
Lectures in 2012:
December 6, 2012 Snellius 174.

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

November 22, 2012 Snellius 174.

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

November 8, 2012 Snellius 312.

Hermen Jan Hupkes (Leiden):
Differential equations with delayed and advanced terms: where, why & how
Abstract.
We discuss the modelling motivation for using MFDEs (functional
differential equations of mixed type, also known as
delayadvanced differential equations) and
illustrate their distinguishing mathematical
features.

May 24 Snellius B1.

Filmvertoning:
Late style  Yuri I. Manin Looking Back on a Life in Mathematics
Studievereniging De Leidsche Flesch zal de onlangs verschenen
biografische documentaire "Late Style  Yuri I. Manin Looking Back on a
Life in Mathematics" vertonen gemaakt door Agnes Handwerk en Harrie
Willems. Late Style vertelt het verhaal over Yuri Manin tijdens de
gouden jaren van de wiskunde in Moskou tijdens de jaren zestig en
zeventig  een periode die niet alleen in het teken van de wiskunde
stond, maar zeker ook beïnvloed werd door de plitieke situatie in
Rusland. Voorafgaand aan de filmvertoning zal Frans Oort een introductie
geven.
Programma: 15:45 koffie en thee, 16:00 inleiding Frans Oort, 16:30 vertoning "Late Style"

May 10, 2012 Snellius 174.

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

April 26, 2012 Snellius 174.

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

April 19, 2012 Snellius 174.

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

March 22, 2012 Snellius 174.

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

February 23, 2012 Snellius 174

John F. Bukowski (Juniata College & Leiden):
The Diverse Interests of Christiaan Huygens: Mechanics and Music
Abstract.
Christiaan Huygens contributed to the early history of the problem of
the hanging chain when he proved at age 17 that the chain did not take
the shape of a parabola. We will examine his proof in detail. Huygens
was also one of many seventeenthcentury mathematicians interested in
the tuning of the musical scale. We will see how he used logarithms to
divide the octave into a 31tone scale, and we will compare his tuning
to other tunings of the scale.



Lectures in 2011:
February 24, 2011 Oort building

Abel in Holland
For details and registration click here

March 3, 2011 Snellius

Jaap Top (Groningen)
Schoute's discriminants
Abstract.
On Saturday, May 27, 1893 the Groningen geometer P.H.
Schoute (18461913)
presented three string models of algebraic surfaces during the monthly
meeting in Amsterdam of
the Royal Netherlands Academy of Arts and Sciences (KNAW). In spite of
their trip to
the Trippenhuis, these models have survived to this day and can still
be admired in the
mathematics institute of the university of Groningen. In the talk we
discuss what these models
show, why Schoute designed them, and we explain some of the beautiful
mathematical properties
of the corresponding surfaces.

April 21, 2011 Snellius

Kloosterman lecture
JeanLouis ColliotThélène (CNRS and Université ParisSud, Orsay)
From sums of squares in fields to motivic cohomology
and higher class field theory
Abstract.
L. Euler (1770) proved that any positive rational number is a sum of
four squares of rational numbers. E. Artin (1927) proved that any
rational function in $n$ variables with real coefficients, if positive
on ${\bf R}^n$,
is a sum of squares of such rational functions (Hilbert's 17th
problem), and A. Pfister (1967) proved
that it may then be written as a sum of at most $2^n$ squares.
Artin also showed that positive rational functions in $n$
variables with {\it rational} coefficients are sums of squares
of such functions. That they may be represented by
a bounded number of squares, more precisely $2^{n+2}$,
was predicted in 1991. This relied on two hypotheses,
both of which are now known, one by work of V. Voevodsky,
the other one by work of U. Jannsen.
I shall go through the history of sums of squares in fields.
I shall then try to give a glimpse of the various tools employed
in the proof of the $2^{n+2}$result :
the algebraic theory of quadratic forms (as started by E. Witt),
Milnor KTheory, Galois cohomology, motivic cohomology
and higher class field theory.
Slides [PDF ]

May 12, 2011 Snellius

Peter Spreij (Amsterdam)
Block Hankel confluent Vandermonde matrices
Abstract.
Vandermonde matrices are wellknown. They have a number of interesting
properties and play a role in (Lagrange) interpolation problems, partial fraction
expansions, and finding solutions to linear ordinary differential equations, to
mention just a few applications. Usually, one takes these matrices square,
q × q say, in which case the ith column is given by u(z_{i}), where we write
u(z) = (1,z,…,z^{q1})^{⊤}. If all the z_{
i} (i = 1,…,q) are different, the Vandermonde
matrix is nonsingular, otherwise not. The latter case obviously takes
place when all z_{i} are the same, z say, in which case one could speak of a
confluent Vandermonde matrix. Nonsingularity is obtained if one considers
the matrix V (z) whose ith column is given by the (i  1)th derivative
u^{(i1)}(z)^{⊤}.
We will consider generalizations of the confluent Vandermonde matrix V (z) by
considering matrices obtained by using as building blocks the q × r matrices
M(z) = u(z)w(z), with u(z) as above and w(z) = (1,z,…,z^{r1}), together with its
derivatives M^{(k)}(z). Specifically, we will look at the matrix whose ijth
block is given by M^{(i+j)}(z). This in general nonsquare matrix exhibits a
blockHankel structure. We will answer a number of elementary questions for this
matrix. What is the rank? What is the nullspace? Can the latter be
parametrized in a simple way? Does it depend on z? What are left or
right inverses? It turns out that answers can be obtained by factorizing
the matrix into a product of other matrix polynomials having a simple
structure. The answers depend on the size of the matrix M(z) and the
number of derivatives M^{(k)}(z) that is involved. The talk will be completely
elementary, no specific knowledge of the theory of matrix polynomials is needed.

October 13, 2011 Snellius 174

Mark Peletier (Eindhoven)
Gradient Flows, Optimal Transport, and Fresh Bread
Abstract.
In 1997, Jordan, Kinderlehrer, and Otto pioneered a new way of looking
at ageold equations for diffusion, thus giving an exact mathematical
description of the sense in which `diffusion is driven by entropy'. This
sense revolves around the concept of optimal transport.
Introduced by Monge in 1781, this theory focuses on optimal ways to
transport given quantities from A to B. Its development took off after
Kantorovich improved the formulation in 1942, and in recent years the
theory has exploded, with applications in differential geometry,
probability theory, functional analysis, analysis on nonsmooth spaces,
and many more.
In this talk I will revisit the original connection between the
diffusive problems on one hand and the theory of optimal transport on
the other. I will show how the two are connected, discuss many
consequences of this, and describe recent insight into the deeper
meaning of this connection.

December 1, 2011 Snellius 174

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

December 8, 2011 Snellius 174

Tanja Eisner (Amsterdam)
Arithmetic progressions via ergodic theory
Abstract.
We sketch the development from van der Waerden's theorem on arithmetic
progressions to the recent GreenTao theorem and show how methods from
ergodic theory have been decisive in this field.



Lectures in 2010:
December 16, 2010 Snellius, room 312

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

December 2, 2010 Snellius, room 174

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

November 4, 2010 Snellius, room 174

Dr. Jan Draisma (TU Eindhoven):
Finiteness results in statistics using algebra
Abstract.
I will give three interrelated examples of how polynomial algebra can be
used to settle finiteness questions arising from statistics. I will assume
no prior familiarity with any of these, and I will emphasise the fundamental
algebraic tools that go into the proofs.
The first example is the by now classical DiaconisSturmfels algorithm for
sampling from contingency tables with prescribed marginals, where algebra
proves the existence of a finite Markov basis. The second concerns recent
work by Hillar and Sullivant, where such Markov bases are shown to stabilise
as some of the sizes of the contingency tables tend to infinity. The third
is a proof that Gaussian factor analysis with a fixed number of (latent)
factors stabilisesas far as polynomial equations are concernedas the
number of observed variables tends to infinity.

October 14, 2010 Snellius, room 174

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

September 23, 2010 Snellius, room 174

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

June 17, 2010 Snellius, room 174

Prof.dr. Nicolai Reshetikhin (UC Berkeley / UvA):
Statistical mechanics in the thermodynamic limit:
deterministic limit shapes and fluctuations
Abstract.
The talk will be focused on how deterministic limit shapes
emerge in statistical mechanics in the thermodynamic limit.
In tiling and dimer models this phenomenon is known as Arctic circle
formation. We will also discuss how the partition function and fluctuations
depend on boundary conditions in the thermodynamic
limit.

May 27, 2010 Snellius, room 174

Dr.ir. M.C. Veraar (Delft University of Technology):
Regularity for stochastic PDEs
Abstract.
In this talk we explain how to formulate a partial dierential equation with noise in a functional
analytic way. Moreover, we show how one can apply functional analytic tools to derive sharp
regularity properties of solutions to such equations. Some of the mathematical tools one needs
for this are operator theory, harmonic analysis, probability theory and evolution equations.

April 29, 2010 Snellius, room 174

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



Lectures in 2009:
December 17 Snellius, room 174

Prof. Rob de Jeu (Vrije Universiteit Amsterdam):
Algebraic Ktheory and arithmetic.
Abstract.
The Riemann zetafunction, which encodes information about the integers and
the
prime numbers, has been studied extensively. Its values at 2,4,... are
wellknown, but much less is known about its values at 3,5,... .
This
difference can be explained to an extent by the different behaviour of
certain
groups (algebraic Kgroups) of the rationals.
In this talk, we discuss some basic examples of such Kgroups, and some
links between them and arithmetic.

October 22 Snellius, room 312

Prof. Tanja Lange (Technische Universiteit Eindhoven):
Coppersmith's factorization factory.
Abstract.
RSA, named after its inventors Rivest, Shamir, and Adleman, is the
most widely used publickey 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 RSA512 since n has 512 bits.
For academic teams it is still too expensive to break RSA1024,
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 RSA1024 is estimated to cost a year of computation on
a machine worth 100 million Euros it is commonly believed that real
world attackers will not invest the effort to attack an individual
user (or any other public key worth less than 100 million Euros).
In 1993 Coppersmith suggested a "factorization factory" which factors
many numbers of the same size in significantly less time than handling
each of them individually. This means that there can be financial gain
in attacking users having less than 100 million Euros if many RSA keys
are attacked simultaneously.
The currently best factorization methods are based on the Number Field
Sieve (NFS). Coppersmith's method is a variant of the NFS but requires
different optimizations. In particular, Coppersmith's method generates
many auxiliary numbers that cannot be sieved. For these numbers the
Elliptic Curve Method of factorization (ECM) is optimal. We recently
sped up ECM by choosing Edwards curves instead of general elliptic
curves, choosing curves with larger torsion and better linking the
implementation to the processor it runs on.
In this talk we will review RSA and standard factorization methods and
then explain how Coppersmith's method works. Then we will show the
impacts of Coppersmith's method and faster ECM on concrete parameter
choices.

October 8 Snellius, room 312

Prof. Peter Jagers (Chalmers University of Technology and University of Gothenborg):
Extinction: how often, how soon, and in what way?
Abstract.
Branching processes were born out of the observation that extinction (of
separate families or subpopulations) is ubiquitous in nature and society. This
lead to Galton's and Watson's famous error, as they
claimed that all family lines must die out, even in exponentially growing
populations. We look back at this discussion, and proceed to exhibiting the
time and path to extinction.

September 24 room 312

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

June 25

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

Wednesday May 20 16:3017:30 Room 312

Prof. Harry Kesten (Cornell University):
A problem in onedimensional diffusionlimited aggregation (DLA)
and positive recurrence of Markov chains

May 14 room 312

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

May 7

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

April 2

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

March 26

Prof. Christian Skau (NTNU, Trondheim):
Rendezvous with Abel's and Ruffini's proof of the unsolvability of the
general quintic.
Abstract.
A result that has fascinated generation upon generation of mathematicians is
the theorem that the general quintic can not be solved algebraically
("by radicals").
Today this result follows as a corollary of Galois
theory  this beautiful subject  which requires,however,
for the student to
absorb and understand many abstract new concepts.
Therefore, when at the end
of a course in Galois theory the unsolvability of the quintic is given as a
corollary,
it is often experienced by the students not as a climactic
ending  as it should be  simply because of exhaustion at learning this
extensive theory.
Abel's original proof  with a certain input from Ruffini 
is on the other
hand much more direct and easily available.
Besides,the proof is strikingly
elegant and deserves not to get forgotten.
In the talk I will sketch Abel's proof,and also stress the new ideas he
thereby introduced into algebra, preparing the way for Galois' work.

March 12

Dr. Martijn de Vries (Technische Universiteit Delft):
Unique expansions of real numbers in noninteger bases.
Abstract.
Following a seminal paper of A. Rényi, many works were devoted to
probabilistic, measure theoretical and number theoretical aspects of
representations for real numbers in noninteger bases.
In this talk we consider for each fixed real number q>1 the topological
structure of the set
U_{q}
consisting of those real numbers x for which
exactly one sequence (c_{i}) of integers
belonging to [0,q)
satisfies the equality ∑_{i≥1}
c_{i}q^{i}=x.
If time
permits we will also give a characterization of the sets
U_{q},
arising from a dynamical system
that was recently introduced by K. Dajani and C. Kraaikamp.
The talk is based on joint work with V. Komornik.

February 12

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



Lectures in 2008:
December 11

Prof. Sebastian van Strien (University of Warwick):
On some questions of Fatou and Milnor on iterations of polynomial maps.
Abstract.
This talk is about iterations of polynomials acting on the complex plane
and their associated Julia, Fatou and Mandelbrot sets. I will give a survey of
some recent results in this area.

November 27 16:0016:40

On November 27 there is a special double General Colloquium session,
to celebrate the appointments of
Peter Grunwald and Vladas Sidoravicius as parttime Professors
at our Institute per November 1 in the frame of an
exchange program with the CWI.
Prof. Peter Grunwald (CWI/Universiteit Leiden):
Learning when all Models Are Wrong.
Abstract.
Statistical analysis of data often results in a model that is *wrong yet
useful*: it is wrong in that it is a gross simplification of the process
actually underlying the data; it is useful in that predictions about future
data taken on the basis of the model are quite successful. For example, we
often assume highly dependent variables to be independent (e.g. in speech
recognition); we assume nonlinear relationships to be linear (e.g. in
econometrics), and so on. Yet most existing statistical methods were
designed under the assumption that one of the candidate models is actually
"true". By assuming at the outset that this is not the case, we can design
algorithms that are provably more robust, and that provably learn faster.
Here I will give an overview of some of the remarkable properties of such
algorithms:
 "when ignorance is bliss": sometimes, it is a good idea to ignore some of
the data;
 there exist prediction methods which, magically, (in some sense) perform
well *no matter what data are observed*;
 Bayesian methods perform remarkably well if the model is wrong, if one is
solely interested in prediction  yet they can fail dramatically if one is
also interested in estimation (identifying which model is "closest to being
true"). We explain how this is related to convexity of probability models.

November 27 16:5017:30

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

November 20

Dr. Steven Wepster (Universiteit Utrecht):
On longitude and Tobias Mayer's lunar tables.
Abstract.
During the early modern period, finding the longitude was regarded as
difficult a task for a navigator as squaring a circle for a mathematician.
This changed in mideighteenth 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 (17231762).
After a brief history of the longitude problem, I will show that Mayer's
tables depend on a hybrid of kinematics, dynamics, and relatively
largescale model fitting.

November 13

Dr. Karma Dajani (Universiteit Utrecht):
Betaexpansions revisited.
Abstract.
We give an overview of some of the old and new results describing the
ergodic,
combinatorial and arithmetic
properties of algorithms generating expansions to noninteger base.

October 23

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

October 9

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

September 18

Dr. Bas Spitters (Radboud Universiteit Nijmegen):
A computerverified implementation of Riemann integration 
an introduction to computer mathematics.
Abstract.
(Joint work with Russell O'Connor).
The use of floating point real numbers is fast, but may cause incorrect
answers due to overflows. These errors can be avoided by hand. Better, exact
real arithmetic allows one to move this bookkeeping process entirely to the
computer allowing one to focus on the algorithms instead. For maximal
certainty, one uses a computer to check the proof of correctness of the
implementation of this algorithm. We illustrate this process by implementing
the Riemann integral in constructive mathematics based on type theory.
The implementation and its correctness proof were driven by an
algebraic/categorical treatment of the Riemann integral which is of
independent interest.
This work builds on O'Connor's implementation of exact real arithmetic. A demo
session will be included.

July 10

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

June 5

Prof. Shigeki Akiyama (Niigata University, Japan):
On the pentagonal rotation sequence.
Abstract.
Let (a_{n})
be the sequence of integers defined by the recurrence
0 ≤ a_{n} +
ωa_{n+1}
+ a_{n+2} < 1
and by the initial values
a_{0},a_{1}
∈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 selfinducing structure of a piecewise isometry emerging from
the discretized
pentagonal rotation. One can also define analogues for Sturmian sequences
and βexpansions in this system.

May 22

Dr. Nelly Litvak (Universiteit Twente):
Power law behavior of the Google PageRank distribution.
Abstract.
PageRank is a popularity measure designed by Google to rank Web pages
according to their importance. It has been noticed in empirical studies
that PageRank and indegree 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 indegree follow power laws with the same exponent.
The difference between these two power laws is in a multiplicative
constant, which depends mainly on the settings of the PageRank
algorithm. Our theoretical results are in good agreement with
experimental data.
This a joint work with Yana Volkovich (UTwente) and Debora Donato
(Yahoo! Research)

April 24

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

March 27

Dr. Vladas Sidoravicius (CWI, Amsterdam):
Stochastic Structure of Critical Systems.
Abstract.
What is in common between the Clairvoyant Demon scheduling
problem, the Riemann hypothesis and quasiisometries of large objects? All
these questions can be represented and studied as critical or
nearcritical percolative systems. Can we handle it by stochastic methods?

March 13

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

February 14

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

January 24

Dr. Wieb Bosma (Radboud Universiteit Nijmegen):
Wiskundige Verpoozingen.
Abstract.
Onder de titel `Wiskundige Verpoozingen' schreef P.H. Schoute
vanaf 1882 enkele jaren een rubriek in `Eigen Haard'. Mijn
nieuwsgierigheid werd gewekt door twee verwijzingen hiernaar
in de wiskundige literatuur. In deze voordracht zal ik aan de hand
van een flink aantal plaatjes en citaten verslag doen van mijn
speurtocht naar Eigen Haard, naar P.H. Schoute, en naar de
context waarin deze soms verrassend moderne rubriek verscheen.
De voordracht is zeer geschikt voor studenten.



Lectures in 2007:
December 6

Prof. Barry Koren (CWI, Amsterdam):
Compressible twofluid flow: model, method and results
Abstract.
Multifluid 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,
waterair flows around ship hulls, etc.
To gain better insight in the behavior of multifluid flows,
especially twofluid 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 twofluid flow, with pressures and velocities that are equal on
both sides of the twofluid interface. The model describes the behavior of a
numerical mixture of the twofluids (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 energyexchange model forms a source term.
The spatial discretization of the model uses a monotone, higherorder accurate
finitevolume approximation, the temporal discretization a threestage
RungeKutta scheme.
For the flux evaluation a Riemann solver is constructed. The source term is
evaluated using the wave pattern found with the Riemann solver.
The twofluid model is validated on several shocktube problems and on
two standard shockbubble interaction problems.
STUDENTS ARE WELCOME.

November 15

Dr. Ben Kane (Radboud Universiteit Nijmegen):
The triangular theorem of 8.
Abstract.
The famous ``Eureka" theorem of Gauss states that every integer n can be
written in the form T(x)+T(y)+T(z) for integers x, y, and z, where
T(x):=x(x+1)/2 is the xth triangular number. In this talk, we
will investigate the more general question which sums of triangular
numbers, that is, expressions of the form
a_{1}T(x_{1})
+...+
a_{r}T(x_{r})
where r,
a_{1},...,
a_{r}
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
a_{1}T(x_{1})
+...+
a_{r}T(x_{r})
with integers
x_{1},...,x_{r}
if and only if the integers 1,2,4,5 and 8 can be
represented in this way.

November 1

Prof. Bas Edixhoven (Universiteit Leiden):
How to count vectors with integral coordinates and given length in
R^{n}.
Abstract.
The question will be addressed how fast one can compute the number of ways in
which an integer m can be written as a sum of n squares. At the moment the
answer is not know. I will explain why I think that if n is even and m is given
with its factorisation into primes, this counting can be done in time polynomial
in n.log(m). The proposed method uses a generalisation of the main results of
joint work with JM. Couveignes, R. de Jong and F. Merkl on the complexity of
the computation of coefficients of modular forms.

October 18

Dr. Gunther Cornelissen (Universiteit Utrecht):
Listening to Riemann surfaces.
Abstract.
A Dedekind zeta function doesn't always encode the isomorphism class of a
number field.
The dynamical Laplace operator zeta function doesn't
always
encode the isometry type of a manifold (e.g., a Riemann surface):
you cannot
hear the shape of a Riemannian manifold.
We look at such problems using tools from noncommutative geometry:
to a compact Riemann surface of genus at
least two,
I will associate a finitedimensional noncommutative Riemannian
manifold
(a.k.a. spectral triple) that encodes the isomorphism class of the
Riemann surface
up to complex conjugation.
The encoding lies in the zeta
functions of the spectral triple:
the spectra of various operators in the
spectral triple reconstruct the Riemann surface,
via application of an
ergodic rigidity theorem a la Mostow.
Joint work with Matilde Marcolli.
Paper available at
http://www.arxiv.org/abs/0708.0500

October 4

Prof. Rob van den Berg (CWI, Amsterdam):
Ponds and power laws.
Abstract.
Invasion percolation is a random spatial growth model with very simple
rules but surprisingly rich and complex behaviour. It was introduced around
1980 by reserachers related to the oil industry but soon drew attention from
many others, including theoretical physicists and mathematicians.
After defining the model, I will concentrate on an object called a
'pond', and explain that this object has indeed a natural 'hydrologic'
interpretation. Although there is no special tuning of a parameter in this
model, it turns out that these ponds are, in a sense which will be
explained, critical. Such 'selforganized critical behaviour' seems to be
quite common in nature, but this is one of the very few 'natural' models
where it can be rigorously proved.
The talk is based on joint work with Yuval Peres (Berkeley and
Microsoft), Vladas Sidoravicius (Rio de Janeiro; now CWI) and Eulalia Vares
(Rio de Janeiro), and on joint work with Antal Jarai (Ottawa) and Balint
Vagvolgyi (VU).

September 13

Dr. Ronald de Wolf (CWI, Amsterdam):
Fourier analysis, hypercontractive inequalities,
and quantum computing.
Abstract.
Fourier analysis of realvalued functions on the Boolean hypercube has been
an extremely
versatile tool in theoretical computer science in the last decades.
Applications include the
analysis of the behavior of Boolean functions with noisy inputs, machine
learning theory,
design of probabilistically checkable proofs, threshold phenomena in random
graphs, etc.
The BonamiBeckner hypercontractive inequality is an important result in
this context.
In this talk I will briefly introduce Fourier analysis of realvalued
functions on the Boolean
hypercube, and then prove a generalization of the BonamiBeckner inequality
for
*matrixvalued* 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 BenAroya and Oded Regev, paper
available at
http://arxiv.org/abs/0705.3806.

May 24 room 312

Kloosterman Lecture
Prof. Michael Bennett (University of British Columbia, Vancouver):
Open Diophantine Problems.
Abstract.
This talk will focus on a number of open problems from the field of
Diophantine equations, and related areas.
I will attempt to indicate where these questions arise, why they turn out to be
so difficult, and whether modern methods can provide, if not their complete
resolution, at least a certain amount of insight.
Prof. Michael Bennett was the Kloosterman Professor 2007.
His research focuses on proving results
for Diophantine equations by combining various theoretical and computational
techniques.
Prof. Bennett was visiting our institute during April and May 2007.

May 3

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

April 5

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

March 8

Prof. Richard Gill (Universiteit Leiden): Perfect passion at a distance
(how to win at Polish poker with quantum dice).
Abstract.
I explain quantum nonlocality experiments and discuss how
to optimize them. Statistical tools from missing data
maximum likelihood are crucial. New results are given on
Bell, GHZ, CGLMP, CH and Hardy ladder inequalities. Open
problems  there are indeed many!  are discussed.
Prior knowledge of quantum theory or indeed physics is not
needed to follow the talk; indeed its lack could be an
advantage ;)
It will be difficult to resist discussion of the
metaphysical implications of Bell's inequality.
Slides for a previous version of this talk, and reference
to an overview paper:
http://www.math.leidenuniv.nl/~gill/betterbelltalk.pdf
http://arxiv.org/abs/math.ST/0610115
[the latter to appear in IMS Lecture Notes  Monographs
series; volume on "Asymptotics: particles, processes and
inverse problems"]



Lectures in 2006:
December 7

Dr. Daan Crommelin (CWI): Stochastic modeling for atmospheric flows.
Abstract.
When modeling or analyzing atmospheric flow, a major question is
how to deal with the wide range of spatiotemporal 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
lowfrequency variability, development of reduced models, the study of
atmosphereocean interaction and improvement of parameterization schemes
in models for weather and climate prediction.
I will discuss several approaches that use stochastic methods for studying
atmospheric dynamics; among them are inverse modeling for stochastic
differential equations (SDEs), elimination of fast variables in
stochastic systems,
and the use of Hidden Markov Models.

November 16

Dr. Cor Kraaikamp (Technische Universiteit Delft): On multidimensional subtractive algorithms.
Abstract.
Define the mapping S_{d} on the set of ordered dtuples x of positive
reals as follows: keep the smallest number, subtract it from the
others and reorder the result in a nondecreasing way. Meester and
Nowicky proved that, for d=3 and almost all x, the nth iteration
(S_{d})^{n}(x)
tends to a vector different from the nullvector, as n tends to infinity.
Later,
Meester and K. showed that this result holds for d≥3. In this
lecture, the proof of this results is outlined, and some applications
and generalizations will be given.

November 2

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

September 28

Prof. Anton Bovier (Weierstraß Institut für angewandte Analysis and Stochastik, Berlin): Metastability: a potential theoretic approach.
Abstract. Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems.
In this talk, I describe recent attempts towards a modelindependent
approach to metastability in the context of reversible Markov processes.
I will present an outline of a general theory,
based on careful use of potential theoretic ideas
and indicate a number of concrete examples
where this theory was used very successfully.
I will also indicate some challenges for future work.

September 21

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

June 15
GORLAEUS lecture hall no. 1

Prof. JeanPierre Serre (Collège de France): Bounds for the orders of the finite subgroups of G(k).
Abstract.
A wellknown theorem of Minkowski gives a sharp
multiplicative upper bound
for the order of a finite subgroup of GL(n,Q).
We shall see how this
result can be extended
to other ground fields and to other reductive groups.

June 8 
Prof. Alexander Schrijver (Universiteit van Amsterdam/Centrum voor Wiskunde en Informatica): Tensors, invariants, and combinatorics.
Abstract.
We give a characterization of those tensor algebras that are invariant
rings of a subgroup of the unitary group. The theorem has as consequences
several "First Fundamental Theorems" (in the sense of Weyl) in invariant
theory.
Moreover, the theorem gives a bridge between invariant theory and
combinatorics. It implies some known theorems on selfdual codes, and it
gives new characterizations of graph parameters coming from mathematical
physics, related to recent work with Michael Freedman and Laszlo Lovász
and of Balázs Szegedy.
In the talk we give an introduction to and explanation of these results.

June 1 
Prof. Jun Tomiyama (Tokyo Metropolitan University): The interplay between topological dynamics and the theory of
C^{*}algebras.
Abstract.
Contrary to a long history of the interplay between measurable dynamics
(ergodic theory) and the theory of von Neumann algebras (factors), its
counterpart of the interplay between topological dynamics and the theory of
C^{*}algebras is far from mature yet, although there are many results
available on the side of the C^{*}algebras as part of the general theory of
transformation group C^{*}algebras.
In this talk, taking the simplest case of a dynamical systems where a single
homeomorphism acts on a compact (not neccessarily metrizable) space, we
discuss first how a noncommutative C^{*}algebra is naturally associated to
such a dynamical system, and then show some aspects of the present state of
knowledge of the interplay surrounding the simplicity of this associated
C^{*}algebra.

May 11 
Prof. Jürgen Klüners (Universität Kassel, Germany): On polynomial factorization.
Abstract.
It is well known that the factorization of polynomials
over the integers is in polynomial time. Unfortunately
this algorithm was not useful in practice. Recently,
Mark van Hoeij found a new factorization algorithm
which works very well in practice. We present the
ideas of his algorithm and extend this algorithm
to an algorithm for factoring polynomials in
F[t][x], where F is a finite field. Surprisingly
the algorithm is much simpler and more efficient
in this setting.
We prove (in the rational and the bivariate case) that the
new algorithm runs theoretically in polynomial time. We will
explain why the expected running times should be heuristically
much better than the given worst case estimates.

April 20 
Prof. Joost Hulshof (Vrije Universiteit Amsterdam): The
holefilling problem for the porous medium equation.
Abstract. The porous medium equation is a nonlinear degenerate version
of
the heat equation. It appears in many physical applications such as
flows in
porous media and thin film viscous flow. Compacly supported solutions
of this
equation have expanding supports. Holes in the support are filled in
finite
time. I will discuss radially symmetric holefilling solutions and
their stability
properties under radially and nonradially symmetric perturbations.

April 6 
Prof. Aad van der Vaart (Vrije Universiteit, Amsterdam):
Estimating a function using a Gaussian prior on a Banach space.
Abstract:
After a general introduction to nonparametric statistical estimation
we discuss recent work [joint with Harry van Zanten]
on Bayesian estimation using Gaussian prior distributions.
As a concrete example consider estimating a probability density p
using a random sample X_{1,...,
Xn from this density
(i.e. the probability that Xi falls in a set B is the area under p above
B).
A Bayesian approach would be to model the density x→p(x) "apriori"
as proportional to the function x→eWx for W a Gaussian
process, e.g. Brownian motion, indexed by the set in which the
observations take their values. The Bayesian machine (Bayes, 1764)
then mechanically produces a "posterior distribution", which is
a random measure on the set of probability densities, can
be used to infer the "true" value of p, and anno 2006 is computable.
We investigate the conditions under which this Bayesian approach
gives equally good results as other methods. A benchmark is
whether it works well if the unknown p is known to belong to
a given regularity class, such as the functions in
a Holder or Sobolev space of a given regularity. This depends
of course on the Gaussian process used. It turns out to be
neatly expressible in the reproducing Hilbert space of the process.
} 
March 16 
Prof. Gerard van der Geer (Universiteit van Amsterdam): The Schottky Problem.
Abstract:
An algebraic curve determines an abelian variety, the Jacobian
of the curve. For example, for a Riemann surface the Jacobian is
a complex torus associated to the periods of integrals over the
Riemann surface. Not every abelian variety is the Jacobian of a
curve and the Schottky problem, due to Riemann, aks for a characterization
of the Jacobians among all abelian varieties. Various answers have been
proposed. We shall discuss the problem, its history and some of the
proposed answers to this problem.

March 2 
Dr. Onno van Gaans (Universiteit Leiden): Invariant measures for infinite dimensional stochastic differential
equations.
Abstract:
If a deterministic system is perturbed by noise, it will not settle to a
steady state. Instead, there may exist invariant measures. Existence of an
invariant measure requires tightness of a solution, which is a compactness
condition. A solution of a finite dimensional stochastic differential
equation is tight if it is bounded. Boundedness is not sufficient in the
case of an infinite dimensional state space. We will discuss several
conditions on infinite dimensional stochastic differential equations that
provide existence of tight solutions and invariant measures.



