Given an F-crystal C over a scheme S of positive characteristic p, one can associate many reduced locally closed subschemes T of S defined by the property that a suitable invariant of F-crystals is constant on the fibres of C over points of S. A basic problem is to study different properties of such locally closed subschemes T. In this talk, we present a survey of purity results for locally closed subschemes T of S, including some of them which were obtained by us and by the graduate student Jinghao Li.

A technical ingredient in Faltings’ original approach to p-adic comparison theorems involves the construction of K(π, 1)-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point x ∈ X has an open neighborhood U whose general fiber is a K(π, 1) scheme (a notion analogous to having a contractible universal cover). I will show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory.

We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation.

We prove that despite the chiral nature of the model anomalies in the isometry currents do not appear for CP(N-1) at any N. This is in contradistinction with the minimal heterotic model (with no right-moving fermions) which is anomaly-free only for N=2, i.e. in CP(1). We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the « Konishi anomaly. » This gives us another method for finding exact β functions.

A clear–cut parallel between N=1 4D Yang-Mills and N=(0,2) 2D sigma models is revealed.

By a function field K, we mean a field extension over a finite field with transcendence degree one. In the function field world, by the work of Deligne, Drinfeld, Jacquet-Langlands, Weil, and Zarhin, the « Drinfeld modular parametrization » always exists for every « non-isotrivial » elliptic curve E over K. Suppose E has split multiplicative reduction at a place ∞. Then there exists a unique « Drinfeld type » (with respect to ∞) automorphic cusp form fE such that its L-function coincides with the Hasse-Weil L-function of E over K. These forms can be viewed as analogue of classical weight 2 modular forms. In this talk, we will start with basic properties of Drinfeld type automorphic forms, and use them as tools to obtain explicit formulas for special values of the L-functions coming from non-isotrivial elliptic curves.

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.

Journée autour des régulateurs et des valeurs de fonctions L.

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.

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.

Page web du séminaire

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…