SM Exposés 2022-2023

• Langue: english

• Résumé de l'exposé: Let K be a finite extension of Q_p and h a positive integer. Using the geometry of Lubin-Tate spaces, Scholze has constructed a functor from mod p representations of GL_h K to mod p representations of D*, the group of units of the central division algebra D over K of invariant 1/h. The functor is widely seen as a candidate for a mod p Jacquet-Langlands correspondence, but is still very mysterious.

In this talk I will introduce the level n Lubin-Tate space M_n, which is the moduli space of "deformations of a formal group in char. p of height h together with some n-th level structure". Moving on quickly to the infinite level one obtains a perfectoid space M_infty, which has commuting actions of GL_h K and D*. It has a nice moduli interpretation and it allows one to readily define the Gross-Hopkins period map as well as Scholze’s functor. If time permits, I will try to give an overview of recent work on the functor.

• Langue: english

• Résumé de l'exposé: Classical Fourier transform occupies a major part of the analysis. An analog on abelian varieties is introduced by S. Mukai in 1981, which is now known as Fourier-Mukai transform. Similar to the Fourier inversion formula, Mukai proved a duality theorem for his transform. This result reveals the phenomenon that, the derived category of coherent modules of two non-isomorphic projective varieties can be equivalent. In this talk, I will present the work of O. Ben-Bassat, J. Block and T. Pantev about the analytic version of Fourier-Mukai transform on complex tori.

• Langue: english

• Résumé de l'exposé: The notion of a pseudo-representation was introduced by A.Wiles for GL_2 and generalized by R.Taylor to GL_n. It is a tool that allows us to deal with the deformation theory of a reducible residual Galois representation, a case where the usual techniques fail. G.Chenevier gave an alternative theory of "determinants" extending that of pseudocharacters to arbitrary rings. In this talk we will discuss some aspects of this theory and introduce a similar definition in the case of the symplectic group, which is the subject of a forthcoming work joint with J.Quast. 

• Langue: english

• Résumé de l'exposé: Holonomic D-modules are algebraic analogues of linear differential equations. A Riemann-Hilbert correspondence, classifying these objects in terms of a generalization of perverse sheaves, has been achieved several years ago by D'Agnolo-Kashiwara (building on work by Kashiwara-Schapira, Sabbah, T. Mochizuki, Kedlaya, Tamarkin and many others), and this theory provides a powerful framework for studying questions related to irregular singularities. In particular, we can use it to study the Stokes phenomenon, a certain "refined monodromy" phenomenon occuring around an irregular singular point.

In this talk, I will first explain the main terms and characters in the story. Then, we will see two different applications of the Riemann-Hilbert correspondence to Stokes phenomena: Using it, we can  compute Fourier transformations of Stokes matrices in certain cases. Moreover, we can study fields of definition of solutions of differential equations using a technique called Galois descent.

• Langue: english

• Résumé de l'exposé: Emerton and Gee have defined and studied stacks which interpolate Fontaine's (phi,Gamma)-modules in families. Studying the geometry of these stacks is expected to shed light on deep problems in the p-adic local Langlands program, including the Breuil-Mézard conjecture and the emerging ``categorical” perspective. In this talk, I will explain a generalization of Emerton--Gee’s construction to the Lubin--Tate setting. By working at a perfectoid level, I will then show that the two versions are in fact isomorphic.

• Langue: english

• Résumé de l'exposé: Geometry, in the sense of Riemannian geometry, is strongly related to discrete subgroups of Lie groups. Arithmetic provides a systematic way to construct them. It gives rise to what we call arithmetic subgroups. In this talk, I will start by presenting the construction of arithmetic subgroups. Their classification reduces to various number theoretic classifications such as the classification of quadratic forms over number field for instance. Recently, there have been a good deal of interest in Zariski-dense subgroups of arithmetic groups. In my thesis, I constructed Zariski-dense subgroups of arithmetic groups isomorphic to the fundamental group of a compact topological surface.

• Langue: english

• Résumé de l'exposé: Given a big line bundle L on a projective manifold, Lazarsfeld–Mustată and Kaveh–Khovanskii introduced method of constructing convex bodies associated with L. These convex bodies are known as Okounkov bodies. These Okounkov bodies completely determine the numerical information of L.

When L is endowed with a singular positive Hermitian metric h, I will explain how to construct smaller convex bodies from the datum (L,h). As we will see, these convex bodies give complete information about the singularity of h modulo the so-called I-equivalence.

• Langue: english

• Résumé de l'exposé: Braverman, Finkelberg and Nakajima defined Coulomb branches for d=3, N=4 gauge theories. From a purely mathematical view, it is a loop version of Springer theory. In this talk, I will recall the classical Springer theory first, and introduce the construction of Coulomb branches by BFN. Then I will show that the Coulomb branches of quiver gauge theories of type ADE can be identified with the generalized affine Grassmannian slices. If time permits, I will survey some conjectures on the dualities between Higgs branches and Coulomb branches. 

• Langue: english

• Résumé de l'exposé: Let K be the function field of a p-adic curve, a field of cohomological dimension 3. If X is a smooth geometrically integral K-variety, we are interested in the following arithmetic questions for X:

- Local-global principle (LGP): If X has K_v-points for all closed points on a smooth projective model of K, does X have K-points?

- Weak approximation (WA): If X has K-points, is X(K) dense in the topological product of the X(K_v)'s?

Generalizing the Brauer-Manin obstruction over number fields, we may use the group H^3_nr(X, Q/Z(2)) of unramified degree 3 cohomology to detect the failure of LGP and WA ("reciprocity obstruction"). It is natural to ask if this obstruction is the only one.

Using global duality Poitou-Tate style duality theorem and parts of Poitou-Tate sequences, Harari, Scheiderer, Szamuely, and Izquierdo provided the positive answer for tori. Tian established the same result for certain reductive groups.

In my talk, I shall present similar results for homogeneous spaces of SLn with geometric stabilizers of type umult (extension of a group of multiplicative type by a unipotent group), obtained by the same techniques. This is my latest preprint https://arxiv.org/abs/2211.08986. 

• Langue: english

• Résumé de l'exposé: Ever since the pioneering work of Green-Lazarsfeld, generic vanishing theorems have been widely studied by numerous authors in the literature. Up to now, such theorems are known for compact Kähler manifolds. In this talk, we shall review the historical development and then talk about how to extend generic vanishing to a broader class of complex manifolds, namely Fujiki class C.  The tool that we lean on is Krämer-Weierssauer's vanishing theorem for perverse sheaves on complex tori.

• Langue: english

• Résumé de l'exposé: A conjecture of Grothendieck and Serre states that a principal homogeneous space for a reductive group scheme over a smooth integral scheme is locally trivial if it is generically trivial. Recently, this conjecture has seen progress through the work of Fedorov, Panin and Česnavičius. We shall see the historical background of this conjecture, followed by the standard techniques that go into some of the proofs. If time permits, we shall also discuss a few of its generalisations.