
Lectures in 2005:
(the lectures from FebruaryJune were organized
in cooperation with TU Delft)
December 15

Prof. Philippe Clément (TU Delft, Univ. Leiden):
Maximal regularity for differential equations in Banach spaces and Rboundedness.
Abstract:
Vectorvalued extensions of the classical Lpmultiplier theorems of
Marcinkiewicz ('39), Mikhlin ('56) and others have been
established by Bourgain ('86) for an important class of Banach spaces
(UMD). Operatorvalued extensions obtained in the sixties
are valid only for operators acting on a space isomorphic to a Hilbert
space. Very recently Lutz Weis ('01) proved a "non
Hilbertian" operatorvalued version of the Mikhlin theorem by using in
an essential way the notion of an Rbounded family
of operators. At the same time he was able to solve a longstanding
problem concerning maximal Lpregularity for abstract
parabolic equations in Banach spaces. The aim of this talk is to
introduce and discuss these results following the approach
of Arendt and Bu ('02).

December 8

Dr. Patsy Haccou (Institute of Biology, Univ. Leiden):
Effects of deleterious mutations on the evolution of reproductive
modes.
Abstract:
The prevalence of sexual reproduction is still one of the great mysteries of
evolutionary biology, since, for all else being equal, asexual populations
have a twofold fitness advantage over their sexual counterparts. Thus,
whenever the two reproductive strategies compete, the elimination of the
sexual mode of reproduction is expected, unless there are factors that
counterbalance its disadvantages. Nevertheless, most eukaryotes reproduce
sexually. Several theories have been developed to explain this. One of the
proposed explanations is that under certain conditions sexual reproduction
may lower the average load of deleterious mutations, thus increasing the
viability of females.
This argument, as well as the lion's share of arguments used in the
discussion concerning competition between sexual and asexual modes of
reproduction, is, however, based on comparison of long term population
growth rates. The general idea is that when an initially small population
of asexually reproducing females has a positive chance to invade in a large,
stable population of sexually reproducing individuals, it will eventually
replace the resident population. On the basis of this reasoning it is for
instance predicted that a clonal population, which has the same expected
viability as a sexual diplodiploid population, but a twice as large
fecundity, will eventually outcompete the sexual one. This argument,
however, disregards the fact that, although the probability of such a
takeover may be positive, it can still be extremely small, thus making it a
very rare event indeed, even on an evolutionary scale.
An additional danger of comparing population growth rates is that it does
not necessarily give correct predictions concerning invasion probabilities.
Finally, focusing on growth rates entails the danger that effects of initial
conditions are ignored. The state of the first female that starts to
reproduce asexually is determined by the distribution of female states in
the resident population. Establishment chances of asexual reproduction turn
out to be considerably affected by such initial conditions.
I examine the probability of succesful invasion of a resident
sexual population by an asexually reproducing mutant when the viability of a
female is determined by the number of deleterious mutations that she
carries.

November 10

Prof. Jaap Korevaar (Universiteit van Amsterdam):
Tauberian Theory: History and Highlights.
Abstract:
Roughly following the historical development, and
noting the motivation provided by number theory, we
discuss various aspects of Tauberian theory. Results
from the HardyLittlewood era (19101935) will lead to Wiener
theory (developed in 19271932).
Complex Tauberian theory (19052005) will be considered,
and a bit of remainder theory (19502000) if time permits.
Some open questions will be indicated.

November 3

Prof. Frank den Hollander (Universiteit Leiden):
The incipient infinite cluster for oriented percolation.
Abstract:
In this talk we consider oriented percolation on
Z^{d}×Z_{+},
where (x,n) is connected to (y,n+1)
for all x,y∈Z^{d}
with xy≤ L and all n∈Z_{+}.
Each bond is open with probability p and
closed with probability 1p.
We consider the case d> 4, L large
and p=p_{c}, the critical
percolation threshold. We are interested in the probability
θ_{n}
that the origin (0,0) is connected to the plane
at time n, i.e., to
Z^{d}×{n}. With the help
of the socalled lace expansion technique, we derive a recurrence
relation for θ_{n} up to quadratic order, from which we deduce
that
lim_{n→∞}
n^{2}(θ_{n}
θ_{n+1})=C ∈ (0,∞).
This limit shows that a key critical exponent for oriented percolation
assumes the meanfield value above the upper critical dimension.
This fact, in turn, has a number of interesting consequences for the
geometry of the incipient infinite cluster, i.e., the cluster that
is about to become infinite at criticality.

October 20

Dr. Robin de Jong (Universiteit Leiden):
Moduli spaces of curves.
Abstract:
The study of complex curves is one of the central topics in algebraic
geometry. We
briefly describe the various viewpoints that one can take for this study
 one
more algebraic, one more geometric and the other more analytic. On the
level of
topology, complex
curves are distinguished by their genus, that is, the number of holes
that they have when embedded in real three dimensional space as a compact
topological surface. Curves of low genus admit a very concrete description
and
we will discuss some examples. We will see that all curves of a fixed low
genus
can be naturally parametrised, and that in fact such natural parameter
spaces exist
for all genera. What we get is the socalled moduli space of curves.
One of the surprising facts is that it itself has the structure
of a (higher dimensional) analytic variety. We advocate the idea that it
is very
interesting to study this variety from the analytic point of view.
In particular we ask
if one can construct natural nontrivial nonvanishing analytic functions
on this
space, like for
example the classical discriminant modular form in the case of moduli of
elliptic curves (that is, curves of genus one).

June 16
Delft

Dr. Marcel de Jeu (Universiteit Leiden)
Nonconvex and real PaleyWiener theorems for the Fourier transform.
Abstract:
Theorems which give information about the support of a function on basis
of the growth rate of an integral transform of that function are commonly
known as PaleyWiener theorems. These theorems are named after the first
and prototypical result by Paley and Wiener in 1934, who characterized the
Fourier transforms of compactly supported
L_{2}functions on the real line
as being precisely the
L_{2}functions
on the real line which have an entire
extension to the complex plane of exponential type. In the mean time, more
theorems in this vein have become available, also in higher dimension,
typically establishing a link between the convex hull of the support of a
function in real nspace on one side and the growth rate of its transform
on complex nspace on the other side.
After giving a brief introduction to Fourier analysis, the classical
PaleyWiener theorems and their relevance, we will present some results in
this direction which are of a different flavour. These PaleyWiener type
theorems, which are joint work with Nils Byrial Andersen, show not only
that the support itself (and not just its convex hull) can be recovered
from certain growth rates related to the transform, but also that one does
not have to go beyond real nspace to retrieve this information.

June 2
Leiden

Prof. Roger Cooke (TU Delft):
Model Inference for graphical models.
Abstract:
We develop a theory of model inference for graphical models based
on the vinecopula representation of high dimensional distributions.
This is usually seen as a problem of discovering conditional independences
in multivariate data (as in the work of eg Terry Speed and Joe Whittaker).
Vines are nested sets of trees which encode conditional bivariate information.
When associated with partial correlations,
they are algebraically independent and uniquely determine the correlation matrix.
When associated with conditional rank correlations,
they determine a sampling algorithm.
Further, it is shown that the determinant equals the
product of all terms
(1(ρ_{ihK})
^{2})
where
ρ_{ihK}
is the partial correlation associated with an edge in the vine.
Each regular vine represents a factorization of the determinant.
We seek a factorization which, when logged, dominates all others in the sense of majorization.
That is, a factorization whose terms
(1(ρ_{ihK})^{2})
are closest to 0 or 1.
This enables us to replace insignificant partial correlations with 0's.
Unlike the method of Speed and Whittaker,
there is no need introduce 'corrections' to restore positive definiteness.
An example illustrates the procedure.

May 19,
Delft

Dr. JanHendrik Evertse (Universiteit Leiden):
Recent results on linear recurrence sequences.
Abstract:
A (complex) linear recurrence sequence
U={u_{n}}_{n≥0}
is given by a linear recurrence
(1)
u_{n}=c_{1}
u_{n1}+...+
c_{k}
u_{nk} (n≥k)
with complex coefficients c_{1},...,
c_{k}
and complex initial values u_{0},...,
u_{k1}.
We give a survey of some recent results on linear
recurrence sequences which have been proved
using Diophantine approximation techniques.
In particular, we are interested in the set of zeros of
a linear recurrence sequence U,
that is the set of n such that u_{n}=0.
According to a classic theorem by SkolemMahlerLech,
the number of zeros of U is finite if
U is 'nondegenerate.' Their proof
was by means of padic analysis.
In 1999, Schmidt obtained the following striking result: if U
is nondegenerate, then its number of zeros is bounded
above by a constant C(k) depending only on the length
of the recurrence (1).
Schmidt's proof was by means of some heavy machinery from Diophantine
approximation. We will discuss this and some other results.

May 12,
Leiden

Dr. Ben de Pagter (TU Delft)
Representations, Boolean algebras and Rboundedness.
Abstract:
Relations
between Hilbert spaces and general Banach spaces
have been somewhat pacified by the introduction and systematic
study of socalled Rbounded (=Randomized bounded) collections
of operators. RBounded sets of bounded linear operators in
Banach spaces play an increasingly important role in various
branches of functional analysis, operator theory, harmonic
analysis and partial differential equations. In some special
situations, this notion was actually introduced in the mid1990 s
by E. Berkson and T.A. Gillespie (Rproperty), but was already
implicit in earlier work of J. Bourgain (1983). In this talk
we shall discuss the concept of Rboundedness and present some
recent applications of this notion to Boolean algebras of
projections in Banach spaces and to the representation theory
of groups and of spaces of continuous functions on Banach spaces.

April 21
Delft

Prof. Bas Edixhoven (Universiteit Leiden):
How fast can one compute Ramanujan's taufunction?
Abstract:
Ramanujan's taufunction τ(n) will be defined (very easy). I will
explain why I expect that for prime numbers p, one should be able to
compute τ(p) in time polynomial in log p, and I will indicate how far we
are from having all details written down. I will also explain why this is
of interest.

April 14 Leiden

Prof. Arnold Heemink (Technische Universiteit Delft)
Filtering algorithms for large scale systems.
Abstract:
Data assimilation methods are used to combine the results of a large scale
numerical model with the measurement information available in order to obtain
an optimal reconstruction of the dynamic behavior of the model state. Many
data assimilation schemes are based on solving the EulerLagrange equations.
A recursive algorithm to solve this twopoint boundary value problem
can be derived and results in the wellknown Kalman filtering algorithm. This
standard filter however would impose an unacceptable computational burden for
large scale systems with a state dimension of more then, say, 100 000. In
order to obtain a computationally efficient filter simplifications have to be
introduced. Recently many new algorithms have been proposed in literature,
all of the square root type: Ensemble Kalman filter (EnKF), Reduced Rank
Square Root filter (RRSQRT), SSQRT, RRTKF, SEIK, POENKF, COFFEE, ... .
In the presentation we will first formulate the general data assimilation
problem and will discuss a number of square root filter algorithms. For a
class of algorithms we will present a convergence theorem. The
characteristics and performance of the methods will be illustrated with a
number of real life data assimilation applications in ground water flow,
reservoir engineering, ocean dynamics and air pollution.

March 24
Delft

Prof. Sjoerd Verduyn Lunel (Universiteit Leiden):
Mixedtype functional differential equations, holomorphic
factorization and applications.
Abstract:
In this talk we introduce and motivate the mixedtype
functional differential equation
x'(t) = ax(t) + bx(t1) + cx(t+1)
defined on the real axis. Such equations arise naturally in
various contexts, for example, in the study of travelling waves
in discrete spatial media such as lattices.
Since this mixedtype equation is not an initial value problem,
it is our goal to decompose solutions of this equation as sums
of "forward" solutions and "backward" solutions. We show that
the set of all forward solutions defines a semigroup which can
be realized by a retarded functional differential equation except
for possibly finitely many modes, and similarly for the set of
backward solutions as an advanced functional differential equation.
Holomorphic factorizations play a crucial role in our results.
Finally, we study the boundary value problem on intervals of
long but finite length in the spirit of the finite section
method.

March 17
Leiden

Dr.ir. Kees Oosterlee (Technische Universiteit Delft):
Evaluation of European and American options with grid stretching and
accurate discretization.
Abstract:
In this talk, we present several numerical issues, that we currently
pursue, related to accurate approximation of option prices. Next to the
numerical solution of the BlackScholes equation by means of accurate
finite differences and an analytic coordinate transformation, we present
results for options under the Variance Gamma Process with a grid
transformation. The techniques are evaluated for European and American
options.

February 17 Leiden

Dr. Klaas Pieter Hart (Technische Universiteit Delft):
Embeddability of the measure algebra
Abstract:
The purpose of this talk is to compare the Boolean algebra
P(N)/fin and the Measure algebra, i.e., the
algebra of Borel sets of the real line modulo the ideal of Lebesgue
null sets.
The first algebra is one of the most widely studied by Boolean algebraists
and general topologists alike.
The second algebra is wellknown among analysts and
topologists have used its Stone space in the construction of
(counter)examples.
I shall discuss just one question: which of the two is embeddable
into the other.
It is relatively easy to see (but surprising to some) that the first
is not embeddable into the second.
The Measure algebra sometimes can and sometimes cannot be embedded into
P(N)/fin.
We shall see what this means and how one goes about proving it.

February 10 Delft

Prof. Sara van de Geer (Leiden):
Adaptive learning
Abstract:
Let (X,Y) be random variables, with X subset of X
a feature and
Y a label. We study the problem of predicting Y
given X.
We restrict ourselves to the case where Y takes
only two values, say in {±1}.
An example is the case where X represents features
of a mushroom (size, shape, color) and Y
indicates whether it is edible or not.
A classifier is a subset G of X.
It predicts the label 1 if X is in G and else the label
1. Our aim is to find a classifier that
produces small prediction error. The optimal classifier is
Bayes rule, which is to predict the most likely label
given X.
However, the distribution of (X,Y) is generally not
at all known.
A strategy is learned using a training set
(X_{1},Y_{1})
,...,(X_{n},
Y_{n})
of independent copies of (X,Y).
In this talk, we will consider support vector
machines (SVM's). Roughly speaking, the idea is to
find the classifier which minimizes the
number of errors in the training data among a certain collection
of classifiers G. But this is generally computationally
infeasible. As convex relaxation one instead minimizes
the hinge loss (1Y_{1}
f(X_{1}))_{+}+...+
(1Y_{n}
f(X_{n}))_{+}
over a collection F of realvalued functions.
Denoting the minimizer by ff_{n}, we use
{ff_{n} ≥0} as classifier.
We will show that by applying a particular complexity regularization
method, closely related to soft thresholding,
the SVM procedure can be adaptive.
For example, when some variables in X are irrelevant for
the prediction, the SVM will behave as if it knew this a priori.
Similarly, if the boundary of Bayes classifier is smooth,
SVM will mimic this.
To prove such behavior, the main effort lies in handling the
random part of the problem: the socalled estimation
error. Here, we use some nice probabilistic tools:
symmetrization, contraction and concentration of measure.



Lectures in 2004:
December 16 
Dr. Vivi Rottschäfer (Universiteit Leiden):
Blowup solutions of the Nonlinear Schrödinger equation
Abstract:
In my talk, I will study the cubic Nonlinear
Schrödinger equation (NLS), this equation arises as a model equation
in a variety of problems coming from physics, biology and
chemistry. Numerical simulations show that there exist solutions
of the NLS that become infinite in a finite time, hence these
solutions blow up. Blowup solutions for the NLS have been studied
extensively via numerical methods and asymptotic analysis
including the socalled multibump solutions. Building on these results
we proved the existence of these multibump blowup solutions
under certain conditions. I will give an overview of the known results
for these solutions starting with the numerics.

December 2 
Prof. Frans Oort (Universiteit Utrecht):
Conjectures in Mathematics
Abstract:
In my talk I will discuss the stimulating aspects of conjectures
in modern mathematics. Examples of questions and conjectures in
number theory and algebraic geometry will be discussed in order
to illustrate the concepts mentioned. I will present and propose
what criteria could be given for a question to be called a conjecture.
Most material will be presented in a way accessible for a general
mathematical audience. Although part of my lecture is "about
mathematics" there will be enough mathematical contents to
satisfy those of you who also want to see proofs and facts,
beautiful structures, expectations, and problems to work on.

November 4
GORLAEUS, C2
4:005:00 pm

Prof. Willem van Zwet (Universiteit Leiden): Statistics and the law: the case of nurse Lucia de B
Abstract:
In a hospital in The Hague a number of unexpected cases of death or
reanimation of patients occurred.
When it was found that in all of these cases
a nurse named Lucia de B was present and caring for these patients,
she was arrested and tried for multiple murder.
The case for the prosecution rested on toxicological evidence as well as
a statistical analysis showing that her presence in all cases could not be
attributed to chance.
On the basis of this Lucia was sentenced to life imprisonment.
The case was appealed by the defendant and considered by the Appellate
Court (Gerechtshof) in The Hague.
The defense now produced another expert witness who claimed that the
statistical evidence presented earlier was unconvincing.
In an interview in the newspaper NRCHandelsblad a third expert
went quite a bit farther and claimed that the statistical analysis
was completely
wrong and when performed correctly, should have lead to a verdict of not guilty.
In the resulting confusion the court made it clear that it could not credit
any of the statistical arguments anymore and confirmed the life sentence
without mentioning the word statistics at all,
though clearly still impressed by the unlikely presence of the defendant
in all cases.
It seems that the various experts have succeeded only in confirming the wellknown
distrust of statistical arguments that goes back to Disraeli's dictum
that there are lies, damned lies and statistics.
In this talk I'll explain that if the ordinary rules for statistical
consulting had been followed, it would have been quite clear what statistics
could and could not contribute to this case.

September 30 
Prof. Ronald Cramer (CWI Amsterdam/Universiteit Leiden):
Informationtheoretically secure cryptography
Abstract:
The security of cryptographic methods often relies on assumptions
about the computational hardness of certain mathematical problems. For
example, the RSA cryptosystem relies on the difficulty of factoring large
integers. Interestingly, there exist several meaningful tasks that can
be performed reliably and privately and yet are not based on
computational infeasibility.
Multiparty computation, which involves a network of collaborating
processors whose goal it is to evaluate a given function on secret
preimages but without revealing the function value, can be achieved
securely under the sole assumption that at most a certain quorum of
the processors are under the control of a malicious adversary. As
another example, consider a scenario where a sender and a receiver are
connected by n independent communication channels.
Even if a malicious
adversary corrupts a minority of the channels and subsequently
eavesdrops on their traffic and modifies it in any way he pleases, it
is still possible for the sender to securely pass messages to the
receiver. This means that, except with very small probability, the
receiver correctly recovers the original messages, whereas the
adversary has no Shannon information about these messages.
This talk addresses solutions that (nearly) minimize the required
traffic on each channel.
In order to securely pass an lbits message,
the data sent on each channel
amounts to approximately l + k bits.
The error probability is 2^{k}.
This works by an interplay of
algebraic error correcting techniques and combinatorial authentication
codes.

March 25 
Dr. Robbert J. Fokkink (Technische Universiteit Delft):
Egbert van Kampen
Abstract:
During his short life the Dutch mathematician
Egbert van Kampen produced
some results that are still of interest today, but his name is sadly
missing from the mathematical history books. In an attempt to set that
right, the aim of this talk is to shed some light on the life and work of
one of our most prolific mathematicians, touching upon various aspects of
nationalistic interest.

March 11 
Dr. Pieter Moree (Universiteit van Amsterdam, Max Planck Institut für Mathematik, Bonn):
The hexagonal versus the square lattice
Abstract:
SchmutzSchaller formulated in 1995 a conjecture concerning lattices
of dimensions 2 to 8 and proved its analogue in hyperbolic
geometry. As a particular case he mentioned that the hexagonal
lattice ought to be 'better' than the square lattice. This statement
is equivalent with the statement that for every x
the number of integers
n smaller or equal than x
that can be written as a sum of two squares is not less than the number
of integers m smaller or equal than x
that can be written as a sum of a square and three times
a square.
Together with Herman te Riele (CWI, Amsterdam) I recently proved this
by methods from computational number theory and the asymptotic theory of
arithmetic functions.
As a by product I disproved some claims on the divisibility of the
taufunction Ramanujan made in his unpublished intriguing manuscript on the
partition and tau function (two famous functions in number theory).

February 19 
Dr. Peter Grünwald (CWI, Amsterdam):
Two Theories of Information: Shannon & Kolmogorov
Abstract:
We introduce, compare and contrast the theories of Shannon information
and Kolmogorov complexity.
We investigate the extent to which these theories have a common purpose
and where they are fundamentally different. We discuss the fundamental
relations 'entropy = expected Kolmogorov complexity' and 'Shannon mutual
information = expected algorithmic information'. We show how 'universal
coding/modeling' (a central idea in practical data compression) may be
viewed as a middle ground between the two theories, and how it leads to
the 'minimum description length principle', a practically useable theory
for statistical inference with arbitrarily complex models.
(joint work with P.M.B. Vitányi)

January 29 
Dr. Wim Couwenberg (Universiteit van Amsterdam):
Reflection arrangements and ball quotients.
Abstract:
In 1986 Deligne and Mostow showed that under certain simple conditions the
moduli space of weighted ntuples of points on
P^{1}
is a
quotient of a complex ball by a discrete group of automorphisms. The
covering map is described in terms of Lauricella hypergeometric functions
F_{D}
on the geometric quotient
(P^{1})^{n}//SL(2,C).
This envelops (for n=4) work by Schwarz on the Gauss
hypergeometric function. In this talk I will discuss a further
generalisation in terms of weights on arrangements of finite (complex)
reflection groups. The DeligneMostow results now correspond to
arrangements of type A_{n}.
For certain weights we find a projective
variety (just like the geometric quotient)
that is again a ball quotient.
Presently, only a few of those are identified as moduli spaces.



Lectures in 2003:
December 4 
Dr. Jaap Top (Rijksuniversiteit Groningen):
Quartic curves over a finite field.
Abstract:
This lecture is about the "why" and "how" of a very elementary question:
how many (or how few) rational points can a plane curve
such as in the title have?

October 30 
Dr. Jan Hogendijk (Universiteit Utrecht):
The determination of π and sin 1^{o}
in ca. 16 decimals by
AlKashi (ca. 1420).
Abstract:
In the 1410s, the Iranian mathematician AlKashi
determined π and sin 1^{o}
far more accurately than any
of his (Greek and Islamic) predecessors. AlKashi's
pidetermination is extant; his
treatise on sin 1^{o}
is lost, but survives in mutilated form in
a commentary by QadiZadeh alRumi
(died 1437).
These two mathematicians were colleagues with a somehat complicated
relationship at the court of Ulugh Beg in
Samarkand. I will discuss new information on AlKashi's
determination of π and sin 1^{o}
which can be
derived from Arabic manuscripts of the two treatises
that have recently become available.
Some of the results are joint work with Boris A. Rosenfeld.

October 9 
Dr. Derk Pik (Universiteit Leiden):
Minimal representations of a contractive operator as a product of two
bounded linear operators.
Abstract:
Given a contraction K acting between Hilbert spaces U
and Y we
present a classification of al pairs of bounded linear operators
B: U > X and C: X > Y (where X is an auxiliary
Hilbert space) such that K = C B.

September 18 
Prof. Joe P. Buhler (Reed College, Portland, Oregon):
A problem in symmetric functions arising from phase
determination in crystallography.
Abstract:
The determination of the structure of a crystal by Xray diffraction
requires that the complex Fourier transform be estimated merely
from data about its absolute value, together with natural physical
constraints on the crystal structure. In practice, stochastic
techniques are used, and these techniques are sometimes
successful even when the number of atoms in the unit cell
of a crystal is on the order of several thousands.
Motivated by a question raised by the crystallographer
Herbert Hauptman, we consider an exact algebraic version
of the phase determination problem. This gives rise to a
question in symmetric functions of two sets of variables
whose solution leads to further interesting algorithmic
and mathematical problems.
These results are joint work with Zinovy Reichstein.

May 22 
Dr. F. Merkl (Universiteit Leiden):
Statistics of Riemann zeta zeros.
Abstract:
The Riemann hypothesis conjectures that all nontrivial
zeros of the Riemann zeta function have real part 1/2.
Assuming this hypothesis, one may ask more refined questions
about the empirical local distribution of these zeros.
There is strong numerical and theoretical evidence  but still no full
proof  that the zero distribution, appropriately rescaled, is described
by the local eigenvalue distribution of certain random matrices.
In the talk, I will review some rigorously known facts on the
distribution of Riemann zeta zeros, proven by Montgomery,
Hejhal, Rudnick, Sarnak, and others. Furthermore, I will report about
recent examinations of the distribution of Riemann zeros on other scales
than the "typical" distance between neighboring zeros.

April 17 
Prof.dr. P. Stevenhagen (Universiteit Leiden):
Primes is in P
Abstract:
In August 2002, the Indian computer scientists Agrawal, Kayal and Saxena
proved that primality of an integer can be tested by means of a deterministic
algorithm that runs in polynomial time. For several decades, this had been
an outstanding problem. We discuss the importance of the result in theory
and in practice, and give an impression of the mathematics that goes
into it.

March 13 
Dr. H. Matzinger (Universität Bielefeld):
Some Ideas in Scenery Reconstruction
Abstract:
We present a few ideas used in scenery reconstruction. A
scenery is a coloring of the integers, that is a map
ξ from Z to a finite set {0,1,...,C1}.
Let {S(k)}=(S(0),S(1),S(2),...)
be a recurrent random walk on Z
starting at the origin.
We will denote by X the color record
obtained by observing the
scenery ξ along the path of the random walk {S(k)},
X := (ξ(S(0)),ξ(S(1)),ξ(S(2)),... ).
The scenery reconstruction problem can be described as follows:
given an unknown
scenery ξ, can we "reconstruct" ξ if we can only observe
X?
Does one path realization of the process
{X(k)}=(X(0),X(1),...) a.s.
uniquely determine ξ? Not every scenery can be reconstructed but a lot
of typical sceneries can.
For this we take the scenery to be random itself
and show that with the right measure
almost every scenery can be reconstructed up to shift and reflection.
How to solve such a reconstruction problem
depends a lot on the specific properties of the random walk
and the distribution of the scenery.

February 27 
Dr. J. van den Berg (CWI, Amsterdam):
Percolation and related phenomena.
Abstract:
Suppose we have an n x n square grid,
and interpret the line segments
between adjacent nodes as streets. If each street is randomly blocked
with probability p (independent of the other streets), is there, when
n
is large, still a reasonable probability that we can cross the grid from
the left to the right (or, for instance, from the southwest to the
northeast corner)? It turns out that the answer is yes if p
is above, and
no if p is below, a socalled critical value.
This is a typical example of
a percolation problem.
Percolation plays an important role in Statistical Physics,
the theory of Interacting Particle Systems and other fields. I will show various
examples and briefly discuss some modern developments.

January 23 
Prof.dr. S.J. Edixhoven (Universiteit Leiden):
Counting solutions of systems of equations over finite fields.
Abstract:
I will first explain what the problem means
(i.e., what is a finite field,
what kind of equations do we consider),
give some examples, and explain
how it is related to cryptography.
Then I will discuss the currently known
algorithms and their complexity and limitations.
Finally, I will describe
my research plan for the next years concerning
the problem of getting rid
of at least one limitation:
that the characteristic of the field should be
small.



Lectures in 2002:
December 12 
Dr. A.J. Lenstra (Universiteit van Amsterdam, EURANDOM Eindhoven):
On Information bounds.
Abstract:
In empirical sciences, the question at hand is often
not a question regarding the available data themselves,
but a question regarding the (at least partially)
random mechanism that, one thinks, produced these data.
In such situations statistical procedures are invoked
to extract an answer from the data. Information bounds
are bounds for the precision with which this can be
done; as such, they provide optimality criteria for
statistical procedures.
The best known classical information bound is the
CramérRao inequality. It has a history of eighty
years, but the customary proofs do little to reconcile
us to its truth. We present a viewpoint from which it
is as obvious as the observation that in a rightangled
triangle the hypotenuse is the longest side. Another
famous bound, the van Trees inequality, then follows
from Pythagoras' theorem.

Wednesday December 4
Gorlaeus, C1
4:005:00 pm

In cooperation with the Seminar on Recreational Mathematics
organized by Chris Zaal
Prof.dr. H.W. Lenstra (Universiteit Leiden, University of California
at Berkeley, USA):
Escher and the Droste effect.
Abstract: Many newspapers reported recently that a team of Leiden
mathematicians worked for two years on completing Escher's
lithograph `Print Gallery' from 1956. However, not much
publicity was given to the techniques from algebraic
topology and complex function theory by which this purpose
was achieved. In the present lecture, both the mathematical
methodology and the pictures produced by it will receive
ample attention.

November 14 
Prof.dr. T.P. Hill (Georgia Tech, Atlanta USA):
The Significantdigit Phenomenon, or Benford's Law.
Abstract:
A centuryold empirical observation now called Benford's Law says that
the significant digits of many real datasets are logarithmically distributed,
rather than uniformly distributed, as might be expected. This talk will
briefly survey some of the colorful history of the problem, including recent
proofs and applications to fraud detection, and will then describe some
very new discoveries that Benford sequences are typical in many
deterministic sequences such as 1dimensional dynamical systems and
differential equations. For example, the orbit of iterates of almost every
rational function obeys Benford's Law. A number of open Benfordrelated
problems in dynamical systems, probability, number theory, and differential
equations will be mentioned.

October 31 
Prof.dr. S.M. Verduyn Lunel (Universiteit Leiden):
Calculating Hausdorff dimensions of invariant sets using spectral theory.
Abstract:
The dimension of an invariant set of a dynamical system is
one of the most important characteristics. In this talk we
present a new approach to compute the Hausdorff dimension of
conformally selfsimilar invariant sets. The approach is based
on a direct spectral analysis of the transfer operator associated
with the dynamical system. In the case that the maps defining the
dynamical system are analytic, our method yields a sequence of
successive approximations that converge to the Hausdorff dimension
of the invariant set at a superexponential rate. This allows us
to estimate the dimension very precisely. We illustrate our
approach with examples from dynamical systems and from number
theory via Diophantine approximations.

October 10 
Dr. F. Wiedijk (NIII, Katholieke Universiteit Nijmegen):
Formalizing mathematics in the computer.
Abstract:
I will present "formal mathematics". This field studies
how to verify the correctness of mathematical theories
with the computer. I will discuss the reliability and the
feasibility of this endeavor. At the end of the talk I
will show a small formalization in the Mizar proof system.

April 25 
Dr. M. de Jeu (Universiteit van Amsterdam):
Subspaces with equal closure
Abstract:
We take a unifying and radically new approach towards polynomial and
trigonometric approximation in an arbitrary number of variables and in
arbitrary topological vector spaces. The key idea is to show in
considerable generality that a module, which is generated over the
polynomials or trigonometric functions by some subset of a topological
vector space, necessarily has the same closure as the module which is
generated by this same subset, but now over the compactly supported smooth
functions. The particular properties of the ambient topological vector
space or of the generating subset are to a large degree irrelevant. This a
priori translation  which goes in fact beyond modules  simplifies
matters considerably and allows one, by what is now essentially a
straightforward check of a few properties, to replace many classical
results by more general and stronger statements of a hitherto unknown
type.
The method ultimately rests on harmonic analysis and can be formulated for
Lie groups in general. The underlying conceptual framework then shows that
many classical approximation theorems are "actually" theorems on the
unitary dual of ndimensional real space. Polynomials then correspond to
the universal enveloping algebra and trigonometric functions correspond to
the group algebra.



Lectures in 2001:
November 29 
Prof. dr. P. Groeneboom (Delft University of Technology,
Vrije Universiteit Amsterdam):
Ulam's problem and Hammersley's process
Abstract:
Let L_{n}
be the length of the longest subsequence of a random permutation
of the numbers 1,...,n for the uniform distribution on the set of
permutations. Hammersley's interacting particle process, implicit in
[4], has been used in Aldous and Diaconis [1] to provide a "soft"
hydrodynamical argument for proving that
lim_{n→∞}
n^{1/2}EL_{n}
=2.
I will show that the latter result
is in fact an immediate consequence of properties of a random 2dimensional signed
measure, associated with Hammersley's process.
References:
[1] Aldous, D. and Diaconis, P. (1995). Hammersley's interacting
particle process and longest increasing subsequences.
Probab. Th. Relat. Fields 103, 199213.
[2] Deift, P. (2000). Integrable systems and combinatorial theory.
Notices of the AMS 47, 631640.
[3] Groeneboom, P. (2001). Ulam's problem and Hammersley's process.
Annals of Probability 29, 683690.
[4] Hammersley, J.M. (1972). A few seedlings of research.
In: Proc. 6th Berkeley Symp. Math. Statist. and
Probability, vol. 1, 345394.

November 15 
Prof. dr. F. Beukers (Universiteit Utrecht):
Spectral problems in Number Theory
Abstract:
Consider an ordinary linear differential equation whose
coefficients are polynomials in Z[x].
Consider a Taylor series solution of this
equation with rational coefficients. The question for which differential
equations these coefficients are arithmetically wellbehaved is
one that has arisen in algebraic geometry, padic differential
equations and recently in studies on mirrorsymmetry. In this
lecture we deal with a particular case of differential equations.

October 18 
Prof. dr. R.J. Schoof (Universitá di Roma
"Tor Vergata"):
Class numbers of cyclotomic fields
Abstract: No good algorithms are known to compute class
numbers of cyclotomic fields with large conductor.
In this lecture we describe
an experimental approach to this problem.

September 27 
Dr. M. Fiocco (Universiteit Leiden):
Statistics for the contact process
Abstract:
A ddimensional contact process is a simplified model for
the spread of an infection on the lattice
Z^{d}.
At any given
time t > =0, certain sites x in
Z^{d}
are infected while the
remaining ones are healthy.
Infected sites recover at constant rate 1, while
healthy sites are infected at a rate proportional to the number
of infected neighboring sites. The model is parametrized
by the proportionality constant $\lambda$. If $\lambda$ is
sufficiently small, infection dies out (subcritical
process), whereas if $\lambda$ is
sufficiently large infection tends to be permanent (supercritical process).
In this talk we present an estimation
problem for the parameter $\lambda$ of the supercritical contact process
starting with a single infected site at the origin.
Based on an observation of this process at a single time t,
we obtain an
estimator for the parameter $\lambda$ which is consistent
and asymptotically normal as t tends to infinity.

May 17 
Prof. dr. R.D. Gill (Univ. Utrecht):
Teleportation into Quantum Statistics
Abstract:
Using the example of the Delft Qubit, or Schrodinger SQUID
(Mooij et al, Science 1999), I sketch the
basis ingredients of quantum measurement theory: states, evolutions,
entanglement, and measurement. I illustrate these by the example of
quantum teleportation. In order to prove that experimenters really did
do, what they claim, one must repeatedly measure and estimate quantum
states. Then the question arises, what is the best experimental design,
and an important issue is whether joint measurement of many quantum systems
together is advantageous compared to separate measurements on separate systems.
I give some recent results on this, connecting to the recently discovered
phenomenon (in the fields of quantum information, quantum computation) of
"nonlocality without entanglement".
An extended abstract is available at
http://www.math.uu.nl/people/gill/Preprints/abstract_gill.pdf.

April 26 
Dr. E. Belitser (Univ. Leiden):
Exact asymptotics in nonparametric estimation
Abstract:
In nonparametric estimation, contrary to parametric, the quantity to be
estimated is no longer a finite dimensional vector, but an infinite
dimensional object  density, regression function, distribution
function, image, to name a few. We discuss the basic nonparametric
minimax estimation problem and consider several models (regression,
white
noise model, random censorship model, blurred signal) where it has been
possible to derive the exact asymptotic behaviour of the minimax risk.

March 29 
Prof. dr. G.J. Heckman (KU Nijmegen):
Hypergeometric functions and some moduli spaces
Abstract:
In a number of exceptional (but not irrelevant)
cases a moduli space of algebrogeometric objects of some
kind can be identified via a period map with a ball quotient.
During the lecture three examples will be discussed:
1. 4 points on a line
2. 12 points on a line
3. pencils of cubics in the plane
The first example is classical 19th century mathematics, the
second example is due to Deligne and Mostow (1986), and the
last example is recent joint work with E. Looijenga. 
February 15 
Dr. W.H. Hundsdorfer (CWI, Amsterdam):
Implicit and Explicit Time Stepping for Convection Problems
Abstract:
Numerical discretization for convectiondiffusion problems
with dominating convection remains one of the major problems
in numerical analysis. In this talk we shall mainly consider
time discretization for such problems by implicit and explicit
methods and by suitable combinations.
A fully implicit treatment of convection terms is often not
very efficient. This is mainly due to the fact that monotonicity
properties of implicit time stepping methods are in general comparable
to those of their explicit counterparts (unlike stability properties).
On the other hand, in spatial regions where the solution is smooth and
the convective velocity is large, implicit methods seem preferable.
We shall consider combined methods that are implicit only locally
in space. Test results are presented for some simple twophase flow
problems.



Lectures in 2000:
October 19 
Dr. N. Bruin (Clay Mathematics Institute, MSRI, Berkeley):
Solving generalised Fermat equations
Abstract:
In this talk we will explain a method that can be used to bound the number
of rational solutions of a bivariate polynomial or, more precisely, the
rational points on an algebraic curve of genus at least 2. We use this
method to solve some instances of the generalised Fermatequation.
The generalised Fermatequation is defined to be
x^{r}
+y^{s}
=z^{t}
where r,s,t are integers >1.
A solution in integers x,y,z is considered
to be trivial if xyz=0 and primitive if gcd(x,y,z)=1. While
we expect no primitive nontrivial solutions for big enough r,s,t, there
are some surprisingly big solutions, such as
43^{8}
+96222^{3}
=30043907^{2}.
In fact, using the method mentioned above, we can prove that for
{r,s,t}={2,3,8}, this is the biggest primitive solution and we can
list the others.
This lecture should appeal to anyone with a mathematical background.

September 14 
Prof. S. Angenent (University of Wisconsin, Madison):
The hole filling problem in the Porous Medium Equation
Abstract:
We describe the singularities of solutions to the porous medium equation
which fill a concavity. We will discuss similarities with a number of
other problems such as Mean Curvature Flow and the HeleShaw problem.

May 18 
Prof. J. Gonzalo (Universidad Autónoma de Madrid):
Isoperimetric problems
Abstract:
The statement of the isoperimetric problem is this:
"enclose a specified amount of volume using the least possible boundary area."
Already for Euclidean space it is a nontrivial problem, where the solution
is the round ball. The problem is even more interesting in Riemannian
manifolds, where the shape (and the topology) of the solution changes
as we change its volume. It is deeply connected with questions in Partial
Differential Equations
and Geometric Measure Theory.
I will review the subject and explain my results concerning the change
of shape in product spaces.

May 4 
Prof. dr. R. Tijdeman (Universiteit Leiden):
Sturmian sequences and words
Abstract:
Sturmian sequences were introduced by Morse and Hedlund around 1940 in the
context of Symbolic Dynamics. They are closely connected with the Beatty
sequences which occur in Number Theory, also known as cutting sequences.
The corresponding words play a role in Theoretical Computer Science
because of their low complexity, in Operations Research (because of
balancedness) and in Quasicrystallography (because of being almost
periodic. They form the class of nonperiodic sequences which are
nearest to periodic ones.
The survey provides the basic properties of Sturmian sequences,
some variants and some applications.

April 6 
Dr. M. Zieve (Universiteit Leiden):
Nice equations for nice groups
Abstract:
Every finite group occurs as the Galois group of a Galois extension
of K(t), where K is an algebraically closed field of positive
characteristic. More precisely, the Abhyankar Conjecture (proved by
Raynaud and Harbater) describes the Galois groups of Galois extensions
of K(t) which are only ramified at a prescribed set of points.
However, little is known about Galois extensions of k(t), where k
is a finite field. I will present work of Abhyankar and Elkies which
yields (by two completely different methods) explicit polynomials in
K(t)[x] having certain classical groups as Galois groups, and the
implications of their results for the Inverse Galois Problem/Abhyankar
Conjecture over k(t).

March 9 
Prof. dr. S.A. van de Geer (Universiteit Leiden):
Entropy and estimation
Abstract:
Consider the problem of estimating an object, a curve or
an image, in statistical terms: a parameter. If a priori
little is known about the parameter, it will in
general be hard to estimate it. We will quantify this by
relating the speed of estimation to the entropy of
parameter space.
The idea is best explained for the regression problem.
Suppose we recorded a signal with additive Gaussian noise,
at n discrete time instants. To recover the signal, let
us suppose it is smooth, say it is of bounded variation.
The rate of convergence of the least squares estimator is
then O(n^{1/3}).
This follows from the calculation of
the entropy of the set of all functions with total variation
bounded by some constant. A somewhat less general
description of the complexity of parameter space is in terms of
a smoothness index. The general theory then says:
suppose we know that the parameter is in
a class with smoothness index s (bounded variation is
s=1), then the rate of convergence is
O(n^{s/(2s+1)}).
For example, k times differentiable
functions of d variables have smoothness index s=k/d.
We will present some nice entropy results: the entropy
of convex spaces with "few" extreme points (bounded
variation being a special case), and the entropy of
Besov spaces. We also consider penalized estimation, with the
penalty being e.g. the total variation or a Besov norm.
By taking a soft thresholding penalty, we obtain
estimators that are (almost) adaptive, i.e.
(apart from some logarithmic factors) they attain the rate
O(n^{s/(2s+1)}),
without knowing a priori the smoothness index s.
Finally, as an illustration that the theory can be extended
to other nonparametric estimation problems, we briefly consider
the least absolute deviations estimator.

February 17 
Prof. dr. E.G.F. Thomas (Rijksuniversiteit Groningen):
Path distributions
Abstract:
An attempt at a rigorous theory of path integrals in which
summable distributions on the space R^{n},
regarded as a finite
dimensional space of `paths', forms the starting point.

January 20 
Prof. dr. T. Koornwinder (Univ. Amsterdam, Kortewegde Vries instituut):
On the work of 1998 Fields medal winner Richard Borcherds
Abstract:
The work of Richard Borcherds brings together a number of beautiful
and significant mathematical structures: automorphic forms, the Leech
lattice, the monster group, generalized KacMoody algebras, vertex
algebras. The theory of the last two concepts was developed by him. In
this work ideas from quantum field theory and string theory were
helpful, and conversely Borcherds' results further stimulate these
theories.
By combining the various tools Borcherds succeeded to prove the
socalled moonshine conjectures of Conway and Norton. These state that
the monster group has an infinite dimensional graded representation
such that the traces of elements of the monster on this representation
are given by certain Hauptmoduls for some genus zero subgroups of
SL(2,Z). The lecture will explain part of this work.



Lectures in 1999:
October 14

Prof. dr. R. van der Hout (AkzoNobel, Universiteit Leiden):
Singularities and nonuniqueness in cylindrical flow of nematic liquid crystals
Abstract:
This is joint work with E. Vilucchi (Roma II).
A nematic liquid crystal is a simple fluid, equipped with a
director field, which is a unit vector field, representing the local
direction of chain molecules. Associated with this field is an energy
density, and the problem of determining the steady director field in a
given cylindrical flow may be formulated as a variational problem.
Minimizing sequences for the total energy are in principle unbounded in
an appropriate Hilberts space. In the case without flow, this
unboundedness gives rise to smooth solutions (harmonic mappings in this
case) with, dependent on the boundary condition, a discrete
finiteenergy singularity on the axis of the cylinder. The level of
energy, stored in the singularity, determines the homotopy type of the
solution. For a given boundary condition, there are at most two
"admissible" homotopy classes, and in each class the solution is
unique.
In the case where flow is present, the questions whether such
singularities do indeed occur, and whether solutions in a given
homotopy class are unique, have been left open up till quite recently.
We shall prove that the first question can be answered affirmatively,
and the second question not: we give an example of three solutions in
the same homotopy class.

September 16

Dr. V. Berthé (Univ. Luminy, France):
Sequences with low complexity function
Abstract:
One can classically measure the ``disorder'' of a sequence with finitely
many values, by introducing its complexity function: this function
counts the number of factors of given length. The aim
of this lecture is to exhibit, through the study
of sequences of low complexity function, connections
between Combinatorics on words, Ergodic Theory, Arithmetic,
Diophantine approximation, and geometry of tilings.
We will first recall some classical properties of low complexity sequences.
We will
focus on
a remarkable family of sequences,
the socalled Sturmian sequences:
these are the sequences with minimal complexity function among
nonperiodic sequences.
We will then see how to extend these results to twodimensional sequences.

May 6

Prof. dr. H.W. Lenstra (Univ. Leiden, Univ. California, Berkeley):
Exceptional polynomials
Abstract:
The subject of this lecture was originally suggested
by the study of permutation polynomials over finite fields.
We will discuss some recent results and open problems.
Further, we will go into the connection with the theory
of finite groups.



Lectures in 1998:
December 10

Prof. dr. R. Dijkgraaf (Universiteit van Amsterdam):
On the work of Maxim Kontsevich
Abstract:
This year Maxim Kontsevich was awarded the Fields Medal. I will review
his work, that ranges from moduli of algebraic curves, knots and
threefolds, enumerative geometry, to deformation quantization. In all
of this the influence of ideas from theoretical physics has been
crucial.

November 12

Dr. B. Gaujal (Inria, France):
Regular sequences in high dimension
Abstract:
In the first part of this talk we concentrate on binary sequences.
We define the notion of regular binary sequences and give several
characterizations of such sequences.
In the second part, we present several extensions to higher dimensions
and present the Fraenkel Conjecture.
We also define an order of regularity that formalizes the fact that one
sequence is more regular than another.
Applications to optimal control problems will be mentioned
in the last part of the talk.

October 22

Dr. J. Melissen (Hogeschool 'sHertogenbosch):
Kepler's conjecture: close packings of
round things
Abstract:
Last August Thomas Hales announced his proof of Kepler's conjecture.
This very old conjecture (part of Hilbert's 18th problem) states that
spheres in three dimensions cannot be packed more densely than the
obviously densest packing. In spite of the obviousness of the result,
the fact that the conjecture has rested unproven for almost 400
years just gives some indication of the magnitude of the problem.
Indeed, it took about 250 pages of proof in combination with several
Gigabytes of computer code and data to complete this Herculean quest.
In this lecture I will put Kepler's conjecture in a historical
perspective and I will treat a selection from the plethora of related
interesting results from discrete geometry.

April 23

Prof. D.W. Masser (Univ. Basel):
Equalities and inequalities for elliptic functions
Abstract:
The equalities of the title are numbertheoretic in nature. In 1975
Brownawell and I proved (independently) that if p(z) is a Weierstrass
elliptic function with complex multiplication then under a suitable normalization
there are two essentially distinct linear relations with algebraic coefficients
between the periods and quasiperiods. One of these coefficients seems to be
slightly unpredictable; for example, if the
period quotient of p(z) is 2×(2)^{½}
this coefficient is equal to
6(2)^{½}(343 + 225×2^{½})/4991
when the invariants are normalized to be equal.
I discuss how this and other examples were recently calculated.
The inequalities of the title are analytic in nature. In 1990 Wüstholz
and I proved the existence of an absolute constant C such that
p'(z)<C.d(z)
for the derivative of any Weierstrass function, where d(z) is
the distance from z to the nearest period.
I discuss my recent proof that the sharp
value of C is 5.513701577... .

February 26

Dr. G. Koole (Vrije Universiteit):
Manpower scheduling in call centers
Abstract:
Call centers form a fast growing industry with many interesting
optimization problems. In this talk we focus on the optimal
scheduling of parttime employees in a call center with one type
of arriving calls.
The objective is to minimize the number of scheduled employees
under an overall service level constraint.
This overall service level is a weighted average of
the service level of each quarter of an hour, which on its turn
depends on the number of scheduled employees and the estimated
number of arriving calls.
A local search technique is presented, for which we can show that
it leads to the optimal schedule. The results are illustrated
with numerical experiments.



Lectures in 1997:
May 1

Prof. dr. T.A. Springer (Universiteit Utrecht):
Some results on complex reflection groups
Abstract:
A complex reflection group is a finite group of invertible
linear maps of a complex vector space C^{n} which is generated by
reflections, i.e. maps with n1 eigenvalues 1.
In the talk I shall discuss some old and new results about eigenvalues of
elements of such groups. The new results are joint work with G. Lehrer.

April 17

Dr. B. de Pagter (Technische Universiteit Delft):
Unconditional Decompositions and their Applications
Abstract:
The notion of unconditional decomposition (and random unconditional
decomposition) has turned out to be of importance for a number of
results in analysis and operator theory.
In the first part of this talk the main concepts concerning unconditional
convergence and decompositions in infinite dimensional spaces will be
introduced and illustrated with some classical examples.
In the second part some applications will be given to certain problems
in operator theory, in particular concerning Schur multipliers and
commutator estimates (for matrices and operators in infinite dimensional
spaces).

March 20

Prof. dr. C.A.J. Klaassen (Universiteit van Amsterdam):
BonusMalus in Quality Inspection (Not all that glitters is gold)
Abstract:
Our BonusMalus system is completely different from classical approaches
to quality inspection and based on the concept of credit. The credit of a
producer is defined as the total number of items accepted since the last
rejection. In this BonusMalus system the sample size for a lot will depend
on the lot size and on the credit of the producer. We will sketch the
simple, basic ideas underlying this approach, which resulted from a consultancy
project for the Waarborg Platina, Goud en Zilver N.V. at Gouda. 
February 20

Dr. R.W. van der Waall (Universiteit van Amsterdam):
Modular Frobenius Groups and Homogeneous Character Induction
Abstract:
The representation theory of finite groups was initiated by Georg Frobenius
a century ago. His socalled Frobenius groups satisfy nice properties in this
respect, when working over characteristiczerorepresentation theory.
To give an idea, let G be a finite group
with proper nontrivial subgroup H.
Assume that the intersection of any two conjugates of H is trivial. Frobenius
proves by means of representation theory, that (G minus the settheoretic
union of the conjugates of H) together with the unit element constitutes a
nontrivial normal subgroup of G.
An elementary proof of this fact is unknown.
About ten to five years ago Modular Frobenius Groups were described in
analogy to the classical Frobenius groups, and their investigation is still
going on.
It is our aim to present a survey of results thereoff to a general
mathematical audience, starting around 1895 and ending in 1996.

January 23

Prof. dr. A. Blokhuis (Vrije Universiteit, TU Eindhoven):
Polynomials in Combinatorics and Finite Geometry
Abstract:
It is illustrated how elementary properties of polynomials
can be used to attack extremal problems in finite and
Euclidean geometry and in combinatorics.



Lectures in 1996:
November 7

Prof. dr. M. van der Put (Rijksuniversiteit Groningen):
Differential equations in characteristic 0 and
characteristic p
Abstract:
An ordinary linear differential equation with coefficients in the field
Q(x) can have Liouvillian solutions, i.e. solutions which can be expressed
in rational functions by means of exponentials, algebraic equations and logarithms.
It is not so easy to find those expressions. One can reduce the equation modulo
a suitable rational prime p. The reduced equation is much easier to solve and this
leads to a guess for the possible solutions in characteristic 0.
On the background of this is a conjecture of Grothendieck. We will give some
examples which will lead to some problems in elementary number theory. 
October 10

Prof. dr. J.J. Duistermaat (Universiteit Utrecht):
Counting eigenvalues of higher
order SturmLiouville problems
Abstract:
Some time ago, Boris Levit asked me
about the eigenvalues of the operator d^{2m}/dx^{2m}
on a real interval, with Neumanntype boundary conditions.
It turned out that for this special case an
asymptotic expansion of the eigenvalues can be
given with exponentially decreasing error term.
For very general SturmLiouville problems for
higher order operators one can give asymptotic
expansions of the eigenvalues which suffice
to give explicit short intervals far away on the
real axis, each of which contains exactly one
eigenvalue (or two, for a special subclass
of boundary conditions). However, the asymptotics
does not give any information on the number of
eignvalues which precede these far away intervals.
In this talk we propose to use topological intersection
theory for curves in a Grassmann manifold in order
to determine this number. This may even be useful
for the numerical determination of the small
eigenvalues.

May 9

Dr. J. Brinkhuis (Erasmus Universiteit Rotterdam):
On linear optimization
Abstract:
In practice many optimization problems are solved by linear programming. The
wellknown standard way to solve these combinatorial problems is by using a
combinatorial algorithm, called the simplexmethod. A new algorithm is being
developed which comes down to the calculation of a limit. This has become
widely known through a publication of Karmarkar in 1984.
Aim of this lecture is to give a geometrical treatment of the basic principles
of this new algorithm.

April 18

Dr. B.M.M. de Weger (Erasmus Universiteit Rotterdam, Rijksuniversiteit Leiden):
Equal binomial coefficients
Abstract:
In the Pascal Triangle, many numbers occur several times. To a large
extent this is due to trivialities, but nontrivial solutions to
C(n,k)=C(m,l) (where C(a,b)=a(a1)...(ab+1)/b!)
do exist. We will formulate a precise
conjecture, and present partial results. If k and l are fixed, the
equation becomes a polynomial equation, which in some cases can be
solved completely. The only case known until recently was k=2, l=3.
Now we can add to this the cases k=2, l=4 and k=3, l=4. The
case k=2, l=4 is far from easy, and requires considerable efforts,
both theoretical and computational. In contrast the case k=3, l=4
is, surprisingly, a lot easier. We will try to explain for a
nonspecialist audience some of the ideas that go into these proofs.

March 7

Prof. dr. F. Takens (Rijksuniversiteit Groningen):
Chaotic dynamics in variations of the Hénon attractor
Abstract:
We consider dynamical systems which are given by a map φ: P→
P
 one thinks of P as the space of possible states, and of
φ as the transformation transforming the present state to the next
state (one unit of time later).
In our examples P will be finite dimensional
(mostly R^{2}), φ
differentiable and all evolutions bounded in the future in the
sense that {φ^{n}(p)}_{
n>0} is bounded (or has a compact closure)
for each p∈P. Such an evolution
{φ^{n}(p)}_{
n>0} is called
chaotic if it is not asymptotically stationary or (quasi)periodic  often
other, more complicated, definitions are used but that need not concern
us here.
We say that a dynamical system exhibits chaos if there is a set of
initial values with positive (Lebesgue) measure, such that the evolutions
starting in that set are all chaotic.
It is not easy to give dynamical systems, say with P equal to R^{n}
given by simple explicit maps for which one can prove that they exhibit
chaos (in a `persistent way'). On the other hand there are many examples of
systems which exhibit chaos, but for which we have no (simple explicit)
equation and there are also
dynamical systems with simple explicit maps for which numerical simulation
suggests that they exhibit chaos, but for which we have no proof.
One of the first (hard) examples where one has proved a simple map
to exhibit chaos was the Hénon example in dimension
2: for a set of parameter values (a,b) with positive measure,
(x,y)→(1  ax^{2} + y, bx)
exhibits chaos.
In the lecture I want to
explain how this example, and variations of it, can be used to show that
many other explicit examples also exhibit chaos.

February 22

Dr. R.W.J. Meester (Universiteit Utrecht):
Ergodic theory and dynamical systems in probability
Abstract:
It can be very useful to study stationary stochastic (spatial) processes from
the viewpoint of measurepreserving dynamical systems and ergodic theory.
I will explain the connections between probability and ergodic theory, starting
from scratch. This will be illustrated with one or two detailed examples
arising from random walks and percolation theory.

January 25

Prof. dr. S.J. van Strien (Universiteit van Amsterdam):
Topology, geometry and robustness of Julia sets
Abstract:
For a long time it was conjectured that the Julia
set of a polynomial could not have positive Lebesgue measure.
Recently a counterexample to this conjecture was given.
In this talk I want to discuss this and related results.



