Algèbre et géométrie

Let k be a field and let P be a lattice polygon, i.e. the convex hull in R^2 of finitely many non-collinear points of Z^2. Let C/k be the algebraic curve defined by a sufficiently generic Laurent polynomial that is supported on P. A result due to Khovanskii states that the geometric genus of C equals the number of Z^2-valued points that are contained in the interior of P. In this talk we will give an overview of various other curve invariants that can be told by looking at the […]

I will discuss joint work with Martin Bridson and Martin Liebeck which addresses the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for a collection of simple groups of fixed Lie […]

P-minimality is a concept that was developed by Haskell and Macpherson as a p-adic equivalent for o-minimality. For o-minimality, the cell decomposition theorem is probably one of the most powerful tools, so it is quite a natural question to ask for a p-adic equivalent of this.In this talk I would like to give an overview of the development of cell decomposition in the p-adic context, with an emphasis on how questions regarding the existence of definable skolem functions have complicated things. The idea of p-adic cell decomposition was first developed […]

The TAC (torsion anomalous conjecture) states that for an irreducible variety V embedded transversaly in an abelian variety A there are only finitely many maximal V-torsion anomalous varieties. It is well know that the TAC implies the Mordell-Lang conjecture. S. Checcole, F. Veneziano and myself were trying to prove some new cases of the TAC. In this process we realised that some methods could be made not only effective but even explicit. So we analysed the implication of this explicit methods on the Mordell Conjeture. Namely: can we make the […]

In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof

Let G be a connected reductive group over C. One can associate with every spherical homogeneous space G/H its lattice of weights X^*(G/H) and a subset S of M of linearly independent primitive lattice vectors which are called the spherical roots. For any subset I of S we define, up to conjugation, a spherical subgroup H_I in G such that dim H_I = dim H and X^*(G/H_I) = X^*(G/H). We call the subgroups H_I the satellites of the spherical subgroup H. Our interest in satellites H_I is motivated by the […]