Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
2024-02-02 14:30 - 15:30 (Europe/Berlin) | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (Second meeting Winter Semester 2023/24)
Andreas Bernig (Goethe-Universität Frankfurt)
Abstract: The hard Lefschetz theorem and the Hodge-Riemann relations have their origin in the cohomology theory of compact Kähler manifolds. In recent years it has become clear that similar results hold in many different settings, in particular in algebraic geometry and combinatorics (work by Adiprasito, Huh and others). In a recent joint work with Jan Kotrbatý and Thomas Wannerer, we prove the hard Lefschetz theorem and Hodge-Riemann relations for valuations on convex bodies. These results can be translated into an array of quadratic inequalities for mixed volumes of smooth convex bodies, giving a smooth analogue of the quadratic inequalities in McMullen’s polytope algebra. Surprinsingly, these inequalities fail for general convex bodies. Our proof uses elliptic operators and perturbation theory of unbounded operators.

Gluing sheaves along Harder-Narasimhan strata of Bun_2.
2024-02-02 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
Jonathan Miles (University of Frankfurt)
We explain how to glue sheaves on the moduli stack of G-bundles on the Fargues-Fontaine curve. In the case of prime-to-p torsion coefficients, the category D_ét(Bun_G) can be thought of as an approximation of the automorphic data appearing in the geometrization of the local Langlands correspondence due to Fargues-Scholze. The stratification of Bun_G arising from the Harder-Narasimhan slope formalism on G-isocrystals yields a semi-orthogonal decomposition of D_ét(Bun_G) into the derived categories of smooth representations of inner forms of Levi subgroups of G. Between such categories there is a (partial) six functor formalism that can be used to compute how sheaves arising on a quasi-compact open substack interact with sheaves on higher strata via nearby cycles functors, which can be interpreted as a derived analogue of Jacquet restriction functors for parabolic subgroups of G (up to inner twisting). We eventually restrict to G=GL_2 and to sufficiently nice coefficients (notably this includes an algebraic closure of F_ell and Z/ell^n Z for almost all ell prime to p), and we will explain how these computations fundamentally reduce to the étale cohomology of p-adic analogues of locally symmetric spaces, such as the Bruhat-Tits building and moduli spaces of mixed p-adic Hodge structures.
Zoom (635 7328 0984, Password: [REDACTED] six digit prime).

Category of matroids with coefficients
2024-02-02 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (Second meeting Winter Semester 2023/24)
Manoel Zanoelo Jarra (Universität Groningen)
Abstract: Matroids are combinatorial abstractions of the concept of independence in linear algebra. There is a way back: when representing a matroid over a field we get a linear subspace. Another algebraic object for which we can represent matroids is the semifield of tropical numbers, which gives us valuated matroids. In this talk we introduce Baker-Bowler’s theory of matroids with coefficients, which recovers both classical and valuated matroids, as well linear subspaces, and we show how to give a categorical treatment to these objects that respects matroidal constructions, as minors and duality. This is a joint work with Oliver Lorscheid and Eduardo Vital.

Evaluating the wild Brauer group
2024-02-06 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Rachel Newton (King’s College London)
The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer–Manin pairing cuts out a subset of the adelic points, called the Brauer–Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer–Manin set is empty then we say there’s a Brauer–Manin obstruction to the existence of rational points on X. Computing the Brauer–Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato. Applications of our work include addressing Swinnerton-Dyer’s question about which places can play a role in the Brauer–Manin obstruction. This is joint work with Martin Bright.
You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits).

Motivic tt-geometry
2024-02-09 14:00 - 15:30 | Zoom
Seminar on Arithmetic Geometry
Martin Gallauer (University of Warwick)
Commutative rings are profitably studied from a geometric point of view, giving rise to the field of algebraic geometry. “Categorified commutative rings” (e.g. tt-categories) can similarly be studied geometrically, leading to a field called tt-geometry. I will introduce these ideas and focus on recent progress regarding tt-categories arising in the motivic theory. (Based on joint work with Paul Balmer).
Zoom (635 7328 0984, Password: [REDACTED] six digit prime).

The Two Lives of the Grassmannian
2024-04-11 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Bernd Sturmfels (MPI-MiS Leipzig)
Abstract:
The Grassmannian parametrizes linear subspaces of a real vector space.
It is both a projective variety (via Plücker coordinates) and an affine
variety (via orthogonal projections). We examine these two representations,
through the lenses of linear algebra, commutative algebra, and statistics.

Lifting differential equations: an introduction
2024-04-18 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Vasily Golyshev (ICTP Trieste)
Abstract. I will introduce a program aiming at constructing lifts of differential equations by using generalized Taylor’s formula in a systematic way. The talk is based on joint work with Ilia Gaiur.

Balanced triple product p-adic L-functions and classical weight one forms
2024-04-19 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Luca Dall’Ava (Universität Mailand)
This talk aims to introduce a new balanced triple product p-adic L-function and discuss its application to the equivariant Birch & Swinnerton-Dyer conjecture; we state a conjecture in a rank-1 situation analogous to the Elliptic–Stark conjecture formulated by Darmon–Lauder–Rotger in rank-2 and prove it in the CM case; this work fits in the general framework studied by Darmon–Lauder–Rotger and Andreatta–Bertolini–Seveso–Venerucci. The main feature of this new p-adic L-function is to allow classical weight one modular forms in the chosen families. Its construction crucially relies on an extension of Chenevier’s p-adic Jacquet–Langlands for quaternionic modular forms with level structure given by Pizer orders. Time permitting, we will explain the intriguing technical difficulties one has to deal with constructing this morphism of eigenvarieties. That is joint work with Aleksander Horawa.

Seminar on Arithmetic Geometry
2024-04-19 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Catrin Mair (TU Darmstadt)
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Motivic cohomology of cyclic coverings
2024-04-26 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Tariq Syed (Mainz/Los Angeles)
Many examples of topologically contractible smooth affine complex varieties are given by cyclic coverings. In this talk, we present new results on the motivic cohomology of such cyclic coverings.

Seminar on Arithmetic Geometry
2024-04-26 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Chirantan Chowdhury (TU Darmstadt): Applications Of Six-Functor Formalisms in Motivic Homotopy Theory of Algebraic Stacks
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl)

A new zero-free region for Rankin-Selberg L-functions
2024-04-30 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Jesse Thorner (UIUC)
https://tu-darmstadt.zoom.us/j/68048280736

Nobodies are perfect, semigroups are not.
2024-05-03 14:30 - 15:30 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2024)
Karin Schaller (FU Berlin)
Abstract: NObodies are asymptotic limits of certain valuation semigroups. Their construction depends on a given flag of subvarieties. We investigate toric surfaces together with non-toric flags and determine when the associated valuation semigroups are finitely generated. This is a joint work with K. Altmann, C. Haase, A. Küronya, and L. Walter.

Seminar on Arithmetic Geometry
2024-05-03 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Tariq Syed (Johannes Gutenberg-Universität Mainz): Motivic cohomology of cyclic coverings
Many examples of topologically contractible smooth affine complex varieties are given by cyclic coverings. In this talk, we present new results on the motivic cohomology of such cyclic coverings.
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

The realization space of a matroid
2024-05-03 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Summer Semester 2024)
Lukas Kühne (Universität Bielefeld)
Abstract: A matroid is a fundamental and widely studied object in combinatorics. Following a brief introduction to matroids, I will showcase parts of a new OSCAR module for matroids using several examples. My emphasis will be on the computation of the realization space of a matroid, which is the space of all hyperplane arrangements that have the given matroid as their intersection lattice.
In the second part, I will discuss an application in the realm of algebraic geometry, namely a novel connection between matroid realization spaces and the elliptic modular surfaces.

Quadratic reciprocity in a polynomial ring
2024-05-07 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
William Duke (UCLA)
I will give a characterization of when a kind of quadratic reciprocity holds for irreducible polynomials whose coefficients are in a number field. The method is based on Gauss’s second proof of classical quadratic reciprocity using binary quadratic forms.
https://tu-darmstadt.zoom.us/j/68048280736

Gan-Gross-Prasad cycles and derivatives of p-adic L-functions
2024-05-14 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Daniel Disegni (Aix-Marseille University)
Certain Rankin-Selberg motives of rank n(n+1) are endowed with algebraic cycles arising from maps of unitary Shimura varieties. Gan-Gross-Prasad conjectured that these cycles are analogous to Heegner points, in the sense that their nontriviality should be detected by derivatives of L-functions.
I will propose another nontriviality criterion, based on p-adic L-functions. Under some local conditions, this variant can be established in a refined quantitative form, via the construction and comparison two p-adic relative-trace formulas. (Joint work with Wei Zhang.)
https://tu-darmstadt.zoom.us/j/68048280736

Density of rational points near manifolds
2024-05-17 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Damaris Schindler (Universität Göttingen)
Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre’s dimension growth conjecture as well as applications in Diophantine approximation. For this I’ll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau.

Seminar on Arithmetic Geometry
2024-05-17 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Nathan Tiggemann (TU Darmstadt)
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Chowla-Selberg phenomenon over function fields
2024-05-21 08:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Fu-Tsun Wei (National Tsing Hua University)
In this talk, I will first determine the algebraic relations among various special gamma values over function fields. The result is based on the intrinsic relations between gamma values in question and periods of CM dual t-motives, which are interpreted in terms of their “distributions”. This enables us to express every “abelian” CM period by a suitable product of special gamma values (up to an algebraic multiple), and derive a Chowla–Selberg-type formula in the function field case.
https://tu-darmstadt.zoom.us/j/68048280736

Algebraic geometry of the classical Yang-Baxter equation
2024-05-22 16:00 - 17:30 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Igor Burban (Universität Paderborn)
Abstract:
Classical Yang-Baxter equation (CYBE) plays an important role in the modern theory of integrable systems. In a seminal work of Belavin and Drinfeld from the year 1983 it was proven that solutions of CYBE can have one of the following three types: elliptic, trigonometric or rational. Moreover, Belavin and Drinfeld gave a complete classification of all elliptic and trigonometric solutions.
In my talk, I am going to explain a geometric description of solutions of (CYBE). Namely, starting with any geometric datum (E, A), where E is a Weierstraß cubic curve and A a torsion free sheaf of Lie algebras (whose generic fiber is a given complex simple Lie algebra) with vanishing cohomology, one can associate to it in a canonical way a solution of CYBE. It turns out that all solutions of CYBE arise in this way. Moreover, the type of the curve in the datum (E, A) (i.e. smooth/nodal/cuspidal) determines the type of the corresponding solution of CYBE (i.e elliptic/trigonometric/rational).
The developed method will be illustrated by explicit examples. This talk is based on my joint works with Raschid Abedin, Lennart Galinat and Thilo Henrich.

Seminar on arithemtic geometry
2024-05-24 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Torsten Wedhorn (TU Darmstadt): Moduli of truncated shtukas
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

The rhizomic topology and tropical abelian varieties
2024-05-28 16:00 - 17:00 | Frankfurt, RM-Str. 6-8, R. 308
Leo Herr (Universität Leiden)
Abstract:
The log etale topology is a natural analogue of the etale topology for log schemes. Unfortunately, very few things satisfy log etale descent — not even vector bundles or the structure sheaf. We introduce a new rhizomic topology that sits in between the usual and log etale topologies and show most things do satisfy rhizomic descent! As a case study, we look at tropical abelian varieties and give some exotic examples.

Isotropic motivic categories
2024-05-31 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Fabio Tanania (TU Darmstadt)
In this talk, I will introduce the isotropic stable motivic homotopy category, which is a stable homotopic version of Vishik’s category of isotropic motives. These are local versions of the classical motivic categories SH(k) and DM(k), obtained roughly speaking by annihilating motives of anisotropic varieties. When the base field k is flexible, then isotropic motivic categories have a lot of remarkable properties. As an example, I will describe the structure of cellular isotropic spectra in terms of well-known objects coming from classical homotopy theory. Then, I will discuss what is known about the non-cellular part of isotropic motivic categories and state some open questions.

Seminar on Arithmetic Geometry
2024-05-31 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Rizacan Ciloglu (TU Darmstadt): Perverse sheaves on twisted affine flag varieties
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

CRC-Colloquium
2024-06-06 15:30 - 18:30 | Darmstadt Gebäude S1|15, Hörsaal 133
15:20 Coffee (or earlier)
15:30 – 16:30 Georg Tamme (Universität Mainz): Higher algebra and K-theory
16:30 Coffee and Cake
17:15 – 18:15 Otmar Venjakob (Universität Heidelberg): Explicit Reciprocity Laws in Number Theory
18:45 Dinner
Abstract G. Tamme:
Algebraic K-theory, more precisely the group K_0(X) of a scheme X, was first introduced by Grothendieck in order to formulate and prove a very general Riemann-Roch theorem. Subsequently, lower (i.e. K_i for i<0) and higher (i.e. K_i for i>0) K-groups have been introduced by Bass, Milnor, Quillen, and Thomason. They have turned out to be a rich invariant, playing an important role in various fields such as number theory and topology. Although explicit computations of K-groups remain challenging, modern methods from higher algebra and derived algebraic geometry have led to several breakthrough structural results concerning these groups. In the talk, I will give an introduction to these topics and, time permitting, discuss some recent applications.
Abstract O. Venjakob:
The quadratic Reciprocity Law for the Legendre or Jacobi-Symbol forms the starting point of all Reciprocity Laws as well as of class field theory. It is closely related to the product formula of the quadratic Hilbert-Symbol over local fields. Various mathematicians have established higher explicit formulae to compute higher Hilbert-Symbols. Analogs were found for formal (Lubin-Tate) groups. Eventually Perrin-Riou has formulated a Reciprocity Law, which allows the explicit computation of local cup product pairings by means of Iwasawa- and p-adic Hodge Theory. In this talk I shall try to give an overview of these topics, at the end I will explain recent developments in this regard.

Zagier’s conjecture on polylogarithms: from function fields to number fields
2024-06-07 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Quentin Gazda (CMLS Palaiseau)
Zagier’s conjecture is a certain formulation of the following slogan: linear relations among polylogarithms evaluated at algebraic numbers arise from relations between K-theory symbols. By identifying the different attributes of this conjecture, one can state and prove a similar version in arithmetic of function fields. Classical polylogarithms are then replaced by those of Carlitz. The proof, very different from the techniques developed so far, uses ingredients from the theory of difference equations. It involves deformations of Carlitz polylogarithms where a new variable $t$ appears. This results from a joint work with A. Maurischat.
While we currently lack the technology to reproduce this argument in number theory, it is amusing to speculate on a hypothetical transcription. $q$-deformations of polylogarithms then replace these « $t$-deformations ». With T. Bouis, we recently encountered versions of $q$-polylogarithms in the syntomic Chern class introduced by Bhatt-Lurie. This is encouraging…! I will mention these works in a second part of the presentation.

Seminar on Arithmetic Geometry
2024-06-07 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Minjia Zhang (Princeton University): Igusa stacks and cohomology of Shimura varieties
Shimura varieties have been playing an important role in the Langlands program. Recently, following a conjecture of Peter Scholze, we have constructed certain p-adic geometric objects, which we call ​”Igusa stacks” (in the PEL type case), which encode prime-to-p information of Shimura varieties. Together with certain flag varieties (encoding information of Shimura varieties at p), they give p-adic uniformization of Shimura varieties. In this talk, I will explain construction of Igusa stack for Hodge type Shimura varieties, which leads to new proofs of Mantovan’s product formula, Eichler-Shimura relations, and vanishing type results about torsion cohomology of Shimura varieties. This is based on joint work with Patrick Daniels, Daniel Kim and Pol van Hoften.
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

On the denominators of the special values of the partial zeta functions of real quadratic fields
2024-06-11 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Hohto Bekki (MPIM Bonn)
It is classically known that the special values of the partial zeta functions of real quadratic fields, or more generally, of totally real fields at negative integers are rational numbers. In this talk, I would like to discuss the denominators of these rational numbers in the case of real quadratic fields. More precisely, Duke recently presented a conjecture which gives a universal upper bound for the denominators of these special values of the partial zeta functions of real quadratic fields. I would like to explain that by using Harder’s theory on the denominator of the Eisenstein class for SL(2,Z), we can prove the conjecture of Duke and moreover the sharpness of his upper bound. This is a joint work with Ryotaro Sakamoto.
https://tu-darmstadt.zoom.us/j/68048280736

A gentle introduction to non-abelian Hodge and P=W conjecture
2024-06-12 16:00 - 18:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Alexandre Minets (MPI Bonn)
Abstract:
Title: A gentle introduction to non-abelian Hodge and P=W conjecture
Abstract: Let C be a Riemann surface, and M_B the character variety of C; in other words, the points of M_B parameterise representations of the fundamental group of C. Up to a _non-algebraic_ homeomorphism, M_B can also be realized as the variety M_H of algebraic vector bundles on C equipped with a twisted endomorphism (otherwise known as Higgs bundles). The P=W conjecture of de Cataldo-Hausel-Migliorini postulates that some algebraic data survives the homeomorphisms above; more precisely, the number of F_q-points of M_B is controlled by an algebraic integrable system on M_H. I will give a gentle introduction to these objects, focusing on one explicit example. If time permits, I will sketch some ideas that went into our recent proof of P=W (joint with T. Hausel, A. Mellit, O. Schiffmann).
