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.

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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Geometric spaces are described by spectral triples (A, H, D) in non-commutative geometry. In this context, A is an involutive noncommutative algebra represented by bounded operators on a Hilbert space H, and D is an unbounded selfadjoint operator acting in H which plays the role of the Dirac operator, namely that it contains the metric information while interacting with the algebra in a bounded manner. The local geometric invariants such as the scalar curvature of (A, H, D) are extracted from the high frequency behavior of the spectrum of D and the action of A via special values and residues of the meromorphic extension of zeta functions ζa to the complex plane, which are defined for a in A by

ζa (s) = Trace (a ⎜D⎜-s),        ℜ(s) >> 0.

Following the seminal work of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for the canonical translation invariant conformal structure on the noncommutative two torus Tθ2, there have been significant developments in understanding the local differential geometry of these C*-algebras equipped with curved metrics. In this talk, I will review my joint works with M. Khalkhali, in which we extend this result to general translation invariant conformal structures on Tθ2 and compute the scalar curvature. Our final formula for the curvature matches precisely with the independent result of A. Connes and H. Moscovici. I will also present our recent work on noncommutative four tori, in which we compute the scalar curvature and show that the metrics with constant curvature are extrema of the analog of the Einstein-Hilbert action.

In this talk I will explain how to realize some powers of the Dedekind eta function as a trace formula via different quantum coordinate algebras associated to semi-simple Lie algebras. If time permits, I will mention several perspectives, including relations to the Lehmer’s conjecture on Ramanujan’s tau function, the quantum dilogarithms, etc…

La conjecture d'Ax-Lindemann hyperbolique est un énoncé de transcendance fonctionnelle concernant les morphismes d'uniformisation des variétés de Shimura par des espaces symétriques hermitiens. Ces derniers sont munis d'une structure semi-algébrique naturelle et la conjecture d'Ax-Lindemann hyperbolique décrit l'adhérence de Zariski des "flots algébriques" dans la variété de Shimura. Nous expliquerons la preuve récente de cette conjecture obtenue dans un travail en commun avec Bruno Klingler et Andrei Yafaev. Nous expliquerons aussi la place de cet énoncé dans la stratégie de Pila-Zannier pour une preuve inconditionnelle de la conjecture d'André-Oort. Nous montrerons en particulier comment on obtient la preuve de la conjecture d'André-Oort pour une puissance arbitraire du module des variétés abéliennes principalement polarisées de dimension 6.

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.

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.

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.

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.

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