Workshop: Ramification in geometric Langlands and non-abelian Hodge theory
2024-09-09 - 2024-09-13 | Heidelberg, MATHEMATIKON, Hörsaal

Junior Retreat
2024-09-16 - 2024-09-19 | Jugendherberge Würzburg

Initial degenerations of homogeneous varieties
2024-09-17 16:00 - 18:00 (Europe/Berlin) | Frankfurt, Rober-Mayer-Str. 10, Raum 711 klein
Victoria Schleis (Durham University):
Abstract: We study the geometry of initial degenerations of homogeneous subvarieties of (multi-)projective space using polyhedral geometry. In 2017, Corey characterized initial degenerations of the Grassmannian through limits of strata in a matroid stratification of the Grassmannian corresponding to the weight vector of the initial degeneration. Since then, similar results have been proven for the flag variety and Spinor varieties. In my talk, I will discuss recent work in progress extending this study beyond specific examples to homogeneous subvarieties of (multi)-projective space. The talk is based on joint work in progress with George Balla, Dan Corey, and Igor Mahklin.

Plenary Meeting
2024-09-19 16:15 - 23:30 | Darmstadt
16:15 Plenary Meeting, at location of barbecue.
17:00 Barbecue at https://tsg-1846.de/angebote/abteilung/fussball/grillhuette

Conference: AMH 2024 (Anabelian and Motivic Homotopy Theory)
2024-09-23 - 2024-09-28 | Heidelberg

Workshop Tropical geometry: Moduli spaces and matroids
2024-10-07 - 2024-10-11 | Frankfurt, Hilbertraum, Rober-Mayer-Str. 6-8, Raum 302

Workshop: Methods in Mixed Characteristic Geometry
2024-10-07 09:00 - 2024-10-11 14:00 | Mainz, Schulz-Horner-Gebäude, Lecture Hall C02
https://sites.google.com/view/mmcg2024

GAUS Junior Symposium
2024-10-14 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß

Vector bundles on curves
2024-10-15 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Timo Richarz (TU Darmstadt): Introduction and distribution of talks
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

The direct summand theorem
2024-10-17 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 1: Immanuel Klevesath (Universität Heidelberg): Almost modules and almost commutative algebra

Seminar on Arithmetic Geometry
2024-10-18 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Joakim Faergeman (Yale University): Motivicity of rigid G-local systems on curves
Abstract: A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. While a classification of motivic local systems is evidently out of reach, Simpson conjectured that for a reductive group G, rigid G-local systems with suitable finiteness conditions at infinity are motivic. This was proven for curves when G=GL_n by Katz who classified such rigid local systems. In this talk, we discuss our generalization of Katz’ theorem to a general reductive group. Our proof goes through the (tamely ramified) categorical geometric Langlands program in characteristic zero.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

On the cohomology of $SL(n,mathbb Z)$ beyond the "stable range"
2024-10-22 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Harald Grobner (University of Vienna): On the cohomology of $SL(n,mathbb Z)$ beyond the “stable range”
The cohomology of the group $SL(n,mathbb{Z}), n>1$, plays a fundamental role in geometry, topology and representation theory, while yielding many number theoretical applications: For instance, Borel used his description of $H^*(SL(n,mathbb Z))$ to compute the algebraic K-theory of the integers; whereas the (non-)vanishing of $H^*(SL(n,mathbb Z))$ tells a lot about the existence of certain automorphic forms. In this talk we will study the cohomology of $SL(n,mathbb Z)$, „right outside“ of what one calls the stable range. More precisely, we will show new non-vanishing results in degrees n−1 and n. As a byproduct, we will also answer a question, recently asked by F. Brown for n=6 and explain a phenomenon for n=8, which has been considered by A. Ash. (This is joint work with N. Grbac.)
https://tu-darmstadt.zoom.us/j/68048280736

Oberseminar Algebra und Geometrie
2024-10-23 16:00 - 17:00 | Frankfurt, RM-Str. 6-8, R. 308
Marius Leonhardt (Universität Frankfurt): Affine abelian non-abelian Chabauty
Abstract: The central question of this talk is how to find all integral points on affine hyperbolic curves. For example, which integers x,y satisfy y^2 = x^3 + x^2 + x + 1?
We approach this problem by introducing Kim’s non-abelian Chabauty method, which constructs p-adic analytic functions that have the integral points among their zeroes. In joint work with M. Lütdke and J.S. Müller, we showed that the abelian version of this method succeeds if the curve satisfies a certain inequality involving its genus and the Mordell-Weil rank of its Jacobian.
This talk is an introduction to the Chabauty–Kim method. I will present the above result and report on work in progress with M. Lüdtke that turns it into an algorithm determining the integral points on the curve.

The direct summand theorem
2024-10-24 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 2: Nils Tietke (Universität Heidelberg): Almost étale extensions and the almost purity theorem in characteristic

On the algebraic K-theory of algebraic tori
2024-10-24 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Florian Riedel (Kopenhagen)
Abstract:
I will describe work in progress joint with Bai, Carmeli and Juran. A classical computation by Quillen expresses the algebraic K-theory spectrum of the ring of Laurent polynomials as the group ring of S^1 over the K-theory of the base field. We generalize this by showing that the algebraic K-theory spectrum of a not-necessarily split algebraic torus is given by the group ring of the delooping of the character lattice of the torus, thus showing that a version of Cartier duality between algebraic and topological tori holds on the level of K-theory. We do this by constructing a motivic Fourier transform which is of independent interest and also recover an explicit formula of Merkujev-Panin for K_0 in a straightforward way.

A moduli-theoretic approach to Sen theory
2024-10-25 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Ben Heuer (Universität Frankfurt)
Classical Sen theory describes C_p-semilinear representations of Galois groups of p-adic fields in terms of linear algebra data called Sen modules. I will first explain how this theory can be reinterpreted geometrically in terms of v-vector bundles, creating a relation to the p-adic Simpson correspondence. I will then explain how one can use this to upgrade Sen’s Theorem to a comparison of moduli spaces of Galois representations and Sen modules: This explains known subtleties in classical Sen theory in a geometric fashion. Finally, I will describe how we can use this to study moduli spaces of (phi,Gamma)-modules. This is joint work-in-progress with Eugen Hellmann.

Vector bundles on curves
2024-10-29 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Julian Born: Vector bundles
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Exact formulae for ranks of partitions
2024-10-29 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Qihang Sun (University of Lille): Exact formulae for ranks of partitions
Dyson’s ranks provided a new understanding of the integer partition function, especially of its congruence properties. In 2009, Bringmann used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this talk, we will prove that the asymptotic formula, when summing up to infinity, converges and becomes a Rademacher-type exact formula for the rank of partitions.

https://tu-darmstadt.zoom.us/j/68048280736

Moduli of twisted maps to smooth pairs
2024-10-29 16:15 - 17:15 | Heidelberg, MATHEMATIKON, SR 5, INF 205, Heidelberg, Germany
Robert Crumplin (Heidelberg)

The direct summand theorem
2024-10-31 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 3: Nils Witt (Universität Heidelberg): Non-Archimedean Banach Algebras

Anabelian geometry - Mochizuki’s proof of the Hom-Conjecture [d’après Faltings]
2024-10-31 14:00 - 17:30 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
14:00 – 15:30 Talk 1: Jakob Stix (Goethe Universität): Introduction
Coffee break
16:00 – 17:30 Talk 2: Jonathan Miles (Goethe Universität): Galois cohomology and Hodge-Tate decomposition

On the K-theory of curves over number fields.
2024-10-31 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Rob de Jeu (Amsterdam)
Abstract: Borel defined regulators for the odd degree higher K-groups of a number field k and proved a relation between these and the values of the zeta-function of k at 2, 3, 4, …, generalising the classical relation between its residue at s=1 and the regulator of the unit group of the ring of integers. Similar results were proved and/or conjectured by Bloch and Beilinson for the K-groups of varieties over number fields. After a review of the background, we discuss some recent joint work with François Brunault, Liu Hang, and Fernando Rodriguez Villegas on K_2 of elliptic curves over certain cubic or quartic number fields, and, time permitting, how one can try to describe the K_4 of curves over number fields.

Seminar on Arithmetic Geometry
2024-11-01 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Pol van Hoften (VU Amsterdam): Igusa stacks and the cohomology of Shimura varieties
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 ?0(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”. More precisely, we prove a compatibility result between the cohomology of Shimura varieties of Hodge type and the Fargues?Scholze semisimple local Langlands correspondence, generalizing the Eichler–Shimura relation of Blasius–Rogawski. Our proof makes crucial use of the Igusa stacks that we construct, generalizing earlier work of Zhang in the PEL case.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Vector bundles on curves
2024-11-05 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Aaron Rauchfuß : Algebraic curves
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Counting characters on algebraic tori according to their Langlands L-functions
2024-11-05 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Ian Petrow (UCL): Counting characters on algebraic tori according to their Langlands L-functions
Given a connected reductive group G over a global field, Langlands introduced the automorphic L-function L(s, π, r) of a cuspidal automorphic representation π of G and a complex representation r of the L-group of G. While in general very little is known about Langlands L-functions, if G = T is a torus the properties of these L-functions can be obtained from class field theory and one can attempt to study analytic problems pertaining to them. In this talk I will describe some analytic results on automorphic characters of tori with respect to the analytic conductor of L(s, π, r), attempting to focus on the interplay of analytic and algebraic ideas that arise in the proofs.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

The direct summand theorem
2024-11-07 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 4: Anna Blanco Cabanillas(Universität Heidelberg): Perfectoid algebras

Moduli of Quiver Representations and GIT Quotients
2024-11-07 14:00 - 16:30 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
14:00 – 15:00 Talk 1.1: Kevin Kühn (Goethe Universität): Introduction to Moduli
Coffee break
15:30 – 16:30 Talk 1.2: Yiu-Man Wong (Goethe Universität): Algebraic Group Actions and Quotients

From SL(2) to SO(2): Rank-2 Vafa-Witten invariants
2024-11-08 11:15 - 12:15 | Heidelberg, MATHEMATIKON, SR 11, INF 205, Heidelberg, Germany
Simon Schirren (Rom)

Rigid cocycles and geodesics on Shimura curves
2024-11-08 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Håvard Damm-Johnsen (University of Oxford /MPIM Bonn)
Darmon and Vonk’s theory of rigid meromorphic cocycles gives a conjectural description of abelian extensions of real quadratic fields akin to the classical theory of complex multiplication. While their conjectures currently seem out of reach, the work of Darmon-Pozzi-Vonk uses p-adic families of Hilbert modular forms to give unconditional results in this direction. I will explain how a reinterpretation in terms of the so-called Kudla programme suggests an approach to extending their results. This point of view also gives a new proof of a theorem of Rickards on intersections of geodesics on Shimura curves, which is related to forthcoming work of Darmon-Vonk.

Seminar on Arithmetic Geometry
2024-11-08 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Siyan Daniel Li-Huerta (MPIM Bonn): Close fields and the local Langlands correspondence
There is a heuristic that, as ramification goes to infinity, p-adic fields tend to function fields. For Galois representations, this was made precise by Deligne, and for representations of p-adic groups, a similar result was shown by Kazhdan and Ganapathy. We present a proof that this is compatible with Fargues–Scholze’s local Langlands correspondence. The proof relies on carrying out Fargues–Scholze’s construction over the one-point compactification of the natural numbers.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl
