CRC 326 - GAUS

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

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.

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

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

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.

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

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)

Topographs and some infinite series
2024-12-03 16:00 - 17:00 | 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)

Workshop: Combinatorial Christmas Geometry without Geometry
2024-12-04 - 2024-12-06 | Frankfurt am Main

The direct summand theorem
2024-12-05 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 8: Marvin Schneider (Universität Heidelberg): Perfectoid spaces II: Tate acyclicity

On the transcendental part of K3 surfaces associated with 3D Fano polytopes
2024-12-05 14:15 - 15:15 | Mainz, Hilbertraum (05-432)

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2024-12-06 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Sriram Chinthalagiri Venkata (Universität Heidelberg): Tate resolutions

Tropical refined curve counting and mirror symmetry
2024-12-06 11:00 - 12:00 | Heidelberg, MATHEMATIKON, SR 10, INF 205, Heidelberg, 69124, Germany
Dr. Qaasim Shafi, postdoctoral research associate at Heidelberg University

Chow-Heegner Points and Artin Formalism for triple product p-adic L-functions
2024-12-06 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Kazim Büyükboduk (University College Dublin)
I will discuss the factorization of a certain triple product p-adic L-function whose interpolation range is empty. The said factorization reflects the Artin formalism for the underlying family of motives (that decompose as the sum of 2 motives of respective degrees 2 and 6). I will explain how this factorization problem can be recast as the comparison of two families of arithmetic GGP conjectures (and can be proved in some cases using this reduction).

Seminar on Arithmetic Geometry
2024-12-06 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Christopher Lang (TU Darmstadt): Ekedahl-Oort Stratification of Deligne-Lusztig Varieties
When developing a stratificaion of Rapoport-Zink spaces, Vollaard and
Wedhorn constructed a decomposition of a certain Deligne-Lusztig variety
for a unitary group using smaller Deligne-Lusztig varieties. We will show
that this decomposition can be obtained by pullback of the Ekedahl-Oort
stratification of G-Zips. With this method one gets an Ekedahl-Oort
stratification of flag varieties for reductive groups, which refines the
usual stratification by Deligne-Lusztig varieties.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Vector bundles on curves
2024-12-10 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Christopher Lang: Vector bundles in families
Zoom (612 2072 7363, Password: [REDACTED] six digit prime number)

Hodge Theory
2024-12-11 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Tom Bachmann (Universität Mainz): Pure Hodge structures

The direct summand theorem
2024-12-12 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 9: Christian Dahlhausen (Universität Heidelberg): The almost purity theorem

Moduli of Quiver Representations and GIT Quotients
2024-12-12 14:00 - 16:30 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
14:00 – 15:00 Talk 3.1: Miguel Prado (Goethe Universität): Introduction to Quivers and Properties I
Coffee break
15:30 – 16:30 Talk 3.2: Jeonghoon So (Goethe Universität): Introduction to Quivers and Properties II

The heart fan of an abelian category
2024-12-12 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
David Ploog (Stavanger)
Abstract: To an abelian category such as coherent sheaves on a projective variety or modules over a finite-dimensional algebra, I associate a fan of convex cones. This fan reflects homological properties of the category. It contains the g-fan of representation theory and is related to the stability conditions.

Congruence Modules and the Wiles–Lenstra–Diamond Numerical Criterion in Higher Codimension
2024-12-13 09:00 - 11:00 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Gebhard Böckle (Universität Heidelberg): Congruence modules and Wiles defect under surjections

Wall crossing for equivariant CY3 categories
2024-12-13 11:00 - 12:00 | Heidelberg, MATHEMATIKON, SR10, INF 205, Heidelberg, Germany
Nikolas Kuhn (University of Oxford)
The Joyce-Song wall-crossing formulas for Donaldson-Thomas invariants of Calabi-Yau threefolds have proven to be a crucial and versatile tool. In the presence of a torus action, there are interesting threefold geometries in which the Calabi-Yau condition only holds up to an equivariant twist – examples include Vafa-Witten invariants, local curves and surfaces and the threefold vertex. In these cases, invariants are defined using localization, and Joyce-Song’s theory doesn’t apply. I will explain how ideas from Joyce’s recent work on wall-crossing in abelian categories can be used to prove wall-crossing in this situation, and which difficulties arise. This is joint work with Henry Liu and Felix Thimm.

Resolution of non-singularities and anabelian applications
2024-12-13 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Emmanuel Lepage (IMJ Paris)
Abstract: In various anabelian settings over p-adic fields, one can reconstruct from the fundamental group of a hyperbolic curve the dual graph of the stable reduction of the curve, and one can get more anabelian information on the curve by applying it to various finite étale covers. For each such finite étale cover, this graph defines a retract of the analytic space associated to the curve (in the adic or Berkovich sense), and resolution of non-singularities predicts that the Berkovich space is homeomorphic to the inverse limit of all these retracts. This was proven in 2023 by Mochizuki and Tsujimura over finite extensions of Q_p. I will try to give a sketch of their proof and explain how to deduce a characterization of geometric Galois sections.

Seminar on Arithmetic Geometry
2024-12-13 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Thomas Nikolaus (Universität Münster): (Relative) Prismatic cohomology, K-Theory and Topology
We will explain the theory of relative prismatic cohomology (relative to a delta ring) and how this is an essential tool in computations of prismatic cohomology. If time allows we will exlain how this connects to K-Theory and other Homotopy-theoretically defined invariants (such as TP and TR) and to the relative de Rham Witt complex.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

p-adic higher Green's functions for Stark-Heegner Cycles
2024-12-17 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Hazem Hassan (McGill)
Heegner Cycles are higher weight generalizations of Heegner points on Modular curves. As such, one expects them to capture similar arithmetic and modular properties to Heegner points. The higher dimensional nature of Heegner cycles makes them less amenable to algebro-geometric and deformation theoretic approaches. I will introduce Stark-Heegner Cycles, which are a conjectural analogue to Heegner Cycles in the theory of Real Multiplication. They are defined through p-adic analytic means. Then, I will describe a p-adic pairing on these cycles which behaves as a local height pairing. When one of the cycles is principal, the pairing computationally seems to produce algebraic integers living in class fields of real quadratic fields.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Hodge Theory
2024-12-18 12:00 - 13:30 | Mainz, Hilbertraum (05-432)
Nutsa Gegelia (Universität Mainz): Variations of Hodge structures I

The direct summand theorem
2024-12-19 11:15 - 12:45 | Heidelberg, Mathematikon, SR 8 and Zoom, INF 205, Heidelberg, Germany
Talk 10: Max Witzelsperger (Universität Heidelberg): The direct summand theorem

Moduli of Quiver Representations and GIT Quotients
2024-12-19 14:00 - 16:30 | 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).
