issues with respect
to convergence, numerical stability and smoothness of the approximation
We conclude with a discussion on adaptive and nonlinear lifting schemes
and a few examples.
Dr. Vladas Sidoravicius (CWI, Amsterdam):
Stochastic Structure of Critical Systems.
What is in common between the Clairvoyant Demon scheduling
problem, the Riemann hypothesis and quasi-isometries of large objects? All
these questions can be represented and studied as critical or
near-critical percolative systems. Can we handle it by stochastic methods?
Dr. Hans Maassen (Radboud Universiteit Nijmegen):
Quantum measurement, purification of states and protected subspaces.
Quantum systems under repeated or continuous observation can be considered
as Markov chains on the space of quantum states. The gain of information
by the observer is reflected by a tendency towards pure states on the
part of the system.
In certain subspaces of the Hilbert space the quantum system may be
from observation. We study such spaces and relate them to the problem of
information against decoherence in some future quantum computing device.
Prof. Eric Opdam (Universiteit van Amsterdam):
The spectrum of an affine Hecke algebra.
Hecke algebras arise in a surprisingly wide
variety of situations in algebra, geometry, number theory,
and mathematical physics. An affine Hecke algebra has a
natural harmonic analysis attached to it, depending on a set
of continuous parameters. In recent joint work with Maarten
Solleveld the spectra of the affine Hecke algebras were
completely determined. We will discuss some basic aspects
of these results.
Dr. Wieb Bosma (Radboud Universiteit Nijmegen):
Onder de titel `Wiskundige Verpoozingen' schreef P.H. Schoute
vanaf 1882 enkele jaren een rubriek in `Eigen Haard'. Mijn
nieuwsgierigheid werd gewekt door twee verwijzingen hiernaar
in de wiskundige literatuur. In deze voordracht zal ik aan de hand
van een flink aantal plaatjes en citaten verslag doen van mijn
speurtocht naar Eigen Haard, naar P.H. Schoute, en naar de
context waarin deze soms verrassend moderne rubriek verscheen.
De voordracht is zeer geschikt voor studenten.
Lectures in 2007:
Prof. Barry Koren (CWI, Amsterdam):
Compressible two-fluid flow: model, method and results
Multi-fluid flows are found in many applications:
flows of air and fuel droplets in combustion chambers,
flows of air and exhaust gases at engine outlets,
gas and petrolea flows in pipes of oil rigs,
water-air flows around ship hulls, etc.
To gain better insight in the behavior of multi-fluid flows,
especially two-fluid flows, numerical simulations are needed.
We assume that the fluids do not mix, but remain separated
by a sharp interface. With this assumption a model is developed for unsteady,
compressible two-fluid flow, with pressures and velocities that are equal on
both sides of the two-fluid interface. The model describes the behavior of a
numerical mixture of the two-fluids (not a physical mixture). This type of
interface modeling is called interface capturing. Numerically, the interface
becomes a transition layer between both fluids.
The model consists of five equations: the mass, momentum and energy equation for
the mixture (the standard Euler equations), the mass equation for one of the
two fluids and an energy equation for the same fluid. In the latter, a novel
model for the energy exchange between both fluids is introduced.
The energy-exchange model forms a source term.
The spatial discretization of the model uses a monotone, higher-order accurate
finite-volume approximation, the temporal discretization a three-stage
For the flux evaluation a Riemann solver is constructed. The source term is
evaluated using the wave pattern found with the Riemann solver.
The two-fluid model is validated on several shock-tube problems and on
two standard shock-bubble interaction problems.
STUDENTS ARE WELCOME.
Dr. Ben Kane (Radboud Universiteit Nijmegen):
The triangular theorem of 8.
The famous ``Eureka" theorem of Gauss states that every integer n can be
written in the form T(x)+T(y)+T(z) for integers x, y, and z, where
T(x):=x(x+1)/2 is the x-th triangular number. In this talk, we
will investigate the more general question which sums of triangular
numbers, that is, expressions of the form
are given positive integers,
will indeed also generate all integers.
A recent theorem of Bhargava shows that
a positive definite quadratic form of a certain type will generate every
positive integer if and only if it generates the integers
1,2,3,5,6,7,10,14 and 15.
We shall find that an equally simple result holds for
sums of triangular numbers.
Indeed every positive integer is represented by
if and only if the integers 1,2,4,5 and 8 can be
represented in this way.
Prof. Bas Edixhoven (Universiteit Leiden):
How to count vectors with integral coordinates and given length in
The question will be addressed how fast one can compute the number of ways in
which an integer m can be written as a sum of n squares. At the moment the
answer is not know. I will explain why I think that if n is even and m is given
with its factorisation into primes, this counting can be done in time polynomial
in n.log(m). The proposed method uses a generalisation of the main results of
joint work with J-M. Couveignes, R. de Jong and F. Merkl on the complexity of
the computation of coefficients of modular forms.
Dr. Gunther Cornelissen (Universiteit Utrecht):
Listening to Riemann surfaces.
A Dedekind zeta function doesn't always encode the isomorphism class of a
The dynamical Laplace operator zeta function doesn't
encode the isometry type of a manifold (e.g., a Riemann surface):
hear the shape of a Riemannian manifold.
We look at such problems using tools from noncommutative geometry:
to a compact Riemann surface of genus at
I will associate a finite-dimensional noncommutative Riemannian
(a.k.a. spectral triple) that encodes the isomorphism class of the
up to complex conjugation.
The encoding lies in the zeta
functions of the spectral triple:
the spectra of various operators in the
spectral triple reconstruct the Riemann surface,
via application of an
ergodic rigidity theorem a la Mostow.
Joint work with Matilde Marcolli.
Paper available at
Prof. Rob van den Berg (CWI, Amsterdam):
Ponds and power laws.
Invasion percolation is a random spatial growth model with very simple
rules but surprisingly rich and complex behaviour. It was introduced around
1980 by reserachers related to the oil industry but soon drew attention from
many others, including theoretical physicists and mathematicians.
After defining the model, I will concentrate on an object called a
'pond', and explain that this object has indeed a natural 'hydrologic'
interpretation. Although there is no special tuning of a parameter in this
model, it turns out that these ponds are, in a sense which will be
explained, critical. Such 'self-organized critical behaviour' seems to be
quite common in nature, but this is one of the very few 'natural' models
where it can be rigorously proved.
The talk is based on joint work with Yuval Peres (Berkeley and
Microsoft), Vladas Sidoravicius (Rio de Janeiro; now CWI) and Eulalia Vares
(Rio de Janeiro), and on joint work with Antal Jarai (Ottawa) and Balint
||Dr. Ronald de Wolf (CWI, Amsterdam):
Fourier analysis, hypercontractive inequalities,
and quantum computing.
Fourier analysis of real-valued functions on the Boolean hypercube has been
versatile tool in theoretical computer science in the last decades.
Applications include the
analysis of the behavior of Boolean functions with noisy inputs, machine
design of probabilistically checkable proofs, threshold phenomena in random
The Bonami-Beckner hypercontractive inequality is an important result in
In this talk I will briefly introduce Fourier analysis of real-valued
functions on the Boolean
hypercube, and then prove a generalization of the Bonami-Beckner inequality
*matrix-valued* functions. Time permitting, I will also describe an
application of this
new inequality to a problem in quantum information theory.
The talk is based upon joint work with Avi Ben-Aroya and Oded Regev, paper
Prof. Michael Bennett (University of British Columbia, Vancouver):
Open Diophantine Problems.
This talk will focus on a number of open problems from the field of
Diophantine equations, and related areas.
I will attempt to indicate where these questions arise, why they turn out to be
so difficult, and whether modern methods can provide, if not their complete
resolution, at least a certain amount of insight.
Prof. Michael Bennett was the Kloosterman Professor 2007.
His research focuses on proving results
for Diophantine equations by combining various theoretical and computational
Prof. Bennett was visiting our institute during April and May 2007.
||Dr. Cristian Giardina (Technische Universiteit Eindhoven):
Hamiltonian and stochastic models for heat conduction.
Non-equilibrium statistical mechanics aims at describing the
properties of systems which are in contacts with two thermal baths
starting from simple microscopic models made of interacting particle
systems. We will give an introduction to this largely open problem by
considering models in one spatial dimension, i.e., chains of particles
with nearest neighbor interaction. We will present both Hamiltonian and
The role of conservation law (energy, momentum,etc.) will be discussed.
||Dr. Joost Batenburg (Universiteit Antwerpen):
Japanese Puzzles for Experts.
Japanese puzzles, also known as "nonograms", are a form of logical
drawing. Initially, the puzzle consists of a grid of small,
empty squares, along with certain logical descriptions for every row
and column in the grid. The puzzler gradually fills the grid by
colouring the squares, using either black or white, based on the row
and column descriptions. Although Japanese puzzles have never reached
the level of popularity of the more recent Sudoku puzzles, they are
still highly ranked on the favourite puzzle list of many people in
The Netherlands and around the world.
All of the Japanese puzzles in regular puzzle magazines can be solved
by repeatedly applying a relatively small set of logical rules. All
such puzzles have a unique solution. However, the general Japanese
puzzle problem is NP-hard. It is possible to construct puzzles that
cannot be solved using simple rules and that have many different
In this talk, I will present an approach to solving Japanese puzzles
which is far more powerful compared to the simple logical rules used
by most human puzzlers. The approach is based completely on logical
reasoning and can be used to find, with proof, all solutions of a
This is joint work with Walter Kosters (LIACS, Leiden).
||Prof. Richard Gill (Universiteit Leiden): Perfect passion at a distance
(how to win at Polish poker with quantum dice).
I explain quantum nonlocality experiments and discuss how
to optimize them. Statistical tools from missing data
maximum likelihood are crucial. New results are given on
Bell, GHZ, CGLMP, CH and Hardy ladder inequalities. Open
problems - there are indeed many! - are discussed.
Prior knowledge of quantum theory or indeed physics is not
needed to follow the talk; indeed its lack could be an
It will be difficult to resist discussion of the
metaphysical implications of Bell's inequality.
Slides for a previous version of this talk, and reference
to an overview paper:
[the latter to appear in IMS Lecture Notes - Monographs
series; volume on "Asymptotics: particles, processes and
Lectures in 2006:
||Dr. Daan Crommelin (CWI): Stochastic modeling for atmospheric flows.
When modeling or analyzing atmospheric flow, a major question is
how to deal with the wide range of spatio-temporal scales that are active
in the atmosphere.
Stochastic methods have become increasingly important for dealing with
this problem, and are used for topics such as analyzing atmospheric
low-frequency variability, development of reduced models, the study of
atmosphere-ocean interaction and improvement of parameterization schemes
in models for weather and climate prediction.
I will discuss several approaches that use stochastic methods for studying
atmospheric dynamics; among them are inverse modeling for stochastic
differential equations (SDEs), elimination of fast variables in
and the use of Hidden Markov Models.
||Dr. Cor Kraaikamp (Technische Universiteit Delft): On multi-dimensional subtractive algorithms.
Define the mapping Sd on the set of ordered d-tuples x of positive
reals as follows: keep the smallest number, subtract it from the
others and reorder the result in a non-decreasing way. Meester and
Nowicky proved that, for d=3 and almost all x, the n-th iteration
tends to a vector different from the null-vector, as n tends to infinity.
Meester and K. showed that this result holds for d≥3. In this
lecture, the proof of this results is outlined, and some applications
and generalizations will be given.
||Prof. Eduard Looijenga (Universiteit Utrecht): Invariants of the geometric and of the automorphic kind.
One of the highlights of 19th century mathematics is the
identification of the invariants of cubic forms in three variables
with the classical modular forms in one variable (as algebras).
This correspondence comes about by means of what we might now call
the period mapping for polarized elliptic curves.
After a review of this classical fact, we discuss some of its
higher dimensional generalizations, among which are the recently
settled cases of cubic forms in four and five variables.
The last case leads us to consider a natural class of
automorphic forms with poles.
||Prof. Anton Bovier (Weierstraß Institut für angewandte Analysis and Stochastik, Berlin): Metastability: a potential theoretic approach.
Abstract. Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems.
In this talk, I describe recent attempts towards a model-independent
approach to metastability in the context of reversible Markov processes.
I will present an outline of a general theory,
based on careful use of potential theoretic ideas
and indicate a number of concrete examples
where this theory was used very successfully.
I will also indicate some challenges for future work.
||Dr. Frank Redig (Universiteit Leiden): Sleepy walkers and abelian sandpiles.
I will give a short introduction to the abelian sandpile model and
discuss recent results on
its infinite volume limit. Next, I'll discuss applications
to a model of interacting (sleepy, and sometimes activated) random
walkers in which
we can show a phase transition as a function of the initial density of
walkers. This is an example of rigorous connection between
and ordinary criticality, conjectured before by physicists.
lecture hall no. 1
|Prof. Jean-Pierre Serre (Collège de France): Bounds for the orders of the finite subgroups of G(k).
A well-known theorem of Minkowski gives a sharp
multiplicative upper bound
for the order of a finite subgroup of GL(n,Q).
We shall see how this
result can be extended
to other ground fields and to other reductive groups.
||Prof. Alexander Schrijver (Universiteit van Amsterdam/Centrum voor Wiskunde en Informatica): Tensors, invariants, and combinatorics.
We give a characterization of those tensor algebras that are invariant
rings of a subgroup of the unitary group. The theorem has as consequences
several "First Fundamental Theorems" (in the sense of Weyl) in invariant
Moreover, the theorem gives a bridge between invariant theory and
combinatorics. It implies some known theorems on self-dual codes, and it
gives new characterizations of graph parameters coming from mathematical
physics, related to recent work with Michael Freedman and Laszlo Lovász
and of Balázs Szegedy.
In the talk we give an introduction to and explanation of these results.
||Prof. Jun Tomiyama (Tokyo Metropolitan University): The interplay between topological dynamics and the theory of
Contrary to a long history of the interplay between measurable dynamics
(ergodic theory) and the theory of von Neumann algebras (factors), its
counterpart of the interplay between topological dynamics and the theory of
C*-algebras is far from mature yet, although there are many results
available on the side of the C*-algebras as part of the general theory of
transformation group C*-algebras.
In this talk, taking the simplest case of a dynamical systems where a single
homeomorphism acts on a compact (not neccessarily metrizable) space, we
discuss first how a noncommutative C*-algebra is naturally associated to
such a dynamical system, and then show some aspects of the present state of
knowledge of the interplay surrounding the simplicity of this associated
||Prof. Jürgen Klüners (Universität Kassel, Germany): On polynomial factorization.
It is well known that the factorization of polynomials
over the integers is in polynomial time. Unfortunately
this algorithm was not useful in practice. Recently,
Mark van Hoeij found a new factorization algorithm
which works very well in practice. We present the
ideas of his algorithm and extend this algorithm
to an algorithm for factoring polynomials in
F[t][x], where F is a finite field. Surprisingly
the algorithm is much simpler and more efficient
in this setting.
We prove (in the rational and the bivariate case) that the
new algorithm runs theoretically in polynomial time. We will
explain why the expected running times should be heuristically
much better than the given worst case estimates.
||Prof. Joost Hulshof (Vrije Universiteit Amsterdam): The
hole-filling problem for the porous medium equation.
Abstract. The porous medium equation is a nonlinear degenerate version
the heat equation. It appears in many physical applications such as
porous media and thin film viscous flow. Compacly supported solutions
equation have expanding supports. Holes in the support are filled in
time. I will discuss radially symmetric hole-filling solutions and
properties under radially and non-radially symmetric perturbations.
||Prof. Aad van der Vaart (Vrije Universiteit, Amsterdam):
Estimating a function using a Gaussian prior on a Banach space.
After a general introduction to nonparametric statistical estimation
we discuss recent work [joint with Harry van Zanten]
on Bayesian estimation using Gaussian prior distributions.
As a concrete example consider estimating a probability density p
using a random sample X1,...,
Xn from this density
(i.e. the probability that Xi falls in a set B is the area under p above
A Bayesian approach would be to model the density x→p(x) "a-priori"
as proportional to the function x→eWx for W a Gaussian
process, e.g. Brownian motion, indexed by the set in which the
observations take their values. The Bayesian machine (Bayes, 1764)
then mechanically produces a "posterior distribution", which is
a random measure on the set of probability densities, can
be used to infer the "true" value of p, and anno 2006 is computable.
We investigate the conditions under which this Bayesian approach
gives equally good results as other methods. A benchmark is
whether it works well if the unknown p is known to belong to
a given regularity class, such as the functions in
a Holder or Sobolev space of a given regularity. This depends
of course on the Gaussian process used. It turns out to be
neatly expressible in the reproducing Hilbert space of the process.
||Prof. Gerard van der Geer (Universiteit van Amsterdam): The Schottky Problem.
An algebraic curve determines an abelian variety, the Jacobian
of the curve. For example, for a Riemann surface the Jacobian is
a complex torus associated to the periods of integrals over the
Riemann surface. Not every abelian variety is the Jacobian of a
curve and the Schottky problem, due to Riemann, aks for a characterization
of the Jacobians among all abelian varieties. Various answers have been
proposed. We shall discuss the problem, its history and some of the
proposed answers to this problem.
||Dr. Onno van Gaans (Universiteit Leiden): Invariant measures for infinite dimensional stochastic differential
If a deterministic system is perturbed by noise, it will not settle to a
steady state. Instead, there may exist invariant measures. Existence of an
invariant measure requires tightness of a solution, which is a compactness
condition. A solution of a finite dimensional stochastic differential
equation is tight if it is bounded. Boundedness is not sufficient in the
case of an infinite dimensional state space. We will discuss several
conditions on infinite dimensional stochastic differential equations that
provide existence of tight solutions and invariant measures.