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Seminars Calendar
Friday February 7, 2024 - 14:30 - 1BC45 - Alberto Vezzani (Università di Milano)
Berthelot's conjecture via p-adic homotopy theory
Abstract: By drawing parallels to classical work by Monsky-Washnitzer, Elkik, Arabia and others, we motivate the study of (non-archimedean) motivic homotopy theory by showing that it can be used to define/re-define rational p-adic cohomology theories and prove new results about them. For example, we show how to define relative rigid cohomology and deduce finiteness properties for it (joint work with V. Ertl), solving a version of a conjecture by Berthelot for coefficients of geometric origin.
Thursday January 9, 2025 - 14:30 - 7B1 - Johannes Girsch (University of Sheffield)
Degenerate Representations of $\mathbf{GL}_n$ over a $p$-adic field
Abstract: Smooth generic representations of $\mathbf{GL}_n$ over a $p$-adic field $F$, i.e. representations admitting a non-degenerate Whittaker model, are an important class of representations, for example in the setting of Rankin--Selberg integrals. However, in recent years there has been an increased interest in non-generic representations and their degenerate Whittaker models. By the theory of Bernstein--Zelevinsky derivatives we can associate to each smooth irreducible representation of $\mathbf{GL}_n(F)$ an integer partition of $n$, which encodes the "degeneracy" of the representation. By using these "highest derivative partitions" we can define a stratification of the category of smooth complex representations and prove the surprising fact that all of the strata categories are equivalent to module categories over commutative rings. This is joint work with David Helm.
Friday December 20, 2024 - 11:00 - 2AB40 - Abhinandan
Prismatic $F$-crystals and Wach modules
Abstract: For an absolutely unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Wach, Colmez and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a natural generalisation of these results to the case of a "small" base ring.
Friday December 13, 2024 - 14:30 - 1BC50 - Lorenzo La Porta (Università di Padova)
Geometry of special fibres in the Serre's conjectures
Abstract: This talk aims to provide an overview of my research journey, beginning with my graduate thesis and culminating in a discussion of ongoing projects. Key themes include Serre's modularity conjectures, the study of (some) Shimura varieties in positive characteristic, modular forms and associated mathematical structures, like the stack \(G\tt{-Zip}^\mu\). Particular emphasis will be placed on \(\theta\)-operators and their applications, showcasing both classical results and novel insights.
Wednesday November 27, 2024 - 11:00 - 2BC30 - Nefton Pali (University of Paris-Sud)
On maximal totally real embeddings
Abstract: It has been 66 years since the existence of a complex structures on Grauert Tubes was proven for the first time by Bruhat-Whitney. Still, up to now, the explicit form of such structure has remained quite mysterious to the community of experts in the field. This is finally clarified in a recent joint article with Bruno Salvy, where we consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real analytic on the fibers of the tangent bundle. This hypothesis is quite natural in view of the existence result of Bruhat-Whitney. For any torsion-free connection acting on the real analytic sections of the tangent bundle of a real analytic manifold, we provide a simple and explicit expression of the coefficients of the Taylor expansion on the fiber of the associated canonical complex structure. An explicit global expression for the above coefficients is important for applications to analytic micro local analysis over manifolds, as it allows an explicit global construction of the complex extension of a given global Fourier integral operator defined on a real analytic manifold. The above application leads to important consequences in the theory of analytic complex deformations of a given compact complex manifold.
Friday November 22, 2024 - 14:30 - 1BC45 - Quoc Bao Vo
Cohomology of the differential fundamental group of algebraic curves
Abstract: Let $X$ be a smooth projective curve over a field $k$ of characteristic zero. The differential fundamental group of $X$ is defined as the Tannakian dual to the category of vector bundles with (integrable) connections on $X$. This work investigates the relationship between the de Rham cohomology of a vector bundle with connection and the group cohomology of the corresponding representation of the differential fundamental group of $X$. Consequently, we obtain some vanishing and non-vanishing results for the group cohomology.
Thursday November 14, 2024 - 14:30 - 2AB40 - Alexandros Groutides, (University of Warwick)
Integral norm relations for Rankin-Eisenstein classes via unramified harmonic analysis.
Abstract: In recent years, local representation theory of $p$-adic groups and zeta integrals, have been linked to Euler system norm relations. In this talk, we will touch upon this idea, initially introduced by Loeffler-Skinner-Zerbes, and discuss recent developments in the integral version of the theory. Using unramified harmonic analysis within this representation theoretic framework, we establish the conjectured integral behavior of local factors appearing in tame norm relations, between any collection of integral motivic Rankin-Eisenstein classes in the recipe of op.cit. If time permits, we will discuss how, by specializing to one such collection, we can obtain the most general version of the Rankin-Selberg Euler system tame norm relations.
Friday November 8, 2024 - 14:30 - 1BC45 - Otto Overkamp, (Universität Düsseldorf)
Existence of global Néron models beyond semiabelian varieties
Abstract: Let S be an excellent Dedekind scheme with field of fractions K. Let G be a smooth algebraic group over K. A Néron lft-model of G over S is a smooth separated group scheme over S with generic fibre G which satisfies a certain universal property; such an object is called a Néron model of G if it is moreover of finite type over S. Because Néron (lft-)models play a very important role in arithmetic geometry, it is of fundamental importance to understand which algebraic groups admit Néron (lft-)models. If S is a local scheme, this is completely understood, but the global case has turned out to be considerably more delicate; this is the subject of Bosch-Lütkebohmert-Raynaud's Conjectures I and II, formulated at the end of their well-known textbook. I shall report on recent joint work with T. Suzuki, the main result of which is that both conjectures hold true if the residue fields of S are perfect, but that Conjecture I is false in general. I shall focus on the proof of Conjecture I in the case of perfect residue fields and explain a counterexample in the case of p-finite S with imperfect residue fields if time permits.
Friday October 18, 2024 - 12:30 - 2BC30 - Kirill Zaynullin (University of Ottawa)
On the formal Peterson subalgebra and its dual
Abstract: We discuss a generalization of the Peterson subalgebra to a generalized (oriented) cohomology theory which we call the formal Peterson subalgebra. One of our results shows that the localized formal Peterson subalgebra for the extended Dynkin diagram of type $\hat A_1$ provides an algebraic model for `quantum generalized cohomology’ of the projective line. Hence, confirming and extending the Peterson conjecture for these settings. We also prove that the dual of the formal Peterson subalgebra (a generalized cohomology of the affine Grassmannian) is the $0$th Hochschild homology of the formal affine Demazure algebra. Hence, extending the techniques and results on the Hopf algebroids of structure algebras of moment graphs by [Lanini-Xiong-Z.] to the case of affine root systems.
Working Seminars
Condensed mathematics 2022
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