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• Langue: english

• Résumé de l'exposé: I will explain the basic statement of Stark's conjecture in the rank 1 abelian case, where I will sketch Kronecker's limit formula and a proof of the conjecture in the case that the base field is imaginary quadratic. To the extent possible, I will give an exposition of some of the major points of the recent work of Bergeron--Charollois--Garcia about a proposed generalization to the complex cubic case. The point of view taken will be an adelic one due to Shouwu Zhang.

• Langue: english

• Résumé de l'exposé: The program of relative Langlands duality (RLC) proposed by Ben-Zvi, Sakellaridis and Vankatesh studies the duality between spherical varieties and their associated periods/L-functions. In this talk, I will explain some notions that can be viewed as geometric counterparts of RLC. We will introduce both local and global geometric conjectures, where the local case plays the role analogous to geometric Satake correspondence and the global one sheafifies periods and L-functions, in the same spirit as geometric Langlands. If time permits, I will provide some physics explanation to RLC.

• Langue: english

• Résumé de l'exposé: In the theory of automorphic forms, a mysterious duality between period integrals and automorphic L-functions appears. In 2012, Sakellaridis and Venkatesh proposed a conjecture on this duality for spherical varieties. Recently, Ben-zvi, Sakellaridis and Venkatesh extend this conjecture to a wider range of objects. In this talk, we will introduce the relative Langlands duality of BZSV in an automorphic way. We will also investigate two classical problems in representation theory: theta correspondence and Gan-Gross-Prasad conjecture and explain how they are mutually dual in the framework of BZSV.

• Langue: english

• Résumé de l'exposé: An amazing theorem of Iwasawa gives a formula for the p-adic valuation of class number of p-cyclotomic fields. This talk will explain where this formula comes from using basic facts about p-adic L-functions. The prerequisite will be nothing more than basic algebraic number theory. If time permits, I will explain a "horizonal" variant concerned with class number of order p cyclic extensions which was introduced in joint work with Daniel Kriz.

• Langue: english

• Résumé de l'exposé: The question on Poincare duality for p-adic étale cohomology of rigid analytic varieties was raised by Scholze. In some special case, it has been obtained by Lan-Liu-Zhu, Gabber-Zavyalov, Mann via different methods. In this talk, I will start with some examples to motivate Scholze's idea in his 2013's pioneer work. Then I will discuss the results mentioned above and recent progress on geometric (Colmez-Gilles-Niziol) and arithmetic dualities of Stein spaces.

• Langue: english

• Résumé de l'exposé: Suppose that F is a totally real field in which p is inert. A 2-dimensional mod p represention ρ of the absolute Galois group G_F cuts out a mod p representation π of GL_2 (F_p) in the cohomology of Shimura curves. In the work of Breuil-Herzig-Hu-Morra-Schraen, they construct a functor D_A from certain mod p representations of GL_2(F_p) to multivariable (φ, Γ)-modules, and a functor D_A^⊗ from mod p representations of G(F_p) to multivariable (φ, Γ)-modules. In this talk, I will review these constructions and give a proof that D_A(π) is isomorphic to D_A^⊗ (ρ|F_p ) when ρ|F_p is sufficiently generic.

• Langue: français

• Résumé de l'exposé: J.-M. Fontaine a défini les anneaux B_dR et B_cris, qui sont servent à plusieurs théorèmes de comparaison en théorie de Hodge p-adique. Dans ces deux anneaux, on peut trouver l'élément t, sur lequel l'action de Galois est la multiplication par le caractère cyclotomique, appelé période p-adique. Nous chercherons à expliquer ce que l'on entend par "période p-adique" et à justifier pourquoi l'anneau B_dR apparaît naturellement dans cette recherche frénétique de périodes.

• Langue: english

• Résumé de l'exposé: We use the affiness of Hodge Tate stack to give a proof of p-adic Simpson correspondence

• Langue: english

• Résumé de l'exposé: In 1989, Mazur defined the deformation functor associated to a residual Galois representation, which played an important role in the proof by Wiles of the modularity theorem.

• This was used as a basis over which many mathematicians constructed variations both to further specify it or to expand the contexts where it can be applied.

• These variations proved to be powerful tools to obtain many strong theorems, in particular of modular nature.

• In this talk I will give an overview of the deformation theory of Galois representations and describe two variants of Mazur's functor that allow to properly deform reducible residual representations (which is one of the shortcomings of Mazur's original functor).

• Namely, I will present the theory of determinant-laws initiated by Bellaïche-Chenevier on the one hand and an idea developed by Calegari-Emerton on the other

• If time permits, I will also expose results that semm to indicate a possible comparison between the two seemingly unrelated constructions.

• Langue: english

• Résumé de l'exposé: We explain the motivation of plectic conjecture. Following the work of Siyan Daniel Li-Huerta, we give a proof of the local analogue of plectic conjecture, which uses the formalism developed by Fargues-Scholze.  We also try to understand the global version using the work of Mingjia Zhang, following the idea of Matteo Tamiozzo.

• Langue: français

• Résumé de l'exposé: En 2000 Colmez et Fontaine démontrent la conjecture faiblement admissible implique admissible, ce qui permet la construction de représentations du groupe de Galois p-adique à partir de données d'algèbre (semi)-linéaire : Frobenius, monodromie et filtration. Cette construction permet en particulier de définir une famille de représentations irréductibles de dimension 2, appelé la série spéciale, qui ne dépend que de la donnée de poids et d'un invariant L, élément paramétrant la filtration qui définit cette représentation. L'invariant L apparait déjà en 1986, dans les travaux de Mazur, Tate et Teitelbaum sur la conjecture BSD p-adique. Il est associé à une forme modulaire et nommée ainsi parce qu'il permet de faire le lien entre les valeurs spéciales des fonctions L complexe et p-adique de la forme modulaire. L'intuition d'une correspondance de Langlands p-adique invite Breuil à "abolir le privilège galoisien de l'invariant L". Il définit une famille de représentations de GL2(Qp) dépendant de cet invariant de manière naturelle : L encode alors l'extension d'une représentation de Steinberg par une représentation algébrique contenue dans les vecteurs localement analytiques. En 2010 Colmez construit une correspondance de Langlands p-adique qui confirmera cette intuition. La même année, Schraen montre que l'invariant L apparait dans le complexe de de Rham du demi-plan p-adique. Dix ans plus tard, les techniques modernes ont permis à Colmez Dospinescu et Niziol d'étudier la cohomologie étale p-adique de la tour de Drinfeld pour y réaliser une partie de la correspondance de Langlands. Après l'introduction de cette odyssée spéciale, j'expliquerai comment on peut réaliser la correspondance entre la série spéciale galoisienne et automorphe dans la cohomologie étale p-adique du demi-plan à valeurs dans un système local p-adique.

• 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.