Abstract: Given a bounded planar domain, we consider the eigenvalues of the Laplacian with Dirichlet boundary conditions. The question to know which shape (if any !) minimizes a given eigenvalue is an old one : it was conjecture in 1877 by Rayleigh (and proved in 1923 by Faber and Krahn) that the disc is the unique minimizer for the first eigenvalue. Shortly after this, Krahn and Szegö proved that the second eigenvalue is minimized by the union of two identical discs. However, for all other eigenvalues of the Dirichlet-Laplacian, the minimizer is unknown. This naturally leads to the use of finite element methods to approximate a minimizing domain.

Abstract: Linear programming is the problem of finding the maximum of a linear function over a polyhedron. It is widely used in practice and has been very successfully applied in solving combinatorial optimization problems as well as in approximation algorithms. The Simplex method for linear programming proceeds by walking from vertex to vertex of the polyhedron, thus connecting questions about the efficiency of linear programming to questions about the geometry of polyhedra. The polynomial Hirsch conjecture claims that the diameter of the vertex-edge graph is bounded by a polynomial in the number of defining inequalities and the dimension. I will report recent results on both the efficiency of the simplex method as well as the polynomial Hirsch conjecture.

Abstract: Efficient linear and infinitely smooth approximation of functions from equispaced samples is an important problem in practice. Runge showed in 1901 that it is not delivered by the interpolating polynomial. In 2007, Floater and Hormann have introduced a family of linear barycentric rational interpolants which extend a construction by Berrut from 1988. These interpolants yield high theoretical rates of convergence, which depend on the smoothness of the approximated function. We will present these rational interpolants as well as a further extension and look at their condition and some of their applications to differentiation and integration.

Abstract: We will talk about the problem of embedding symplectically an open set of an Euclidean space of even dimension into anoter one. In particular, we will present Gromov's non-squeezing theorem, that gives a necessary and sufficient condition to embed symplectically a ball in a symplectic cylinder. Finally, we will try to give an idea of the proof of the theorem using some symplectic invariants, called symplectic capacities.

Abstract: The tropicalisations of complements of hyperplane arrangements are known as Bergman fans and have a very combinatorial description. In this talk I will study tropical curves contained in these fans. First I will show how to intersect curves and explain how this relates to the intersection product of actual complex curves. Finally, I will present some obstruction theorems to lifting tropical curves in surfaces that arise from this intersection product. This talk is based on joint work in progress with Erwan Brugallé.

Abstract: Stein manifolds admitting a proper holomorphic embedding of the complex line are characterized, among all complex manifolds, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them. Several classes of Stein manifolds admitting a proper holomorphic embedding of the complex line are described.

Abstract: nonpositively curved space is a geodesic metric space such that any triangle is thinner than an euclidean one. Simply connected riemannian manifolds provide a large class of examples. We will aim to understand the dynamical behaviour of isometries of such spaces. A lot of examples will be described to improve our intuition. Moreover, we will pay our attention on the differences between finite dimensional case and infinite dimensional case.

Abstract: In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the 2d Ising model are a convex combination of the two pure phases. After introducing the relevant definitions and concepts needed to understand the physical content of this result, I will present a new approach to it, with a number of advantages:
(i) a finite-volume, quantitative analogue (implying the classical claim) is obtained;
(ii) the scheme of the proof seems more natural and provides a better picture of the underlying physical phenomenon;
(iii) this new approach seems substantially more robust (possible extension to the Potts model).
This is a joint work with Yvan Velenik.

Abstract: If V is a subscheme of Pⁿ and F is a general hypersurface of degree d, then F cuts out on V a subscheme Z = V∩F, which is also a subscheme of F. A natural and interesting question is to study the properties that either Z or V transfers to the other. In this talk we will discuss this problem and will show how to construct a curve C in P³ whose general hyperplane section Z = C∩L in P² has a given degree matrix.

Abstract: Right-angled polyhedra turn out to be an interesting family of (almost) hyperbolic polytopes. They are connected with other various problems and notions, e.g. right-angled Coxeter groups, Loebell manifolds, combinatorial volume estimates and decompositions of acute-angled polyhedra, dimension bounds. In my talk, a survey on the main part of this zoo will be given together with a brief explanation of what I'm doing.

Abstract: In the 1960s Shafarevich introduced ind-varieties in order to explore some naturally occurring groups that allow the structure of an infinite-dimensional analogon of an algebraic group (such as the group of polynomial automorphisms of ℂⁿ ). Shafarevich defined an ind-variety as the successive limit of an increasing chain
X₁ ⊆ X₂ ⊆ X₃ ⊆ . . . of varieties X_n, each one closed in the next. There are essentially two ways of endowing such an ind-variety with a topology. One topology is naturally induced by the increasing chain of varieties and is due to Shafarevich. The other is naturally induced by the regular functions on the ind-variety and is due to Kambayashi. These topologies differ on a rather large class of ind-varieties. The aim of this talk is to give an idea of the proof of this result.

Abstract: We define classical knots and explain how they form a discrete metric space with respect to the Gordian distance. Then we give different descriptions of the subspace of torus knots. Finally we introduce three notions of adjacency for torus knots and conclude with some examples of Gordian adjacency and some of our questions.

Abstract: After a description of Witt groups I will show how they play a role in my current work namely in studying the centralizer of unipotent elements in linear algebraic groups for small characteristic. We will describe how one constructs such subgroups with examples both in the classical and exceptional cases.

Abstract: Elliptic curves defined over finite fields are one of the most important types of groups used in cryptography today. Trace zero varieties arise from certain subgroups of such elliptic curves, namely those points of trace zero. They are interesting from a constructive point of view, because they allow fast arithmetic, and also from a cryptanalytic point of view, since the security of many cryptographic protocols is directly linked to the properties of these varieties. For both constructive and destructive use of trace zero varieties, it is important to be able to efficiently represent their elements. We discuss the geometric construction that leads to the trace zero variety, and how to find an easy and compact representation of trace zero elements.