Seminar of the Research Training Group 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)
14.12.2023	N. N.	tba
21.12.2023	Niklas Müller	tba
11.01.2024	Riccardo Tosi	tba
18.01.2024	Chirantan Chowdhury	tba
25.01.2024	Ravjot Kohli	tba
1.02.2024	Giulio Marazza	tba

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 $G=GL_n$.