Thursday


10:3010:55

arrival and coffee

10:5511:00

opening

Bas Terwijn

Algorithmic randomness

11:0011:55

This is a brief introduction to the theory of algorithmic randomness,
which is a mix of various topics, such as computability, complexity,
probability, and measure theory. The theory of finite randomness
(Kolmogorov complexity) introduces the notion of randomness for finite
strings. There is also a theory for infinite strings (reals), which can
be set up by effectivizing the classical theory of Lebesgue measure.
These two theories are deeply connected in various beautiful ways. In
this talk we review the main definitions of algorithmic randomness,
both of the finite and the infinite theory. We discuss several recent
developments and connections with the theory of computation.

Monique van Beek

Elliptic Curves of the form y^{2}=x^{3}+A(xB)^{2}

12:0012:25

Much is known about elliptic curves of the form y^{2}=x^{3}+Ax^{2}+Bx. In his book 'Rational points on Elliptic Curves', Tate explores the group of rational points on these curves. Can we use the same kind of techniques on curves of the form y^{2}=x^{3}+A(xB)^{2}?

Michiel Kosters

Anisotropic groups and applications 
12:3012:55

Let k be a field and let W be a onedimensional kvector space. Let V be a finite dimensional kvector space and let b:V x V > W be a symmetric bilinear form. Then we call b anisotropic if for all nonzero x in V we have b(x,x)
nonzero. During this lecture, we will give a meaningful generalization
of this definition where we consider finite abelian groups and their
forms. We will also discuss applications of this new definition in
algebraic number theory.

13:0014:00 
lunch

Frank Vallentin

Inhomogeneous extreme forms 
14:0014:55

G.F. Voronoi (18681908) wrote two memoirs in which he describes two
reduction theories for lattices, wellsuited for sphere packing and
covering problems. In his first memoir a characterization of locally
most economic packings is given, but a corresponding result for
coverings has been missing. In this paper we bridge the two classical
memoirs. By looking at the covering problem from a different
perspective, we discover the missing analogue. Instead of trying to
find lattices giving economical coverings we consider lattices giving,
at least locally, very uneconomical ones. We classify the covering
maxima up to dimension 6 and prove their existence in all dimensions
beyond. New phenomena arise: Many highly symmetric lattices turn out to
give uneconomical coverings; the covering density function is not a
topological Morse function. Both phenomena are in sharp contrast to the
packing problem. This is joint work with Mathieu Dutour Sikiric and Achill Schuermann. 
Irene Marquez Corbella 
Computing minimal codewords of certain linear codes

15:0015:25

The set of minimal codewords in linear codes is related with decoding
algorithms and the socalled gradientlike decoding algorithms, which
for binary codes can be regarded as an integer programming with binary
arithmetic conditions. Conti and Traverso proposed an efficient
algorithm which uses Groebner bases to solve integer programming with
ordinary integer arithmetic conditions. Then Ikegami and Kaji extended
the ContiTraverso algorithm to solve integer programming with modulo
arithmetic conditions. It seems natural to consider for those problems
the Graver basis associated to them which turns to be the set of
codewords of minimal support of certain codes, which in the binary case
correspond to the set of minimal codewords. This provides us an
universal test set that allows us gradient decoding in those codes
related to the test set stated considered by ContiTraverso which we
will see that is equivalent to the approach by IkegamiKaji. Additional
interest to the set of minimal codewords is associated to different
topics in cryptography. In particular the set of minimal codewords of a
code is one to one related to the minimal access structure of secret
sharing schemes based on linear codes as J. Massey has shown.

15:3016:00

tea

Tony Huynh

Graph minors for grouplabelled graphs 
16:0016:25

A grouplabelled graph is an oriented graph with its edges labelled
from a group. We present generalizations of some of the results from
the Graph Minors Project of Robertson and Seymour to grouplabelled
graphs.

Filip Najman

Compact representations of quadratic integers and consecutive smooth integers 
16:3017:25 
When working in a real quadratic field (or equivalently solving
quadratic equations in two variables), one of the first things that
needs to be done is to compute the fundamental unit. This is a
notoriously hard computation, that dominates the run times of many
algorithms. The reason of the difficulty of this problem is the size of
the unit, which makes it difficult just to write it down. Compact
representations of the unit are used to overcome this difficulty.
However, to compute a compact representation of the fundamental unit,
one needs to compute the regulator of the quadratic field, and one has
to use either an exponential algorithm, or a subexponential algorithm
whose correctness depends on the GRH. We will show how one can use the
subexponential algorithm to obtain unconditional results concerning
consecutive smooth (having prime factors below some bound) integers,
and greatly extend results of Lehmer, Bauer and Bennett.

17:3019.00

drinks

19:00

dinner 


Friday


David Gruenewald

Humbert surfaces and applications 
9:009:55

In this talk, we present methods for computing Humbert surface
equations using Fourier expansions of Siegel modular forms. We describe
some applications for these equations to improve existing endomorphism
algorithms and point counting algorithms. 
Dion Coumans

Finitely generated free algebras and duality theory

10:0010:25

Algebras that are axiomatized by rank 1 equations (i.e. by equations
where each variable is under the scope of exactly one occurrence of the
operator under consideration) can be represented as algebras for a
functor. This enables one to construct free algebras for rank 1 logics
as a direct limit of a sequence of algebras. It is an open question
whether this construction may be generalized to logics with a rank 01
axiomatization. Using duality theory one may obtain a concrete
description of the algebras in the sequence approximating the free
algebra. This has led to more insight into the rank 01 problem and to
a description of the free Heyting algebra and of the free S4 modal
algebra as a direct limit. We will explain the basic idea of this
construction and illustrate it by working out a concrete example.

10:3011:00

coffee

Sam van Gool

Canonical extensions and Stone duality for strong proximity lattices

11:0011:25

Strong proximity lattices were introduced by Jung and Sünderhauf (1996)
as the finitary algebraic structures dual to stably compact spaces. A
strong proximity lattice is a lattice endowed with a binary relation
satisfying certain axioms. We show that the duality between strong
proximity lattices and stably compact spaces can also be described
algebraically and in a pointfree way, by defining the appropriate
generalisation of canonical extensions of lattices to strong proximity
lattices.

René Pannekoek

Diagonal quartic surfaces, rational points and padic numbers 
11:3011:55 
We give a review of padic numbers, which are an extension of the
rational numbers. Also, we introduce diagonal quartic surfaces. The padic numbers
form a topological space. The same is therefore true for the set of
padic points of a diagonal quartic surface. If inside this set the rational points
lie dense, i.e. every padic open set contains a rational point, then
that is a way of saying that there are many rational points on the
surface. We will next take a particular diagonal quartic surface, and we will
investigate how the rational points are distributed inside the set of
its padic points. The goal will be to prove that they lie dense.

12:0013:30

lunch

13:3017:00

Special afternoon around the abc conjecture


The abc conjecture has originally been formulated in the 1980s by David Masser and Joseph Oesterlé, two of the three afternoon speakers. It has a central role in modern number theory. Roughly the conjecture says that when a, b are coprime positive integers, and c=a+b, then the product abc cannot have, in some precise sense, only a few number of prime factors. The first talk is about the search for triples (a,b,c) that challenge the abcconjecture. The second talk discusses an analogue of the abcconjecture in the polynomial ring over C. The third talk discusses an analogue of the abcconjecture in a function field in characteristic p.

13:3014:20

WillemJan Palenstijn  Enumerating abc triples

14:3015:30

Joseph Oesterlé  abc over C[t] 
15:3016:00 
tea

16:0017:00

David Masser  abcd... over F_{p}(t,u,...) 