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• Langue: english

• Résumé de l'exposé: Non-commutative Hodge structures were introduced by Katzarkov, Kontsevich, and Pantev as a generalization of classical Hodge structures for non-commutative spaces. Their formal (or non-archimedean) analogs are known as F-bundles. Such structures naturally appear in mirror symmetry, enumerative geometry, and singularity theory. In this talk, I will first discuss the spectral decomposition of nc-Hodge structures of exponential type and relate it to the vanishing cycle decomposition after Fourier transforms via certain choices of Gabrielov paths. I will then discuss the spectral decomposition of maximal F-bundles. If time permits, I will also talk about the extension of framings for F-bundles and its applications. This talk is based on joint work with T. Yu, T. Hinault, and C. Zhang.

• Langue: english

• Résumé de l'exposé: The mod p reduction of Shimura varieties can be stratified by loci of points where the corresponding isocrystal is constant, resulting in the Newton stratification. In the Siegel case i.e. if we take the Shimura variety to be the moduli space of principally polarised abelian varieties, this stratification corresponds to the loci of points where the isogeny class of the p-divisible group attached to the abelian variety is constant. Geometric questions about the Newton strata like dimension, number of irreducible components, etc. have been widely studied, starting with the Siegel case. In this talk, I will describe a new approach to study these questions, by using automorphic forms. The approach is a modification of the Langlands-Kottwitz method and involves a comparison of two trace formulas: (i) Fujiwara-Leftschetz-Verdier trace formula, which relates the actions of Frobenius and Hecke operators on the cohomology of a Newton stratum to fixed points of these operators and (ii) Arthur-Selberg trace formula, which links automorphic representations of a group to orbital integrals. I will start with a gentle introduction to the topic before explaining how to establish such a trace formula comparison.

• Langue: english

• Résumé de l'exposé: Arithmetic level raising concerns congruences between automorphic representations of different levels at a prime, playing a crucial role in the construction of Euler systems and the Beilinson–Bloch–Kato conjecture. In this talk, we establish level raising for the unitary group U(1,2r) at an inert prime by analyzing the mod p geometry of the corresponding Shimura variety. This case is more intricate than that of U(1,2r−1) at an inert prime, which was previously treated by Liu, Tian, and Xiao. The key ingredients include a variant of Ihara’s lemma and the observation that the map between the special fibers of Shimura varieties with certain level structures is semi-small. This semi-smallness enables us to determine the monodromy filtration on the nearby cycles of a Steinberg-type local system, which, via Ihara’s lemma, is closely related to the level raising map. This is joint work with Ruiqi Bai. 

• Langue: english

• Résumé de l'exposé: In this talk, I will give an introduction to the theory of motives. More precisely, I will explain how to construct categories of pure and mixed motives, and how these are related to certain standard conjectures on algebraic cycles. 

• Langue: english

• Résumé de l'exposé: Given a family of varieties X over a global field k, one is interested in the sufficiency of the Brauer-Manin obstruction to explain the possible failure of the Hasse principle and weak/strong approximation of adelic points on X. Let G be an affine algebraic group G over k, and our family of interest - the principal homogeneous spaces X of G. In 1981, Sansuc proved this sufficiency for connected G over a number field k by reduction to the case of a torus and an application of Poitou-Tate duality. Since then, arithmetic duality theorems have also proven useful in the study of many similar problems. In this lecture, we briefly recall the significant generalization by Rosengarten of the Poitou-Tate theory to all commutative affine algebraic groups G over a global field k of any characteristic. Then we explain how this theory allows us to extend the stated Brauer-Manin results to (the principal homogeneous spaces of) all such G, not necessarily smooth or connected, highlighting the difficulties which appear in the case when k is a global function field. This talk is based on the speaker's recent preprint, available on arXiv. 

• Langue: english

• Résumé de l'exposé: Cohomological field theories (CohFTs) were defined by Kontsevich and Manin to give an abstraction of the recursive properties of certain curve-counting invariants known as the Gromov-Witten (GW) invariants. The Givental-Teleman theorem provides a classification of CohFTs in the semisimple case and has various applications in enumerative geometry, among which we present two in this talk. The first is the reconstruction by Givental of high-genus GW invariants in terms of genus-zero ones for GKM varieties, using torus fixed point localization. The second is the proof by Pandharipande-Pixton-Zvonkine of the Pixton conjecture on tautological relations, using Witten’s r-spin theory. Much of this talk is based on Pandharipande’s ICM 2018 talk (arxiv:1712.02528).

• Langue: english

• Résumé de l'exposé: We will talk about the property of being affine or Stein for period domains over finite fields and local fields. The Steinness of the Drinfeld space and its dual is proven by Drinfeld-Schneider-Stuhler by covering the Drinfeld space by an increasing filtration of affinoids given as complements of tubes around rational hyperplanes. The finite field analogues of period domains were proven by Orlik and Rapoport to be affine only for the Drinfeld space and its dual, by an Artin-Grothendieck vanishing result combined with an inequality of dimensions. We will give an introduction to period domains, discuss Drinfeld-Schneider-Stuhler's proof of Steinness of the Drinfeld space and sketch a p-adic analogue of Orlik-Rapoport's classification result.

• Langue: english

• Résumé de l'exposé: I will talk about p-adic cohomologies of log rigid analytic varieties over a p-adic field. For a log rigid analytic variety X defined over a discretely valued field, we can compute the Kummer pro-étale cohomology of B+_dR and B_dR. When X is defined over C_p, we introduce a logarithmic B+_dR-cohomology theory, serving as a deformation of log de Rham cohomology. Additionally, I will talk about the log de Rham-étale comparison in this setting and show the degeneration of both the Hodge-Tate and Hodge-log de Rham spectral sequences for X defined over a discretely valued field or admitting a proper semistable formal model over Spf(O_C).

• Langue: english

• Résumé de l'exposé:  Back in 1961, Borel and Haefliger have considered the question of whether the group of algebraic homology classes of a smooth algebraic variety can be generated by homology classes represented by smooth subvarieties. Kollár and Voisin considered a similar question in the rational equivalence setting. To set up, let X be a smooth projective variety over a field of characteristic 0. Then they proved that every cycle of X whose degree is in the Whitney range (i.e. lower than half of the dimension of X) is smoothable in the Chow ring, i.e. is rationally equivalent to a sum of smooth subvarieties of X. Meanwhile, when the degree of the cycle is out of the Whitney range, infinitely many counter-examples have been constructed by other mathematicians, making Kollár and Voisin’s result optimal. This talk will briefly explain Kollár and Voisin’s proof of their result.

• Langue: english

• Résumé de l'exposé:  The local monodromy theorem states that the local monodromy of a (smooth proper) family of complex algebraic varieties is quasi-unipotent, i.e. the eigenvalues are all roots of unity. This theorem can be proved by geometric methods, but I will explain how the result can be proved p-adically, following works of Katz and André-Baldassarri. The key is proving that a certain condition on the p-curvatures (or the p-adic radii of convergence) of vector bundles with connection is stable under taking direct images of D-modules.

• Langue: english

• Résumé de l'exposé:  The Godement-Jacquet L-function is a classical invariant attached to irreducible representations of GL_n. Minguez extended their definition to representations over fields of characteristic l ≠ p. In this talk we will finish the computation of these L-functions for modular representations and check that they agree with the L-function of their respective C-parameter defined by Kurinczuk and Matringe. We approach the problem by extending the theory of square-irreducible cuspidal representations, and their derivatives, of Minguez and Jantzen to modular representations and applying it to our setting.

• Langue: english

• Résumé de l'exposé: Bruhat-Tits buildings are geometric spaces that elegantly encode the combinatorics of "parahoric" subgroups of a semisimple algebraic group over a field with a (discrete) valuation. Their significance and motivation come from many directions: as an indispensable bookkeeping tool in p-adic representation theory, as the p-adic analogue of a symmetric space, as a fascinating example of a metric space of non-positive curvature, and as the "skeleton" of the so-called affine Grassmannian. We shall discuss the construction and basic properties of the Bruhat-Tits building of a split algebraic group, focusing in particular on SL(n). Time permitting, we shall also comment on the utility of the Bruhat-Tits building in representation theory, for example how it allows one to "visualize" the Satake isomorphism.

• Langue: français

• Résumé de l'exposé: Le problème classique de branchement étudie la restriction d'une représentation irréductible à un sous-groupe compact. Les travaux de Gross-Prasad et de Gan-Gross-Prasad ont généralisé ce cadre en une conjecture pour les groupes classiques sur des corps locaux de caractéristique zéro. La première percée a été réalisée par Waldspurger dans les cas spéciaux orthogonaux non archimédiens. Depuis lors, diverses approches ont été développées, menant à la démonstration complète de la conjecture dans tous les cas. Dans cette présentation, je vais introduire une approche qui s'applique de manière uniforme aux cas unitaires et non unitaires, archimédiens et non archimédiens, de Bessel et de Fourier-Jacobi. Cette approche repose sur les travaux fondamentaux de Waldspurger, Moeglin-Waldspurger et Gan-Ichino. Le développement de cette approche inclut certains de mes travaux, ainsi que des travaux en collaboration avec Luo, des travaux en collaboration avec Chen et Zou, et des travaux en collaboration avec Jiang, Liu et Zhang.

• Langue: english

• Résumé de l'exposé: The Corlette-Simpson correspondence is an equivalence of categories between semi-simple local systems and (some) polystable Higgs bundles on a compact Kähler complex manifold. The first part of this talk will be a gentle introduction to this correspondence. In the second part we will deduce from it that rigid local systems support variations of Hodge structures, and explain how this result of Simpson fits into the framework of a non-Abelian analog of the Hodge conjecture.

• 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.