Seminar des Graduiertenkollegs 2553

RTG Seminar Winter term 2023/24

The Thursday morning seminar (10:15-11:45 in WSC-N-U-3.05) will be the “Research Training Group Seminar” where members of the RTG (PhD students, post-docs,…) present their results. Sometimes, we also have speakers from other places. Depending on the number of speakers and on the proposed topic, a speaker could use one or two sessions.

12.10.2023	Andrés Jaramillo-Puentes	Tropical Methods in $\mathbb{A}^1$-Enumerative Geometry
26.10.2023	Ryosuke Shimada (Univ. of Tokyo)	Beyond the cases of Coxeter type
9.11.2023	Luca Marannino	Diagonal classes and explicit reciprocity laws
16.11.2023		General Assembly of the RTG
23.11.2023	Nicolas Dupré	Homotopy classes of simple pro-$p$ Iwahori-Hecke modules
30.11.2023		9am-12pm Trial run for RTG evaluation (talks given by PhD students, poster session)
7.12.2023		9am-12pm Trial run for RTG evaluation (plenary discussion)
21.12.2023	Niklas Müller	Non-Vanishing Results for Anti-Canonical Bundles
11.1.2024	Riccardo Tosi	A geometric approach to irrationality proofs for zeta values
18.1.2024	Chirantan Chowdhury	Applications of Six Functor Formalisms in Motivic Homotopy Theory for Algebraic Stacks
25.1.2024	Ravjot Kohli	Reduction of almost Kähler manifolds and an embedding result
1.2.2024	Giulio Marazza	Normality of integral models of Shimura varieties with $\Gamma_1(p)$-level structure

Abstracts

Andrés Jaramillo-Puentes: Tropical Methods in $\mathbb{A}^1$-Enumerative Geometry

Motivic homotopy theory allows us to tie together the results from classical and real enumerative geometry, and yield invariant counts of solutions to geometric questions over an arbitrary field $k$. The enumerative counts are valued in the Grothendieck-Witt ring ${\rm GW}(k)$ of nondegenerate quadratic forms over $k$ and we call it quadratic enrichment. In this talk, I will detail some examples of these counts and I will present a quadratically enriched version of the Bernstein–Khovanskii–Kushnirenko theorem, as well as a quadratically enriched version of the Correspondence Theorem for counting curves passing through configurations of $k$-rational points and allowing for computations of arithmetic Gromov-Witten invariants.

Ryosuke Shimada (Univ. of Tokyo): Beyond the cases of Coxeter type

The notion of affine Deligne-Lusztig variety (ADLV) was first introduced by Rapoport, which has been applied to number theory such as the study of Shimura varieties and a realization of the local Langlands correspondence. Many of these applications make use of the special cases where the ADLV admits a simple description. One large class of such cases is the ADLV of Coxeter type, which has been already classified by Görtz-He-Nie. However, many people (Chan-Ivanov, Howard-Fox-Imai, Trentin,…) have found examples which are not of Coxeter type but admit a simple description. In this talk, I will talk about recent progress on this kind of new examples, including my recent work for $GL_n$.

Luca Marannino: Diagonal classes and explicit reciprocity laws

Theorems known as reciprocity laws are ubiquitous in number theory. In this talk I will discuss a particular instance of $p$-adic reciprocity law that shall appear in my PhD thesis. This explicit reciprocity law relates certain diagonal classes on a triple product of modular curves to $p$-adic special values of a suitable $p$-adic $L$-function, extending work of Darmon-Rotger and Bertolini-Seveso-Venerucci. I will present this result and explain how it can be applied to address certain cases of the Birch and Swinnerton-Dyer conjecture.

Nicolas Dupré: Homotopy classes of simple pro-p Iwahori-Hecke modules

Let $G$ denote the group of rational points of a split connected reductive group over a non-archimedean local field of residue characteristic $p>0$. Over a field of characteristic $p$, we let $H$ denote the associated pro-$p$ Iwahori-Hecke algebra. In earlier joint work with J. Kohlhaase, we considered Hovey’s so-called Gorenstein projective model structure on ${\rm Mod}(H)$ and used it to study the relationship between ${\rm Mod}(H)$ and the category of smooth representations of $G$. In this talk, I will present new results on the structure of the homotopy category ${\rm Ho}(H)$ of ${\rm Mod}(H)$ with respect to this model structure. In particular, we obtain a classification of the isomorphism classes in ${\rm Ho}(H)$ of the supersingular simple $H$-modules when $G=GL_n$.

Niklas Müller: Non-Vanishing Results for Anti-Canonical Bundles

In this talk I want to present an existence result for anti-canonical sections on projective log pairs $(X, \Delta)$ which are invariant under the action of a commutative algebraic group acting on $X$. The proof relies heavily on methods from geometric invariant theory and I will try to emphasise how to use the methods we learned in this terms research seminar to attack such a problem.

Riccardo Tosi: A geometric approach to irrationality proofs for zeta values

The values of the Riemann zeta function at odd positive integers greater than 1 are conjectured to be transcendental, yet even their irrationality remains a mostly open question. Recently, Brown has drawn some connections with period integrals over the moduli space of smooth projective curves of genus zero with marked points, which have given new inputs to irrationality proofs for zeta values. In this talk, I will discuss Brown’s result and sketch the main methods that appear in the proof. I will then present some work in progress regarding possibile extensions of these methods to more general varieties and thus to irrationality proofs for a broader range of numbers, mainly concerning multiple polylogarithmic values.

Chirantan Chowdhury: Applications of Six Functor Formalisms in Motivic Homotopy Theory for Algebraic Stacks

The six functor formalism was formulated by Grothendieck to give a framework for the basic operations and duality statements for cohomology theories. In this talk, I shall provide an overview of extending the six functor formalism for Motivic Homotopy Theory from schemes to a large class of algebraic stacks. We shall also discuss developing such formalism in non-representable situations and understanding purity for such morphisms (joint work with Alessandro D’Angelo). In the later part of the talk, I shall discuss applications of such formalisms in constructing Chow weight structure for geometric motives on quotient stacks (joint work with Dhyan Aranha).

Ravjot Kohli: Reduction of almost Kähler manifolds and an embedding result

Given an isometric action of a compact Lie group $K$ on a Riemannian manifold $M$, the orbit space $M/K$ is a stratified space with each piece being a Riemannian manifold. In case $M$ is symplectic with a Hamiltonian $K$-action, the reduced space is a stratified space with each piece being a symplectic manifold. I show that when $M$ has both a Riemannian and symplectic structure, i.e., an almost Kähler structure, with a Hamiltonian and isometric group action, the reduced space is a stratified space with each piece being an almost Kähler manifold. Moreover, this reduced space has a stratified embedding into the orbit space such that restricted to each piece, it is a Riemannian embedding. Finally, I show how this embedding can be used to extend a result of Sjamaar for the duality of $L^2$ and intersection cohomology of Riemannian orbit spaces to the case of almost Kähler reduced spaces with two strata.

Giulio Marazza: Normality of integral models of Shimura varieties with $\Gamma_1(p)$-level structure

One aspect of the arithmetic theory of Shimura varieties is the construction of integral models of such objects over the ring of integers of (the completion at a prime of) their reflex field. One tries to find models with good geometric properties, for example smoothness or at least mild singularities, in order to compare the generic and the special fibers: such models exist in almost all cases of Iwahori level, while less is known in the presence of a “deeper” level structure. In this talk I will present the construction of integral models of some PEL Shimura varieties with pro-$p$ Iwahori level structure, due to Shadrach, and discuss some of their local geometric properties, in particular normality.