Oberseminar WiSe 24/25
Die Vorträge finden jeweils donnerstags um 16:45 Uhr im Raum WSC-N-U-3.05 (im Mathematikgebäude ) statt.
Der Tee findet ab 16:15 in Raum O-3.46 statt.
Alle Interessenten sind herzlich eingeladen!
The seminar takes place on Thursday, starting at 4:45pm. The duration of each talk is about 60 minutes. Before the talk, at 4:15pm, there is tea in room O-3.46.
Everybody who’s interested is welcome to join.
Directions from the train station.
| 10.10.2024 | Yu Min (Imperial) | Classicality of derived Emerton—Gee stack for general groups |
| 31.10.2024 | Dimitri Wyss (EPFL) | Non-archimedean integration on quotients |
| 7.11.2024 | |
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| 14.11.2024 | Georg Tamme (Universität Mainz) | A homotopical approach to crystalline cohomology |
| 21.11.2024 | Tasho Kaletha (Bonn) | On the endoscopic classification of representations of classical groups |
| 28.11.2024 | N. N. | tba |
| 5.12.2024 | Alberto Merici (Universität Heidelberg) | Logarithmic TC and saturated descent |
| 12.12.2024 | Riccardo Zuffetti (TU Darmstadt) | Lefschetz decompositions of Kudla-Millson theta functions. |
| 19.12.2024 | reserved | tba |
| 9.1.2025 | Nikolaos Tsakanikas (EPFL) | tba |
| 16.1.2025 | Claudius Heyer (Paderborn) | A 6-functor formalism for smooth mod p representations |
| 23.1.2025 | Ana Maria Botero (Bielefeld) | tba |
| 30.1.2025 | Benoît Cadorel (Nancy) | tba |
Abstracts
Yu Min: Classicality of derived Emerton—Gee stack for general groups.
Abstract: In this talk, we will define the Emerton—Gee stack for a general group scheme using the Tannakian formalism and discuss its representability. Moreover when the group scheme is a generalised reductive group as defined by Paškūnas—Quast, we will define a derived version of the Emerton—Gee stack using prismatic theory and show how it is controlled by its underlying classical stack.
Dimitri Wyss: Non-archimedean integration on quotients.
Motivated by mirror symmerty, Batyrev defines ‘stringy’ Hodge numbers for a variety X with Gorenstein canonical singularities using motivic integration. While in general it is an open question, whether these numbers are related to a cohomology theory, the orbifold formula shows, that if X has quotient singularities, they agree with Chen-Ruan’s orbifold Hodge numbers.
I will explain how to generalize this orbifold formula to quotients of smooth varieties by linear algebraic groups. As an application we obtain identifications of stringy Hodge numbers with enumerative invariants, so-called BPS-invariants, in the case when X is the moduli space of an abelian category of homological dimension 1, for example the moduli space of semi-stable vector bundles on a curve. This is joint work with Michael Groechenig and Paul Ziegler.
Pol van Hoften: A new proof of the Eichler—Shimura congruence relation
Abstract: Associated to a modular form f is a two-dimensional Galois representation whose Frobenius eigenvalues can be expressed in terms of the Fourier coefficients of f, using a formula known as the Eichler—Shimura congruence relation. This relation was proved by Eichler—Shimura and Deligne by analyzing the mod p (bad) reduction of the modular curve of level Γ0(p). In this talk, I will discuss joint work with Patrick Daniels, Dongryul Kim and Mingjia Zhang, where we give a new proof of this congruence relation that happens “entirely on the generic fibre” and works in great generality.
Georg Tamme: A homotopical approach to crystalline cohomology
A classical idea to define a p-adic cohomology theory for smooth varieties X over a perfect field k of characteristic p, due to Grothendieck and Monsky-Washnitzer, is to choose a lift X^ of X over the ring of Witt vectors W(k) of k (if it exists) and consider the de Rham cohomology of X^. The problem with this approach is the non-unicity of such a lift. In this talk, I will present a way to make this idea work using an explicit infinity category of W(k)-algebras whose higher morphisms in some sense take care of the non-unicity mentioned above. Every smooth k-algebra has a lift to this category, unique up to coherent homotopy. One can then define its de Rham cohomology, which easily globalises to smooth, non-affine k-schemes. The cohomology obtained in this way is isomorphic to crystalline cohomology of X. I will also indicate how to use this formalism to define a category of crystals, coefficients for de Rham cohomology. This is joint work with Moritz Kerz.
Tasho Kaletha: On the endoscopic classification of representations of classical groups
The endoscopic classification of representations of quasi-split symplectic and orthogonal groups is a celebrated result of Arthur (extended to quasi-split unitary groups by Mok) which has had wide applications to representation theory and number theory. It is a collection of many interrelated statements that in particular gives a classification of the irreducible admissible representations of such groups over local fields, and the discrete automorphic representations of such groups over number fields, in terms of A-packets.
Until recently this result was conditional on a number of unproven statements, in particular the construction, character identities, and intertwining relations, of co-tempered A-packets over non-archimedean local fields. These statements have now been proved in joint work with Atobe, Gan, Ichino, Minguez, and Shin, rendering Arthur’s result conditional only on the validity of the weighted fundamental lemma.
In this talk I will explain the general statements in Arthur’s classification, the role played by co-tempered A-packets, and the work that supplied the missing results.
Alberto Merici Logarithmic TC and saturated descent
We prove a descent result for logarithmic TC (that we call “saturated descent”) that allows to compute logTC of a log ring as TC of a (usual) stack in many cases of interests.
As an application, we prove a comparison between the site-theoretic prismatic cohomology of Koshikawa-Yao and the graded pieces of the motivic filtration of logTP, generalising Bhatt—Scholze’s comparison, and a log variant of the Beilinson fibre square of Antieau-Mathew-Morrow-Nikolaus. This is a joint work with F. Binda, T. Lundemo and D. Park.
Riccardo Zuffetti: Lefschetz decompositions of Kudla-Millson theta functions
In the 80’s Kudla and Millson introduced a theta function in two variables, nowadays known as the Kudla-Millson theta function. This behaves as a Siegel modular form with respect to one variable, and as a closed differential form on an orthogonal Shimura variety with respect to the other variable. In this talk I show that the Lefschetz decomposition of (the cohomology class of) this theta function provides simultaneously the modular decomposition in Eisenstein, Klingen and cuspidal parts. Time permitting, I will report on geometric applications. This is joint work with J. Bruinier.
Claudius Heyer: A 6-functor formalism for smooth mod p representations
The formalism of six operations was introduced by Grothendieck to show that many phenomena in the étale cohomology of schemes can be formally deduced from a small set of axioms. Since then these six operations have been constructed in many other contexts like D-modules, motives and rigid-analytic geometry. But only recently has there been a formal definition of a 6-functor formalism, mainly due to Liu–Zheng and then further simplified by Mann in his PhD thesis. Also in Fargues–Scholze’s geometrization of the local Langlands correspondence the six operations are a guiding theme.
In this talk I will report on joint work with Lucas Mann where we construct a full 6-functor formalism in the setting of smooth representations of p-adic Lie groups with mod p coefficients (in fact, we allow arbitrary discrete coefficient rings). If time permits we use the formalism to construct a canonical anti-involution on derived Hecke algebras generalizing earlier work by Schneider-Sorensen.