Programme of the DIAMANT/EIDMA symposium, 25-26 November 2010

arrival and coffee
Bas Terwijn
Algorithmic randomness
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 y2=x3+A(x-B)2
Much is known about elliptic curves of the form y2=x3+Ax2+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 y2=x3+A(x-B)2?
Michiel Kosters
Anisotropic groups and applications
Let k be a field and let W be a one-dimensional k-vector space. Let V be a finite dimensional k-vector 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.
13:00-14:00 lunch
Frank Vallentin Inhomogeneous extreme forms
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited 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
The set of minimal codewords in linear codes is related with decoding algorithms and the so-called gradient-like 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 Conti-Traverso 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 Conti-Traverso which we will see that is equivalent to the approach by Ikegami-Kaji. 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.
Tony Huynh
Graph minors for group-labelled graphs
A group-labelled 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 group-labelled graphs.
Filip Najman
Compact representations of quadratic integers and consecutive smooth integers
 16:30-17:25 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.
David Gruenewald
Humbert surfaces and applications
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
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 0-1 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 0-1 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.
Sam van Gool
Canonical extensions and Stone duality for strong proximity lattices
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 point-free way, by defining the appropriate generalisation of canonical extensions of lattices to strong proximity lattices.
René Pannekoek Diagonal quartic surfaces, rational points and p-adic numbers
11:30-11:55 We give a review of p-adic numbers, which are an extension of the rational numbers. Also, we introduce diagonal quartic surfaces. The p-adic numbers form a topological space. The same is therefore true for the set of p-adic points of a diagonal quartic surface. If inside this set the rational points lie dense, i.e. every p-adic 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 p-adic points. The goal will be to prove that they lie dense.
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 abc-conjecture. The second talk discusses an analogue of the abc-conjecture in the polynomial ring over C. The third talk discusses an analogue of the abc-conjecture in a function field in characteristic p.  
Willem-Jan Palenstijn - Enumerating abc triples
Joseph Oesterlé - abc over C[t]
15:30-16:00 tea
David Masser - abcd... over Fp(t,u,...)

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