CRC 326 - GAUS

Hodge Theory
2025-01-15 12:00 - 13:30 (Europe/Berlin) | 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)

Uniform Non-vanishing of Hilbert Modular L-values
2025-01-28 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Liyang Yang (Princeton University): Uniform Non-vanishing of Hilbert Modular L-values
Let ℱ(k,q) be the set of normalized Hilbert newforms of weight k and prime level q. In this talk, we will present a uniform positive proportion of #{π∈ℱ(k,q):L(1/2,π)≠0} as #ℱ(k,q)→+∞. This is joint work with Zhining Wei and Shifan Zhao.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Hodge Theory
2025-01-29 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Georg Tamme (Universität Mainz): Mixed Hodge structures on smooth varieties III

Anabelian geometry - Mochizuki’s proof of the Hom-Conjecture [d’après Faltings]
2025-01-30 14:00 - 17:30 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
14:00 – 15:30 Talk 8: Amine Koubaa (Goethe Universität): Preparations for Proposition 11: the Tate conjecture
Coffee break
16:00 – 17:30 Talk 9: Magnus Carlson (Goethe Universität): Sections geometric up to torsion

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2025-01-31 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Alireza Shavali (Universität Heidelberg): Hecke eigensystems and Galois representations for quaternion
algebras

Multivariable Lubin–Tate Fontaine equivalence
2025-01-31 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Nataniel Marquis (IMJ Paris)
In 1991 J.-M. Fontaine proved an equivalence between continuons representations of $mathcal{G}_{mathbb{Q}_p}$ of finite type over $mathbb{Z}_p$ and the category of étale $(varphi,Gamma)$-modules over the ring of fonctions on a ghost circle. Recent developments in the mod $p$ Langlands program encouraged the search for similar equivalences for modules over multivariable rings. Work by Z’abr’adi and Carter-Kedlaya-Z’abr’adi fulfilled part of this expectation by establishing an equivalence between representations of finite products of $mathcal{G}_{mathbb{Q}_p}$ and multivariable cyclotomic $(varphi,Gamma)$-modules.
The first goal of this talk is to sketch a proof of a Lubin-Tate variant for a $p$-adic local field $K$. Namely, for a finite set $Delta$, we obtain an equivalence between continuous representations of $prod_{Delta} mathcal{G}_K$ and a category called the étale $(Phi_{Delta, q}times Gamma_{Delta,K,lt})$-modules over $mathcal{O}_{mathcal{E}_{K,Delta}}$ with finite projective dévissage. On the way to characterise the essential image of the functor $mathbb{D}_{Delta,lt}$, we will explain which properties of finite type representations over $mathbb{Z}_p$ are preserved by a Fontaine type functor. This will allow to give a theorem similar to the structure of finite type $mathbb{Z}_p$-modules for the underlying $mathcal{O}_{mathcal{E}_{K,Delta}}$ appearing in the previous equivalence. Finally, we will motivate how Lubin-Tate multivariable $(varphi,Gamma)$-modules should be more useful than cyclotomic ones to obtain a Colmez functor for $gl{n}{K}$.

Tropical correspondence theorems for plane curve counts over arbitrary fields
2025-01-31 14:00 - 15:00 | Heidelberg, MATHEMATIKON, SR10, INF 205, Heidelberg, Germany
Jr.-Prof. Dr. Sabrina Pauli (TU Darmstadt)
We study the problem of counting rational curves of fixed degree on a toric del Pezzo surface subject to point conditions. Over algebraically closed fields, this count is invariant under the choice of point conditions. Over non-algebraically closed fields, however, the invariance fails. For real numbers, Welschinger’s groundbreaking work introduced a signed count of real curves that restores invariance.
Building on this, Levine and Kass-Levine-Solomon-Wickelgren have developed curve counts over arbitrary fields that not only generalize Welschinger’s signed counts and classical counts over algebraically closed fields, but also encode much richer arithmetic information.
In this talk I will survey these different approaches to counting rational curves with point conditions and discuss a recent joint result with A. Jaramillo Puentes, H. Markwig, and F. Röhrle. We establish a tropical correspondence theorem for curve counts over arbitrary fields, identifying the count of algebraic curves with point conditions with a weighted count of their tropical counterparts with point conditions. The latter are combinatorial objects and there are several purely combinatorial methods to find all tropical curves with point conditions.

Seminar on Arithmetic Geometry
2025-01-31 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Paul Siemon (TU Darmstadt): Stacks of parabolic vector bundles and Katz’s middle convolution algorithm
I will discuss the geometry of the cuspidal locus inside stacks of parabolic vector bundles occurring in the tamely ramified geometric Langlands program. I will focus on the example of the stack of rank 2 bundles on the projective line with ramification at four points. Finally, I will outline how the considerations in this example could extend to other situations using Katz’s middle convolution algorithm.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Vector bundles on curves
2025-02-04 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Jon Miles: tCohomology of Bun_n
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Lifting of Maass forms to O(1,8n+1) and applications to the sup-norm problem
2025-02-04 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Ameya Pitale (University of Oklahoma): Lifting of Maass forms to O(1,8n+1) and applications to the sup-norm problem
In a joint paper with Yingkun Li and Hiroaki Narita, we had constructed liftings from Maass forms with respect to SL_2(Z) to Maass forms on O(1,8n+1), which violated the Generalized Ramanujan conjecture. These were constructed via Borcherds theta lifts and we were able to give explicit formulas for their Fourier coefficients. In a recent joint work with Simon Marshall and Hiroaki Narita, we first computed the Petersson inner product of the lift using the Rallis inner product formula. This essentially involves an archimedean integral computation. These are usually very complicated and intractable, but in this case we are able to get an exact formula for the Petersson norm. Explicit formulas for the Fourier coefficients and Petersson norm are the essential ingredients of one of the approaches to obtain sup-norm bounds on these Maass forms. Investigations regarding sup-norm bounds for modular forms in the GL(2) case has been recently a very active area of research. Using the method mentioned above, as well as a pre-trace formula approach, we obtain the first sup-norm bounds results for these orthogonal groups.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Hodge Theory
2025-02-05 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Tom Bachmann (Universität Mainz): Mixed Hodge structures on singular varieties

Moduli of Quiver Representations and GIT Quotients
2025-02-06 14:00 - 16:30 | Frankfurt, Rober-Mayer-Str. 10, Raum 711 klein
14:00 – 15:00 Talk 6.1: Andreas Gross (Goethe Universität): Framed Representations
Coffee break
15:30 – 16:30 Talk 6.2: Pedro Souza (Goethe Universität): Symplectic Geometry and Double Quivers

Anabelian geometry - Mochizuki’s proof of the Hom-Conjecture [d’après Faltings]
2025-02-06 14:00 - 17:30 | Heidelberg
14:00 – 15:30 Talk 10: Tim Holzschuh (Universität Heidelberg): Proof of the Hom conjecture

Pro-étale Q_p-cohomology of rigid analytic spaces
2025-02-06 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Arthur-César Le Bras (Strasbourg)
Abstract: The pro-étale Q_p-cohomology of rigid analytic spaces exhibits surprising features. I would like to explain a joint work with Johannes Anschütz and Lucas Mann which provides a conceptual explanation for these phenomena.
