CRC 326 - GAUS

Moduli of Quiver Representations and GIT Quotients
2024-12-19 14:00 - 16:30 (Europe/Berlin) | Frankfurt, Rober-Mayer-Str. 10, Raum 711 klein
14:00 – 15:00 Talk 4.1: Arne Kuhrs (Goethe Universität): Affine Moduli Spaces of Quiver Representations
Coffee break
15:30 – 16:30 Talk 4.2: Johannes Horn (Goethe Universität): Moduli Spaces of Quiver Representations

Anabelian geometry - Mochizuki’s proof of the Hom-Conjecture [d’après Faltings]
2024-12-19 14:00 - 17:30 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
14:00 – 15:30 Talk 4: Marius Leonhardt (Goethe Universität): Construction of 𝔥 and Hodge-Tateness of rational sections
Coffee break
16:00 – 17:30 Talk 5: Morten Lüders (Universität Heidelberg): Bloch-Kato Selmer groups

Explicit methods for enumerative invariants using Siegel and mock modular forms: The K3 case
2024-12-19 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Abhiram Kidambi (MPI Leipzig)
Abstract: A conjecture for nearly two decades on the interface of algebraic geometry, number theory and topological string theory is the OSV conjecture which conjectures a form for the generating function of (certain) rank 0 Donaldson-Thomas invariants for CY threefolds, and this formulation goes hand in hand with understanding wall crossing phenomena. On the automorphic forms side, one often encounters mock modular forms and iterated integrals thereof. To understand the precise relation between mock-modularity, OSV conjecture is an open problem as well.
There are many problems in the same spirit, and we will visit the problem in the case of K3 surfaces and symmetric powers thereof. This case is a particularly nice exercise since the analogue of the OSV conjecture is a bit more restrictive (because there is only one polynomial term in the formal power series expansion which is needed to compute the enumerative invariants). This allows us to compute things rather explicitly using the theory of mock modular/Jacobi forms. Of particular interest are special points (called attractor points) where the generating function for enumerative invariants follow an OSV-type behaviour.
In this (chalk) talk, I will start with the statement of the problem for this particular case, introduce all the necessary objects (Siegel modular forms, mock modular forms, Jacobi forms) and explain how one can construct the generating function for enumerative invariants at special points (and generic points if time permits).
A conjecture for nearly two decades on the interface of algebraic geometry, number theory and topological string theory is the OSV conjecture which conjectures a form for the generating function of (certain) rank 0 Donaldson-Thomas invariants for CY threefolds, and this formulation goes hand in hand with understanding wall crossing phenomena. On the automorphic forms side, one often encounters mock modular forms and iterated integrals thereof. To understand the precise relation between mock-modularity, OSV conjecture is an open problem as well.
There are many problems in the same spirit, and we will visit the problem in the case of K3 surfaces and symmetric powers thereof. This case is a particularly nice exercise since the analogue of the OSV conjecture is a bit more restrictive (because there is only one polynomial term in the formal power series expansion which is needed to compute the enumerative invariants). This allows us to compute things rather explicitly using the theory of mock modular/Jacobi forms. Of particular interest are special points (called attractor points) where the generating function for enumerative invariants follow an OSV-type behaviour.
In this (chalk) talk, I will start with the statement of the problem for this particular case, introduce all the necessary objects (Siegel modular forms, mock modular forms, Jacobi forms) and explain how one can construct the generating function for enumerative invariants at special points (and generic points if time permits).

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2024-12-20 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Junyan Xu (Universität Heidelberg): Wiles defect and free direct summands

Euler characteristics of moduli of twisted sheaves on Enriques surfaces
2024-12-20 11:00 - 12:00 | Heidelberg, MATHEMATIKON, SR10, INF 205, Heidelberg, Germany
Weisheng Wang, Utrecht Geometry Center (Utrecht University)

On local Galois deformation rings
2024-12-20 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Julian Quast (Universität Duisburg-Essen)
In joint work with Vytautas Paškūnas, we show that the universal framed
deformation ring of an arbitrary mod p representation of the absolute
Galois group of a p-adic local field valued in a possibly disconnected
reductive group G is flat, local complete intersection and of the
expected dimension. In particular, any such mod p representation has a
lift to characteristic 0. The work extends results of Böckle, Iyengar
and Paškūnas in the case G=GL_n. We give an overview of the proof of
this main result.

Seminar on Arithmetic Geometry
2024-12-20 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Can Yaylali (TU Darmstadt): A^1-homotopy theory of rigid analytic spaces
In this talk, I will report about work with Christian Dahlhausen (Heidelberg) on A^1-homotopy theory of rigid analytic spaces. The B^1-homotopy category has already been defined and studied by Ayoub and a full six-functor formalism was established by Ayoub-Gallauer-Vezzani. One drawback of the B^1-invariant theory is that analytic K-theory for rigid analytic spaces (as defined and studied by Kerz-Saito-Tamme) is not representable since it is not B^1-invariant. Thus we aim for an A^1-invariant version with coefficients in any presentable category. For the stable theory, we can prove the existence of a partial six-functor formalism for analytifications of schemes and algebraic morphisms between them by using the results of Ayoub’s thesis. Furthermore, using coefficients in condensed spectra, we can represent analytic K-theory as the P^1-ring spectrum Z x BGL. If time permits I will also highlight some of the remaining questions.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Hodge Theory
2025-01-08 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Nutsa Gegelia (Universität Mainz): Variations of Hodge structures II

A new class of maximal hyperelliptic curves
2025-01-09 14:15 - 15:15 | Mainz, Hilbertraum (05-432)

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2025-01-10 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Andrea Conti (Universität Heidelberg): Patching

Towards results on the anticyclotomic Iwasawa theory of modular forms at inert primes via diagonal classes
2025-01-10 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Luca Marannino (IMJ-PRG Paris)
In this talk we outline an approach to the study of anticyclotomic Iwasawa theory of modular forms when the fixed prime p is inert in the relevant quadratic imaginary field. Following ideas of Castella-Do and Alonso-Castella-Rivero for the “p split” case, one can envisage a construction of an anticyclotomic Euler system arising from a suitable manipulation of diagonal cycles (considered in previous works of Darmon-Rotger and Bertolini-Seveso-Venerucci). We will report on this work in progress, trying to underline the main difficulties arising in the “p inert” setting.

Vector bundles on curves
2025-01-14 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Yanik Kleibrink (Goethe Universität Frankfurt): Stacks and examples
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Counting and equidistribution
2025-01-14 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Yiannis Petridis (UCL): Counting and equidistribution
I will discuss how counting orbits in hyperbolic spaces lead to interesting number theoretic problems. The counting problems (and the associated equidistribution) can be studied with various methods, and I will emphasize automorphic form techniques, originating in the work of H. Huber and studied extensively by A. Good. My collaborators is various aspects of this project are Chatzakos, Lekkas, Risager, and Voskou.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Hodge Theory
2025-01-15 12:00 - 13:30 | Mainz, Hilbertraum (05-432)

Anabelian geometry - Mochizuki’s proof of the Hom-Conjecture [d’après Faltings]
2025-01-16 14:00 - 17:30 | Heidelberg
14:00 – 15:30 Talk 6: Ruth Wild (Goethe Universität): Hodge-Tate sections are geometric up to torsion
Coffee break
16:00 – 17:30 Talk 7: Benjamin Steklov (Goethe Universität): Preparations for Proposition 11: 𝑝-divisible groups

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2025-01-17 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Gebhard Böckle (Universität Heidelberg): Galois deformation conditions

Cones of Noether--Lefschetz divisors
2025-01-17 11:00 - 12:00 | Heidelberg, MATHEMATIKON, SR 10, INF 205, Heidelberg, 69124, Germany
Dr. Brandon Williams, Universität Heidelberg

The tropical 1-fold Abel-Prym map
2025-01-17 14:30 - 15:30 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Winter Semester 2024/25)
Giusi Capobianco (Università di Roma Tor Vergata)

Abstract: The algebraic Abel-Prym map relates the geometry of a double cover of algebraic curves with their corresponding Prym varieties. Birkenhake and Lange proved that the map has degree 2 if and only if the cover curve is hyperelliptic.

In the talk I will present joint work with Yoav Len, in which we investigate the 1-fold Abel-Prym map in the tropical setting and prove similar results. I will describe a new combinatorial construction of hyperelliptic double covers of metric graphs and prove that the tropical Abel-Prym map is a harmonic morphism of degree 2. Furthermore, we will see that the Jacobian of the image of this map is isomorphic, as pptav, to the Prym variety of the cover. When the double cover is not hyperelliptic however, contrary to the algebraic result, the tropical Abel-Prym map is almost never injective. I will provide counterexamples and discuss its image.

Seminar on Arithmetic Geometry
2025-01-17 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Jens Eberhardt (Gutenberg Universität Mainz): K-motives and Local Langlands
In this talk, we construct a geometric realization of the category of representations of the affine Hecke algebra and p-adic reductive groups.
For this, we introduce a formalism of K-theoretic sheaves (called K-motives) on stacks.
The affine Hecke algebra arises from the K-theory of the Steinberg stack, and we explain how to “categorify” this using K-motives.
Lastly, we discuss applications of K-motives to the local geometric Langlands program.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Tree spaces in tropical geometry
2025-01-17 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Winter Semester 2024/25)
Shelby Cox (MPI Leipzig)

Abstract: The space of phylogenetic trees on n leaves arises naturally in tropical geometry through the tropical Grassmannian trop Gr(2,n). The space of equidistant trees on n leaves is the tropicalization of M_{0,n}, which is tropically convex. In this talk, I will present recent work using tropical tree spaces for phylogenetic statistics and inference (joint with Curiel, Sabol, Talbut, and Yoshida). I will also discuss a conjectural analogue of the space of equidistant trees for type C (joint with Igor Makhlin).

Junior AG
2025-01-20 | Mainz
Maps between spherical group rings

Vector bundles on curves
2025-01-21 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Thibaud van den Hove (TU Darmstadt): Algebraicity of BunX,n
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Kudla-Millson lift on the symmetric space of SL_N
2025-01-21 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Romain Branchereau (McGill University): Kudla-Millson lift on the symmetric space of SL_N
I will present a construction of a map from the homology in degree N-1 of locally symmetric spaces associated to SL_N, to modular forms of weight N. The image of a cycle C by this map is a modular form whose Fourier coefficients are intersection numbers between C and a family of generalized modular symbols on the locally symmetric space. This map can be seen as a Kudla-Millson theta lift for the dual pair (SL_N, SL_2) and also resembles a construction of Bergeron-Charollois-Garcia.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Hodge Theory
2025-01-22 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Klaus Mattis (Universität Mainz): Mixed Hodge structures on smooth varieties II

Moduli of Quiver Representations and GIT Quotients
2025-01-23 14:00 - 16:30 | Frankfurt, Rober-Mayer-Str. 10, Raum 711 klein
14:00 – 15:00 Talk 5.1: Martin Ulirsch (Goethe Universität): Algebraic Aspects of Stability
Coffee break
15:30 – 16:30 Talk 5.2: Andrei Bud (Goethe Universität): Further Geometric Properties of Quiver Moduli

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2025-01-24 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Andrea Conti (Universität Heidelberg): Hecke eigensystems and Galois representations for PGL_2

Cycles on the moduli space of abelian varieties
2025-01-24 11:00 - 12:00 | Heidelberg, MATHEMATIKON, SR10, INF 205, Heidelberg, Germany
Prof. Dr. Rahul Pandharipande, Department of Mathematics, ETH Zürich
I will explain developments in the study of cycles on the moduli space of abelian varieties with connections to the moduli space of curves, the cohomology of the Lagrangian Grassmannian, modular forms, and the quantum cohomology of the Hilbert scheme of points of the plane.

Abstract six-functor formalisms in Motivic Homotopy Theory
2025-01-24 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Chirantan Chowdhury (TU Darmstadt)
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 give an overview of abstract six-functor formalisms in the language of $infty$-categories followed by its applications in the setting of Motivic Homotopy Theory. After introducing the setup, we shall show how one can extend such formalism from smaller to larger categories (for e.g. : schemes to algebraic stacks). If time permits, I would like to explain further extensions of such results to non-representable situations (joint work with Alessandro D’Angelo).

Seminar on Arithmetic Geometry
2025-01-24 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Maximilian Hauck (MPIM Bonn): Stacks in the p-adic Hodge theory of formal schemes
I will briefly review the stacky approach to p-adic cohomology theories due to Bhatt–Lurie and Drinfeld using a new view on the filtered prismatisation recently found by Gardner–Madapusi. Then I will show how to use these tools to obtain a new proof of the Beilinson fibre square of Antieau–Mathew–Morrow–Nikolaus and a generalisation of a comparison theorem between rational arithmetic étale cohomology and crystalline cohomology of Colmez–Niziol.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Vector bundles on curves
2025-01-28 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Paul Siemon: Geometry of BunX,n
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)
