
September 2 
Special program around PhD defense of
Gabor Wiese.
Location of the talks: Spectrumzaal of Studentencentrum Plexus,
Kaiserstraat 25, Leiden (
directions)

10:00−10:45

Gabor Wiese (Leiden),
Modular Forms of Weight One Over Finite Fields.
Abstract.
The talk having the same title as my thesis to be defended
in the afternoon will start by presenting the main
motivation for my research, namely (some of) the number
theoretic information really and conjecturally provided by
modular forms. Next, there will be a short overview over the
thesis. Finally, I will sketch the proof of one of the
results.

11:00−11:45

Loïc Merel (Paris),
The formula relating modular symbols to Lvalues.
Abstract.
There are well known formulas relating values of Lfunctions of
modular forms to modular symbols. These formulas enable to construct
padic Lfunctions etc. Modular symbols contain a finite
generating set consisting of the socalled Manin symbols. I will
describe how one can express a Manin symbol at level N in terms
of Lvalues obtained by twisting modular forms of level
N by characters of level dividing N and of a few local
invariants. Here is an elementary corollary of this formula : the
regular representation of
Gal(Q(μ_{N})/Q), where
Q(μ_{N}) is the field generated by a
primitive Nth root of unity, does not occur in the group of
Q(μ_{N})rational points of an elliptic curve
E over Q of conductor N.

14:15−15:00
 Thesis defense of Gabor Wiese at the Academiegebouw,
Rapenburg 73 in Leiden.


Special day on the ABCconjecture, September 9 2005
This is the kickoff meeting of an NWO sponsored "Leraar in Onderzoek"
project that will help Kennislink
to take ABC to the masses.
September 9 
Leiden, room 412.
 11:1512:00  Frits Beukers (Utrecht), Introduction to the
ABC conjecture
[PDF]
 12:1512:45  Jaap Top (Groningen), Finding good ABC triples,
part I; notes in Dutch (PDF)
 12:4514:00  Lunch
 14:0014:30  Johan Bosman (Leiden), Finding good ABC
triples, part II;
notes in Dutch (PDF)
 14:4515:30  Hendrik Lenstra (Leiden), Granville's
theorem;
notes in Dutch (PDF)
Abstract.
Barry Mazur defined the `power' of a number to be the logarithm
of the number to the base its radical. For example, every perfect
square has power at least 2. How many integers up to a large
bound have power at least a given number? This question is
answered by Granville's theorem. It is of importance both in
understanding why the ABCconjecture has a chance of being true,
and in analyzing an algorithm for enumerating ABCtriples.
 15:4516:15  Willem Jan Palenstijn (Leiden), Enumerating
ABC triples;
notes in Dutch (PDF)
Abstract.
An ABC triple is a triple of coprime positive integers a,
b, c with a + b = c and c
larger than the radical of abc. In this talk we present an
algorithm that enumerates all ABC triples with c smaller than a
given upper bound N with a runtime essentially linear in
N.
 16:3017:00  Carl Koppeschaar (Kennislink), Reken mee met
ABC
[PPT]


September 23

Utrecht room K11 Joint session with
Intercity Arithmetic Geometry

11:0011:45 
Bas Edixhoven,
Introduction to Serre's conjecture [PDF]
Abstract.
The conjecture will be stated, and put in its historical context and
in the wider context of the Langlands program. Serre's level and
weight of a 2dimensional mod p Galois representation will be defined.
Khare's result will be stated. An overview will be given of what will
be treated in the seminar.

12:0012:45 
Johan Bosman,
Galois representations associated to modular forms [PDF]
Abstract.
Modular forms, Hecke operators and eigenforms will be defined. The
existence of Galois representations associated to eigenforms will be
stated. The construction of these representations will be postponed
until later, in the more general case of Hilbert modular forms.
Something about weight 2 being easier and about weight 1 being special
can be said.

13:3015:15 
Gunther Cornelissen,
Remarks on a conjecture of Fontaine and Mazur [PS]
Abstract.
The following statement will be explained and proved:
Let K be a
number field, and S a finite set of places of K. Let
K_{S} be a maximal algebraic extension of K in which all
places in S are completely split. Let X be a smooth irreducible
quasiprojective scheme over K, such that for every v in
S the
set X(K_{v}) is nonempty. Then
X(K_{S}) is
Zariski dense in X.


October 7

Leiden, room 312.
Joint session with
Intercity Arithmetic Geometry

12:1513:00 
Sander Dahmen,
Lower bounds for discriminants,
Abstract.
Lower bounds for discriminants of number fields will be
roved, in terms of their degree only; see
Odlyzko

14:0014:45 
Frits Beukers,
Upper bounds for discriminants,
Abstract.
Let p be prime. Suppose that r is an irreducible
representation of the absolute Galois group of Q on a
2dimensional vector space over an algebraic closure of
F_{p}, with p in {2,3}. The image of such
a representation is finite, hence determines a number field K
that is Galois over Q. An upper bound for the discriminant of
K will be proved that contradicts the lower bound of the
previous lecture in the case that p is in {2,3} and r is
unramified outside p. The audience will draw the logical
conclusion. For p=2 this result is due to Tate, and for
p=3 it is due to Serre.

15:0015:45 
Bas Edixhoven,
Overview of Khare's proof
Abstract.
An overview will be given of Khare's proof of Serre's conjecture
in level one. Those who will not attend the rest of the seminar
will have an idea of Khare's proof, and it is hoped that those
who will attend the rest of the seminar will now be sufficiently
motivated to digest the more technical parts that are to come.

16:0016:45 
Johan de Jong,
Kloosterman lecture:
Rational points and rational connectivity
Abstract.
Recently, Graber, Harris and Starr proved that any family of
rationally connected projective varieties over a smooth curve has a
section. A complex projective variety (or manifold) M is rationally
connected when every two points in M lie on a rational curve in
M.
In this lecture we will explain a generalization of this
result, joint with Jason Starr, to the case when the base of the
family is a surface.
In the other three lectures (October 14, November 11 and
November 25) we will explain the idea behind the proof of the theorem,
the analogy with Tsen's theorem, the connection with weak
approximation and the connection with the periodindex problem for
Brauer groups.


October 21

Delft, Snijderszaal (1e verdieping laagbouw EWI), Mekelweg 4.

11:0011:45 
Klaas Pieter Hart (Delft),
Ultrafilters and combinatorics
Abstract.
In this talk I describe how ultrafilters may be used to prove
combinatorial
theorems about subsets of the set of natural numbers.
For example, an ultrafilter helps in making various choices in a
standard
proof of Ramsey's theorem.
For more intricate applications one needs to bring in some algebra.
One can extend ordinary addition to the whole set of ultrafilters and
thus obtain a compact semitopological semigroup.
An idempotent in this semigroup is instrumental in proving Hindman's
finitesum theorem and a special idempotent can be used to prove
Van der Waerden's theorem on arithmetic progressions.

12:0013:00 
Valerie Berthe (Montpellier),
Some applications of numeration systems
Abstract.
The aim of this talk is to survey several applications of numeration
systems, first in discrete geometry, and second, in cryptology. We
first detail some natural connections between the study of discrete
lines (and more generally discrete hyperplanes), and betaexpansions
and tiling theory. For the latter application to cryptology, we then
focus
on some redundant numeration systems in mixed bases.

14:0015:00 
Tom Ward (East Anglia), Heights and highly effective
Zsigmondy theorems
Abstract.
Elliptic divisibility sequences are known to eventually have
primitive divisors (indeed, the primitive part is very large).
This talk will describe some families of elliptic divisibility
sequences for which effective bounds can be given. For example,
every term beyond the fourth term in the Somos4 sequence
1,1,1,1,2,3,7,23,... is shown to have a primitive divisor.

15:1516:15 
JanHendrik Evertse (Leiden),
Linear equations with unknowns from a multiplicative group
Abstract.
Let G be a finitely generated multiplicative group contained in
the complex numbers, and let a, b be nonzero complex numbers.
In 1960, Lang (building further on work of Siegel and Mahler),
proved that the equation
has at most finitely many solutions x, y in G.
Two equations ax+by=1, a'x+b'y=1 may be
called isomorphic if a'=au, b'=bv for
certain u, v in G. Clearly, two isomorphic
equations (1) have the same number of solutions. It is not difficult
to construct equations (1) having two solutions. In 1988, Györy,
Stewart, Tijdeman and the speaker proved that for given G, up
to isomorphism there are only finitely many equations of type (1)
having more than two solutions.
In my lecture I will discuss a generalization to linear equations
in more than two variables with unknowns from G. At the end
of my lecture I will explain how this fits into a more general
geometrical setting, and discuss a recent result by Gael Rémond
on semiabelian varieties, of which the results on linear equations
mentioned above are special cases.

16:3017:15 
Robbert Fokkink (Delft),
Minimal sets and Euclidean rings
Abstract.
Recently J.P. Cerri settled some conjectures on the euclidean
minimum of a number field, using results from topological dynamics.
This may seem like a remarkable link between two unrelated fields,
but it really is not.


November 4

Groningen, Room WSN 31, WSNbuilding

11:3012:30 
Lenny Taelman,
An introduction to Drinfeld's shortest paper
Abstract.
The paper referred to in the title is called
"Commutative subrings of certain noncommutative rings"
(Funkcional. Anal. i Prilozen. 11 (1977), no. 1, 1114).
We discuss its results and applications to number theory
and differential equations.

13:1514:15 
Marius van der Put,
A localglobal problem for differential equations
Abstract.
Consider a linear differential operator
L:=a_{n}(d/dz)^{n} + ... +
a_{1}(d/dz) + a_{0}. The question
studied in the lecture is: if the inhomogeneous
equation L(y)=f has a formal solution everywhere locally,
does it follow that a global solution exists?

14:3015:00 
Khuong An Nguyen,
dsolvable linear differential equations
Abstract.
A linear differential equation over a differential
field K is called
dsolvable if all its solutions are in a tower
K⊂K_{1}⊂K_{n} of
differential fields, in which
each K_{i+1}/K_{i} is
either a finite extension or an
extension obtained by adjoining to K_{i} all
solutions
and their (higher) derivatives of some linear diff. eq.
over K_{i} of degree at most d. We
discuss this notion
and give examples.

15:0015:45

Steve Meagher,
Forms of algebraic curves
Abstract.
Consider the Fermat curve
X^{4} + Y^{4} + Z^{4} =
0; this
curve has 96 automorphisms, and thus many `forms' (i.e.,
curves defined over the same ground field, which are not
isomorphic to the given one but which become isomorphic
over an extension field).
We compute these forms and give a formula for their number of points
over a finite field.

16:0017:00 
Heide GluesingLuerssen,
A survey on convolution codes
Abstract.
Convolutional codes are (free) submodules
of F[z]^{n}, for F a
finite field.
They can be interpreted as encoding arbitrarily long sequences
of message blocks into sequences of codeword blocks.
As is the case for classical block codes, a certain weight
function is needed for describing and investigating the
errorcorrecting properties of the code.
In this talk I will explain the basic notions for convolutional
codes.
Thereafter, I will sketch some current research projects.
 
December 16

Leiden, Room 412

11:3012:30

Sven Verdoolaege
(LIACS, Leiden),
Enumerating parametric convex integer sets and their projections
Abstract.
Given a convex set of integer tuples defined by linear
inequalities over a fixed number of variables, where some of
the variables are considered as parameters, we consider two
different ways of representing the number of elements in the set
in terms of the parameters. The first is an explicit function
which generalizes Ehrhart quasipolynomials. The second is its
corresponding generating function and generalizes the classical
Ehrhart series. Both can be computed in polynomial time based
on Barvinok's unimodular decomposition of cones. Furthermore, we
can convert between the two representations in polynomial time.
Finally, we present some ideas on how to handle projections of
convex integer sets.

13:3014:15
 Willemien Ekkelkamp (CWI/Leiden),
A variation of the MPQS factoring algorithm: analysis and
experiments
Abstract.
One of the methods used for factoring is the multiple polynomial
quadratic sieve factoring algorithm. In this talk I will present a
variation of this algorithm and some experimental results. Further,
we will take a look at the (theoretical) ratio of the different types
of relations coming out of the sieve. This ratio will be used in a
simulation program for estimating the time needed for factoring a
number.

14:3015:15
 Jeanine Daems (Leiden),
A historical introduction to mathematical crystallography
Abstract.
What is a crystallographic group? How many crystallographic groups are
there? The first part of Hilbert's 18th problem deals with
crystallographic groups. In 1910 Bieberbach solved this problem by
proving
that the number of crystallographic groups in each dimension is only
finite. In this talk I will discuss Bieberbach's proof and a modern
proof
of the same theorem. In dimensions 2, 3 and 4 the exact number of
crystallographic groups is known, I will talk about the methods used
to find them.

15:3016:30
 Peter Stevenhagen (Leiden),
Algorithmic class field theory
Abstract.
Class field theory `describes' the abelian extensions of a number
field, but does not readily provide generators for these extensions.
Finding a `natural' set of generators is a Hilbert problem that
is still largely unsolved.
We will discuss to which extent the theory furnishes algorithms to
generate class fields, and how this can be applied in algorithmic
practice.



