CRC 326 - GAUS

Topographs and some infinite series
2024-12-03 16:00 - 17:00 (Europe/Berlin) | Zoom
International Seminar on Automorphic Forms
Cormac O’Sullivan (CUNY)): Topographs and some infinite series
The Fibonacci numbers are a familiar recursive sequence. Topographs are a kind of two dimensional version conjured up by J.H. Conway in his study of integral binary quadratic forms. These forms are ax^2 + bxy + cy^2 with integer coefficients, and have a long history in number theory. We’ll review Conway’s classification of topographs into 4 types and look at some new discoveries. Applications are to new class number formulas and a simplification of a proof of Gauss related to sums of three squares. We’ll also see how several infinite series over all the numbers in a topograph may be evaluated explicitly. This generalizes and extends results of Hurwitz and more recent authors and requires a certain Poincare series.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Vector bundles on curves
2024-12-03 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Rizacan Ciloglu : Existence of semi-stable vector bundles and examples
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Seminar on Arithmetic Geometry
2024-11-29 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Felipe Espreafico (IMJ-PRG): Gauss-Main Connection in Disguise: A «quasi-modularity» for Gromov-Witten invariants for the Quintic Threefold
Gromov-Witten invariants and modularity are topics that often come together. In this talk, we will explore a type of quasi-modularity for the genus zero invariants for the quintic threefold. We start by explaining how classical Eisenstein series are related to periods of the Weistrass family of Elliptic Curves. A similar relation may be observed by looking at periods of the mirror quintic family: that generating functions for the genus zero invariants can be written in terms of solutions to certain differential systems coming from the Gauss-Manin connection that generalize the classical Ramanujan equations that give rise to Eisenstein series. This is part of larger program called Gauss-Manin connection in Disguise, that can be also applied in other contexts. We finish by briefly discussing other applications and further questions.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

The degree of algebraic cycles on hypersurfaces
2024-11-29 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Matthias Paulsen (Universität Marburg)
Abstract: Let X be a very general hypersurface of dimension 3 and degree d at least 6. Griffiths and Harris conjectured in 1985 that the degree of every curve on X is divisible by d. Substantial progress on this conjecture was made by Kollár in 1991 via degeneration arguments. However, the conjecture of Griffiths and Harris remained open in any degree d. In this talk, I will explain how to prove this conjecture (and its higher-dimensional analogues) for infinitely many degrees d.

Quantum periods, toric degenerations and intrinsic mirror symmetry
2024-11-29 10:30 - 11:30 | Heidelberg, MATHEMATIKON, SR 11, INF 205, Heidelberg, Germany
Samuel M Johnston, Heilbronn Research Fellow at Imperial College London

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2024-11-29 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Theresa Kaiser (Universität Heidelberg): Cohen–Macaulay modules and complete intersection rings

CRC-Colloquium
2024-11-28 15:30 - 18:30 | Heidelberg, MATHEMATIKON, Konferenzraum, 5. OG, Campus Im Neuenheimer Feld (INF), INF 205, Heidelberg, 69120, Germany
15:20 Coffee (or earlier)
15:30 – 16:30 Timo Richarz (TU Darmstadt): Reduction of Shimura Varieties
16:30 Coffee and Cake
17:15 – 18:15 Jens Eberhardt (Universität Mainz): K-motives and Local Langlands
18:45 Dinner
Abstract – Timo Richarz: Reduction of Shimura Varieties
The general theory of Shimura varieties, first developed by Deligne in the 1970s, extends classical objects such as modular curves, moduli of polarized abelian varieties and Hilbert-Blumenthal varieties. The varieties play a crucial role in the search for higher reciprocity laws within the Langlands program. The arithmetic properties of Shimura varieties, in particular their reduction to positive characteristic, have enabled remarkable advances in arithmetic geometry in recent decades. In this talk, I will give an overview of recent results concerning the geometry of Shimura varieties in their reduction to positive characteristic.
Abstract – Jens Eberhardt: K-motives and Local Langlands
In this talk, we construct a geometric realisation of the category of representations of the affine Hecke algebra. 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.

The direct summand theorem
2024-11-28 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 7: Marlon Kocher (Universität Heidelberg): Perfectoid spaces I: Tilting of Rational Subsets

Oberseminar Algebra und Geometrie
2024-11-27 16:00 - 17:00 | Frankfurt, RM-Str. 6-8, R. 308
Jiaming Chen (Universität Frankfurt): Convex Fujita numbers
Abstract: I will discuss joint work with Alex Küronya, Yusuf Mustopa, and Jakob Stix on the effective global generation of adjoint line bundles on smooth projective varieties. To measure effectiveness, we introduce the concept of the convex Fujita number of a smooth projective variety. I will present several examples and explore its relation to fundamental groups.

Hodge Theory
2024-11-27 12:00 - 13:00 | Mainz, Hilbertraum (05-432)
Luca Passolunghi (Universität Mainz): Kähler manifolds I

Knots, q-series, and modular forms
2024-11-26 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Matthias Storzer (University College Dublin): Knots, q-series, and modular forms
To study knots, we use knot invariants like the colored Jones polynomials (CJP). For alternating knots, it is known that the CJP converge to a well-defined q-series, the tail of the CJP. For several but not all knots with up to 10 crossings, the tail of the CJP can be written as a product of (partial) theta functions and thus has modular properties. In this talk, we present a general formula for a class of knots.Moreover, we argue that the tail of the CJP for some knots does not have any modular properties. We also briefly discuss potential topological interpretations of the (non-)modularity.This is joint work with Robert Osburn.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Vector bundles on curves
2024-11-26 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Saskia Kern : Harder–Narasimhan filtration
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Seminar on Arithmetic Geometry
2024-11-22 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Andrés Jaramillo Puentes (Universität Tübngen): A Wall-Crossing Formula for Motivic Gromov-Witten Invariants
In enumerative geometry, Gromov-Witten invariants play a central role in counting curves on algebraic varieties, and their variations under different conditions provide a rich framework for understanding moduli spaces. In recent years, there has been significant progress in developing enriched versions of these invariants within the framework of motivic homotopy theory, leading to what we now call motivic Gromov-Witten invariants. Motivic invariants encode additional algebraic structure over the Grothendieck-Witt ring of a base field, allowing for finer distinctions in curve counts, particularly over fields with nontrivial real structure.
In this talk, we discuss a wall-crossing formula for motivic Gromov-Witten invariants. Specifically, we explore how variations in point conditions and configurations influence the values of these invariants, and demonstrate how these changes can be systematically tracked using a motivic analogue of classical wall-crossing phenomena. We will illustrate how this formula provides a mechanism to relate invariants associated with distinct configurations by tracking contributions along certain “walls” in the parameter space, which play an analogous role to wall-crossing in real enumerative geometry.
Additionally, we will present applications of this formula to specific enumerative problems, showcasing how the motivic perspective not only recovers known real and complex cases but also opens new pathways for counting problems over arbitrary fields. This development lays the groundwork for future research, providing a powerful tool to bridge combinatorial and motivic techniques in tropical and algebraic geometry.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Tropical refined curve counting and mirror symmetry
2024-11-22 14:15 - 15:15 | Heidelberg
Qaasim Shafi (Heidelberg)
An old theorem, due to Mikhalkin, says that the number of
rational plane curves of degree d through 3d-1 points is equal to a
count of tropical curves (combinatorial objects which are more amenable
to computations). There are two natural directions for generalising this
result: extending to higher genus curves and allowing for more general
conditions than passing through points. I’ll discuss a generalisation
which does both, as well as recent work connecting it to mirror symmetry
for log Calabi-Yau surfaces. This is joint work with Patrick
Kennedy-Hunt and Ajith Urundolil Kumaran.
https://sites.google.com/view/heidelbergag/algebraic-geometry-seminar

Heights of modular polynomials
2024-11-22 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Prof. Dr. Florian Breuer (University of Newcastle)
For every positive integer $N$, the modular polynomial $Phi_N(X,Y)$ has integer coefficients and vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of order $N$. These polynomials have applications in cryptography and define integral (but singular) models for the modular curves $X_0(N)$. Their coefficients grow rapidly with $N$. In this talk, I will explain recent joint work with Fabien Pazuki and Desir’ee Gij’on G’omez obtaining explicit upper and lower bounds on the size of these coefficients. Our methods also lead to explicit bounds on the heights of Hecke images. If time allows, I can also outline analogous results for Drinfeld modular polynomials.

Differential operators on automorphic forms, special functions, and arithmetic applications
2024-11-22 10:30 - 11:30 | Heidelberg, MATHEMATIKON, SR 11, INF 205, Heidelberg, Germany
Tomoyoshi Ibukiyama, Professor Emeritus Department of Mathematics Graduate School of Science, Osaka University
A theory of differential operators on automorphic forms
which preserve automorphy after restrictions of
the domains have a long history and turned out
to include a nice theory of special functions.
We will give rough outline on results
since 1990 (partly a joint work with D. Zagier) and
explain several arithmetic applications.

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2024-11-22 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Alireza Shavali (Universität Heidelberg): Congruence modules and Wiles defect

Anabelian geometry - Mochizuki’s proof of the Hom-Conjecture [d’après Faltings]
2024-11-21 14:00 - 17:30 | Heidelberg
14:00 – 15:30 Talk 3: Leonie Scherer (Goethe Universität): Unipotent Tannakian categories

The direct summand theorem
2024-11-21 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 6: Yanik Kleibrink (Universität Frankfurt): Adic spaces II

Hodge Theory
2024-11-20 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Andreas Gieringer (Universität Mainz): Harmonic forms I

Cones of Noether-Lefschetz divisors and moduli of hyperkähler manifolds
2024-11-19 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Laure Flapan (Michigan State University): Cones of Noether-Lefschetz divisors and moduli of hyperkähler manifolds
We describe how to compute cones of Noether-Lefshetz divisors on orthogonal modular varieties with a particular view towards moduli spaces of polarized K3 surfaces and hyperkähler manifolds. We then describe some geometric applications of these cone computations for these moduli spaces. This is joint work with I. Barros, P. Beri, and B. Williams.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Vector bundles on curves
2024-11-19 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Michelle Klemt : Vector bundles on curves of low genus
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Seminar on Arithmetic Geometry
2024-11-15 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Ludwig Modin (Leibniz Universität Hannover): Moduli spaces for Theta-strata and non-reductive quotients
The U-hat theorem of Bérczi, Doran, Hawes and Kirwan gives conditions for when a linear action of a complex graded unipotent group admits a geometric quotient, it is one of the key results non-reductive geometric invariant theory is built on.
We give a stacky re-interpretation of this theorem in terms of Theta-strata, as introduced by Halpern-Leistner, of algebraic stacks. As a corollary we generalize the U-hat theorem to not necessarily linear actions of graded unipotent groups over a Noetherian base scheme.
Zoom (635 7328 0984), Kenncode: kleinste sechsstellige Primzahl

Antisymmetry in the theory of rigid meromorphic cocycles
2024-11-15 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Sören Sprehe (Universität Bielefeld)
Around six years ago Darmon and Vonk initiated the theory of p-adic singular moduli for real quadratic fields by defining “rigid meromorphic cocycles”. These are elements of the first cohomology group of Ihara’s group SL2(Z[1/p]) with values in the group of rigid meromorphic functions on Drinfeld’s upper half-plane. Using rigid meromorphic cocycles, Darmon and Vonk assign to each pair of real quadratic irrationalities a p-adic number. The two irrationalities play a vastly different role in the construction of this assignment. However, it is expected to behave like the difference of two classical singular moduli – in particular, it should be anti-symmetric in the argument. We will use the recent work of Darmon, Gehrmann and Lipnowski on rigid meromorphic cocycles for higher dimensional orthogonal groups to give a new, symmetric construction of this function.

Jacobi forms, mock modular forms and qMZVs in enumerative geometry
2024-11-15 11:15 - 12:15 | Heidelberg, MATHEMATIKON, SR 11, INF 205, Heidelberg, Germany
Jan-Willem van Ittersum (Cologne)
Abstract: There are several instances where Gromov-Witten invariants can be expressed in terms of (quasi)Jacobi forms. In other examples in enumerative geometry, one also encounters mock modular forms or even q-analogues of multiple zeta values. We explain the origin and properties of these series and provide examples of their occurrences as generating series of geometrical invariants.

https://sites.google.com/view/heidelbergag/algebraic-geometry-seminar

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2024-11-15 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Andrea Conti (Universität Heidelberg): Overview

Moduli of Quiver Representations and GIT Quotients
2024-11-14 14:00 - 16:30 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
14:00 – 15:00 Talk 2.1: Nicole Müller (Goethe Universität): Affine GIT
Coffee break
15:30 – 16:30 Talk 2.2: Felix Göbler (Goethe Universität): Projective GIT

The direct summand theorem
2024-11-14 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 5: Lars Wüste-Schmülling (Universität Heidelberg): Adic spaces I

Hodge Theory
2024-11-13 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Timon Tausendpfund (Universität Mainz): Kähler metrics

On an extension of the Rohrlich-Jensen formula
2024-11-12 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Leila Smajlovic (University of Sarajevo): On an extension of the Rohrlich-Jensen formula
We revisit the Rohrlich-Jensen formula and prove that, in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur-Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass-Selberg relation. In this form, we develop a Rohrlich-Jensen formula associated to any Fuchsian group Γ of the first kind with one cusp by employing a type of Kronecker limit formula associated to the resolvent kernel. We present two examples of our main result: First, when Γ is the full modular group; and second when Γ is an Atkin-Lehner group Γ0(N)+. This work is joint with James Cogdell and Jay Jorgenson.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)
