I will explain some universal inequalities among eigenvalues of Laplacian and how to derive it from the point of view of the theory of concentration of measure.

The aim of this talk is to provide a notion of Poincaré duality for the Chow groups of singular varieties where a torus acts with finitely many fixed points. We relate this concept to the usual notion of Poincaré duality in the smooth and rationally smooth cases (e.g. Betti numbers). Finally, we characterize it in terms of equivariant multiplicities, i.e. certain rational functions having poles along hyperplanes associated to the weights of the action.

A toric arrangement is given by a family A of level sets of characters of a complex torus T. The focus of this talk will be on the topology of the complement M:=T A, and in particular on the extent to which M is determined by the combinatorial data of the arrangement A, a line of research recently revived by work of De Concini and Procesi. 
 
I will first present some basics about toric arrangements, and then hint at how a theoretical framework can be developed in parallel to the theory of hyperplane arrangements. In particular, I will describe combinatorial models for the homotopy type of M and explain the methods we use in our proof of minimality, and thus of torsion-freeness, of M.
 
This is mostly joint work with Giacomo d'Antonio.

Journée autour des régulateurs et des valeurs de fonctions L.

In my talk I will discuss the possible hypothetical generalization of the Langlands correspondence for two-dimensional local fields and two-dimensional arithmetic schemes. A two-dimensional local field appear naturally from a point and a formal stalk of irreducible curve on a surface such that this curve contains the point. An example of two-dimensional local field on an algebraic surface is the field of iterated Laurent series.

I will give an overview of the Geometry of Interaction program and its applications to complexity theory. The Geometry of Interaction program takes its root in the field of logic, and more precisely of proof theory. It can be understood as a refoundation of logic arising from the dynamics of a mathematical model of computation.

This research program provides well-suited frameworks for the study of computational complexity in which both time and space complexity classes can be described and which offer new methods and techniques for the study of those.

Neutron star and black hole magnetospheres, converting rotational energy to Poynting flux, are often modeled using force-free electrodynamics, since the energy of the field dominates that of the charged matter. In this peculiar scheme, the requirement that the 4-force on the current vanish plays the role of a non-linear field equation, with the current defined by the field via Maxwell’s equations. This talk will show how the theory becomes remarkably simple and elegant when treated relativistically with the help of differential forms, and some applications and new exact solutions will be discussed.

Microlocal analysis is a theory that, among many other things, explores the classical/quantum correspondence in the theory of partial differential equations. It also draws upon the many links between analysis and geometry. This talk will start with the basic notions of microlocal analysis, and conclude with a discussion of Hörmander’s propagation of singularities theorem and its applications.

Journée autour des formes modulaires et amplitudes en théorie des cordes.

The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion of height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture, Faltings first proved Tate conjecture for abelian variaties by defining heights of abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. I will explain that his height of an abelian variety is generalized to the height of a motive. This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

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Journée autour des formes modulaires et amplitudes en théorie des cordes.

(ATTENTION CHANGEMENT DE DATE)

We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vinberg's monoid.

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