Quadratically enriched tropical intersections 2
2022-11-25 15:30 - 16:30 (Europe/Berlin) | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiZ-Seminar: Tropical geometry in Zoom (First meeting Winter Semester 2022/23)
Andrés Jaramillo Puentes (Universität Duisburg-Essen)
Abstract:
Tropical geometry has been proven to be a powerful computational tool in enumerative geometry over the complex and real numbers. Results from motivic homotopy theory allow to study questions in enumerative geometry over an arbitrary field k. In these two talks we present one of the first examples of how to use tropical geometry for questions in enuemrative geometry over k, namely a proof of the quadratically enriched Bézout’s theorem for tropical curves.
In the second talk we define the quadratically enriched multiplicity at an intersection point of two tropical curves and show that it can be computed combinatorially. We will use this new approach to prove an enriched version of the Bézout theorem and of the Bernstein–Kushnirenko theorem, both for enriched tropical curves.

Valuative invariants for large classes of matroids
2022-11-25 16:45 - 17:45 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiZ-Seminar: Tropical geometry in Zoom (First meeting Winter Semester 2022/23)
Benjamin Schröter (Universität Frankfurt)
Abstract:
Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid.
In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., sparse paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new. However, this class appears naturally in this context. Moreover, (sparse) paving matroids are split. I will demonstrate the framework by looking at Ehrhart polynomials, relations in Chow rings of combinatorial geometries, and further examples.
This talk is based on the preprint `Valuative invariants for large classes of matroids’ which is joint work with Luis Ferroni.

Vector-valued orthogonal modular forms
2022-11-29 09:00 - 10:00 | Zoom
International Seminar on Automorphic Forms
I will talk about the theory of vector-valued modular forms on domains of type IV, with some emphasis on its algebro-geometric aspects.
Shouhei Ma (Tokyo Institute of Technology)
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).

Bloch-Kato groups, perfectoid fields, and Iwasawa theory
2022-12-02 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Gautier Ponsinet (Université Bordeaux)
The Bloch-Kato Selmer groups associated with a geometric representation of the Galois group of a number field take part in Bloch and Kato's conjecture on the special values of L functions of motives. In Iwasawa theory, we are interested in the structure of these Bloch-Kato Selmer groups over infinite algebraic fields extensions. To do so, we need to study the local Bloch-Kato groups defined via p-adic Hodge theory. In this talk, I will present new results about the local Bloch-Kato groups over perfectoid fields, thereby answering a question by Coates and Greenberg in new cases. These local results allow to describe the Bloch-Kato Selmer groups over many infinite extensions as Selmer groups "à la Greenberg" which are easier to handle. If time allows, I will present immediate consequences of this description in Iwasawa theory.
livestream:
http://129.206.106.240/UzPL29kB/mathematikon-seminarraum.html

Torsors on valuation rings
2022-12-02 14:00 - 15:30 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
A conjecture of Grothendieck and Serre states that a torsor under a reductive group over a Noetherian regular scheme X is Zariski locally trivial if it is generically trivial. Recently, this conjecture has seen progress through the work of Fedorov, Panin and Česnavičius. We shall see the historical background of this conjecture, followed by the techniques that go into the proof of the quasi-split case in the analogous situation when X is a smooth scheme over a valuation ring of rank one.
Arnab Kundu (University Paris-Saclay)
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

Rigidity properties of the cotangent complex
2022-12-02 14:00 - 16:00 | Mainz, Hilbertraum (05-432)
Srikanth Iyengar (Utah)
This talk is about the cotangent complex of a homomorphism of commutative noetherian rings. I will present some new results on this topic, some due to Benjamin Briggs, and others obtained in collaboration with him, that highlight various rigidity properties of the cotangent complex. These are reported in the following article: https://arxiv.org/abs/2010.13314

Arithmetic Quantum Chaos and L-functions
2022-12-06 09:00 - 10:00 | Zoom
International Seminar on Automorphic Forms
In this talk, I will introduce some aspects of the theory of arithmetic quantum chaos, focusing on the quantum unique ergodicity theorem for automorphic forms on the modular surface. Then I will give some results on effective decorrelation of Hecke eigenforms and the cubic moment of Hecke-Maass cusp forms. The proofs are based on the analytic theory of L-functions.
Bingrong Huang (Shandong University)
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).

Depth $0$ local Langlands and cohomology of stacks of global chtoucas
2022-12-09 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
Let $G$ be a split reductive group, $X$ a smooth proper curve over a finite field and $x in X$ a place. Let $F_x$ the completion of the function field of $X$ at $x$. In this setting, Lafforgue and Genestier have constructed a semisimple local Langlands correspondence for $G(F_x)$ by geometric methods. In another direction, DeBacker and Reeder have constructed the depth $0$ part of a local Langlands correspondence by representation theoretic methods. In this talk, I will discuss some compatibility statement between the two constructions and explain the connection to global chtoucas over $X$.
Arnaud Eteve (ENS, Paris)
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

Point-set topology methods in the theory of v-sheaves and diamonds
2022-12-12 13:30 - 15:10 | Darmstadt, Room 315 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
In this lecture series we aim to introduce the audience to the theory of kimberlites and its applications. The theory of kimberlites attempts to single out within Scholze’s category of v-sheaves those objects that “behave” as formal schemes. In this way, kimberlites provide well-behaved candidates for integral models of locally spatial diamonds. In the first two talks we discuss the foundations of the theory, and in the second two talks we discuss its applications to study moduli spaces of B_dR-lattices and moduli spaces of p-adic shtukas.
Talk 1: The purpose of this talk is to discuss the point-set topology subtleties that arise from working with non-analytic adic spaces within Scholze’s framework of v-sheaves. We recall miscellaneous aspects of the theory of adic spaces, perfectoid spaces, diamonds and v-sheaves. We introduce the small diamond and big diamond functors. We define the “diamond” (or olivine) spectrum of a Huber pair and describe it. We explain explicitly the example of the olivine spectrum of valuation rings. Finally, we use this topological considerations to “explain” the reversal of closure relations betwenn |Bun_G| and |Isoc_G|.
Ian Glesaon (Berkeley, USA)
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

Milnor K-theory of p-adic rings and motivic cohomology
2022-12-12 16:00 - 18:00 | Heidelberg, Mathematikon, SR C
Dr. Morten Lüders (Hannover)
Abstract: We explain a joint work with Matthew Morrow on $p$-adic Milnor K-theory. Our main theorem is a comparison of mod $p^r$ Milnor K-groups of $p$-henselian local rings with the Milnor range of a nwly defined syntomic cohomology theory by Bhatt, Morrow and Scholze. We begin by putting our result into context. Then we sketch the proof which builds on an analysis of a filtration on Milnor K-groups and a new technique called the left Kan extension from smooth algebras.

Point-set topology methods in the theory of v-sheaves and diamonds.
2022-12-13 09:50 - 11:30 | Darmstadt, Room 315 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
In this lecture series we aim to introduce the audience to the theory of kimberlites and its applications. The theory of kimberlites attempts to single out within Scholze’s category of v-sheaves those objects that “behave” as formal schemes. In this way, kimberlites provide well-behaved candidates for integral models of locally spatial diamonds. In the first two talks we discuss the foundations of the theory, and in the second two talks we discuss its applications to study moduli spaces of B_dR-lattices and moduli spaces of p-adic shtukas.
Talk 2: The purpose of this talk is to discuss the theory of kimberlites. We introduce the reduction functor, specializing sheaves, prekimberlites, kimberlites, tubular and etale neighborhoods. We discuss the topological specialization map and the v-sheaf theoretic specialization map. We discuss the etale site of a prekimberlite, the formal and analytic nearby cycles functors and the comparison between them. We discuss finiteness condtions and the specialization triples principle.
Ian Gleason (Berkeley, USA)
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

Bias in cubic Gauss sums: Patterson's conjecture
2022-12-13 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846. One important byproduct of our proof is that we show Heath-Brown’s cubic large sieve is sharp under GRH. This disproves the popular belief that the cubic large sieve can be improved. An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term.
Alexander Dunn (Caltech)
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).

Torsors on Valuation Rings
2022-12-14 14:00 - 16:00 | Heidelberg, Mathematikon, SR C
Arnab Kundu (Université Paris-Saclay)
Abstract: A conjecture of Grothendieck and Serre states that a torsor under a reductive group over a Noetherian regular scheme X is Zariski locally trivial if it is generically trivial. Recently, this conjecture has seen progress through the work of Fedorov, Panin and Česnavičius. We shall see the historical background of this conjecture, followed by the techniques that go into the proof of the quasi-split case in the analogous situation when X is a smooth scheme over a valuation ring of rank one.

Der Satz von Belyi
2022-12-14 16:00 - 17:00 | Frankfurt and Zoom
Marie Kassner (Universität Frankfurt)
Bachelorabschlussvortrag

Point-set topology methods in the theory of v-sheaves and diamonds.
2022-12-15 13:30 - 15:10 | Darmstadt, Room 315 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
In this lecture series we aim to introduce the audience to the theory of kimberlites and its applications. The theory of kimberlites attempts to single out within Scholze’s category of v-sheaves those objects that “behave” as formal schemes. In this way, kimberlites provide well-behaved candidates for integral models of locally spatial diamonds. In the first two talks we discuss the foundations of the theory, and in the second two talks we discuss its applications to study moduli spaces of B_dR-lattices and moduli spaces of p-adic shtukas.
Talk 3: The purpose of this talk is to discuss the moduli of B^+_dR-lattices with extra structure. We prove they are kimberlites in general and we sketch the proof of the representability by formal schemes of the “local models” attached to minuscule cocharacters. We introduce the notion of unibranch kimberlites and explain its relation to normality. We prove that these moduli spaces are unibranch.
Ian Gleason (Berkeley, USA)
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

Residual intersections and Witt-valued Euler numbers
2022-12-15 14:00 - 16:00 | Mainz, Hilbertraum (05-432)
Tom Bachmann
Report on joint work with Kirsten Wickelgren. Let K = J:I be a (local) residual intersection satisfying appropriate hypotheses. We show that the conormal module J/JK is free and use this to exhibit a *canonical* isomorphism between I^{t+1}/JI^t twisted by the determinant of J/KJ and the canonical module (the derived dual of R/K). Using this we determine the canonical modules of global residual intersections. As an application we show how to interpret the van Straten–Warmt form of an almost complete intersection as arising via algebraic surgery from the Koszul complex. This yields a new local formula in quadratic enumerative geometry.

A motivic integral p-adic cohomology
2022-12-16 13:30 - 15:00 | Heidelberg, Mathematikon, SR C
Alberto Merici (Oslo)
Abstract: We use the theory of logarithmic motives to construct an integral p-adic cohomology theory for smooth varieties over a field k of characteristic p, that factors through the category of Voevodsky (effective) motives. If k satisfies resolutions of singularities, we will show that it is indeed a “good” integral p-adic cohomology and it agrees to a similar one constructed by Ertl, Shiho and Sprang: we will then deduce many interesting motivic properties.
If time permits, we will explain how the above construction suggests a strategy to prove a conjecture of Hübner and Schmidt on tame motivic cohomology.

Point-set topology methods in the theory of v-sheaves and diamonds
2022-12-16 14:00 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar on Arithmetic Geometry
In this lecture series we aim to introduce the audience to the theory of kimberlites and its applications. The theory of kimberlites attempts to single out within Scholze’s category of v-sheaves those objects that “behave” as formal schemes. In this way, kimberlites provide well-behaved candidates for integral models of locally spatial diamonds. In the first two talks we discuss the foundations of the theory, and in the second two talks we discuss its applications to study moduli spaces of B_dR-lattices and moduli spaces of p-adic shtukas.
Talk 4: The purpose of this talk is to discuss the moduli of p-adic shtukas. We prove these moduli spaces are smelted kimberlites and single out the difficulties to proving that they are kimberlites. We discuss the v-sheaf theoretic “local diagram correspondence” for tubular neighborhoods relating moduli spaces of shtukas to moduli of B^+_dR-lattices. We prove they are unibranch. Finally we expalin how this is used to compute their connected components.
Ian Gleason (Berkeley, USA)
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)

Distributions of Manin's iterated integrals
2022-12-20 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms

We recall the definition of Manin’s iterated integrals of a given length. We then explain how these generalise modular symbols and certain aspects of the theory of multiple zeta-values. In length one and two we determine the limiting distribution of these iterated integrals. Maybe surprisingly, even if we can compute all moments also in higher length we cannot in general determine a distribution for length three or higher. This is joint work with Y. Petridis and with N. Matthes.
Morten Risager (University of Copenhagen)
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).

Homotopical characterization of exceptional complete intersection maps
2022-12-22 14:00 - 16:00 | Mainz, Hilbertraum (05-432)
Janine Letz
A surjective map between commutative noetherian local rings (R, m) → S is exceptional complete intersection (eci) if its kernel is generated by a regular sequence that is part of a minimal generating set of m. I present two characterizations of eci maps: First, a map is eci if and only if the truncated Atiyah class vanishes at the residue field. This establishes a second characterization in terms of the lattices of thick subcategories of complexes of finite length homology. This is joint work with Srikanth Iyengar, Jian Liu and Josh Pollitz.

Deligne-Mostow theory and beyond
2023-01-10 09:00 - 10:00 | Zoom
International Seminar on Automorphic Forms
Ball quotients have been studied extensively in algebraic geometry from the aspect of moduli spaces, and in number theory with emphasis on the relation with modular forms. The Deligne-Mostow theory gives them moduli interpretation through the isomorphism between the Baily-Borel compactifications of them and certain GIT quotients.
In this talk, I will discuss whether the isomorphisms given by the Deligne-Mostow theory are lifted to other compactifications from the viewpoint of modular forms and pursue “better” compactifications. Moreover, I will also clarify their connection with the recent development in the minimal model program. This work is based on a joint work with Klaus Hulek (Leibniz University Hannover).
Yota Maeda (Kyoto University)
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).

Gauged Gromov-Witten theory and affine grassmannians
2023-01-11 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Rescheduled to 10.01.2023, 4pm, room 308
Andres Fernandez Herrero (Columbia University)
Abstract: Objects of interest in algebraic geometry (e.g. curves, vector bundles, or differential equations) are often parametrized by algebraic varieties, called moduli spaces. In this talk I will discuss some recent techniques developed to construct moduli spaces for a broad range of moduli problems which are related to the moduli of vector bundles on a fixed compact Riemann surface.
With time permitting, I will also try to explain what it means to count vector bundles on compact Riemann surfaces, and why such counts are given by combinations of certain special values of transcendental functions. This talk is based on joint work with Daniel Halpern-Leistner.

The Picard-Lefschetz formula for normal crossings
2023-01-13 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Moritz Kerz (Universität Regensburg)
Abstract: In the study of semi-stable degeneration of Lefschetz pencils one is led to a generalization of the classical Picard-Lefschetz formula for certain perverse sheaves on normal crossing spaces. In the talk I will recall the formalism of nearby cycle and vanishing cycle functors and I will explain how Hodge theory allows one to obtain the normal crossing Picard-Lefschetz formula. Joint work with A. Beilinson and H. Esnault.

Sup-norms of automorphic forms on average
2023-01-17 09:00 - 10:00 | Zoom
International Seminar on Automorphic Forms
Bounding the sup-norms of automorphic forms has been a very active area in research in recent times.
Whereas lot of nice results are known for small rank groups, like GL(2), almost nothing is known for, say, Siegel or Jacobi modular forms of higher degrees. In this talk we aim to discuss some conjectures and results in this area. We use either the theory of Poincare series or averages of central values of L-functions to tackle this problem. Our methods have the benefit of having a hands-on approach and fits into many situations.
Soumya Das (Indian Institute of Science)
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).

Expanded degenerations for Hilbert schemes of points.
2023-01-18 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
Calla Tschanz (University of Bath)
Abstract:
Let X –> C be a projective family of surfaces over a curve with smooth generic fibre and simple normal crossing singularity in the special fibre X_0. We construct a good compactification of the moduli space of relative length n zero-dimensional subschemes on XX_0 over C{0}. In order to produce this compactification we study expansions of the special fibre X_0 together with a GIT stability condition, generalising the work of Gulbrandsen-Halle-Hulek who use GIT to offer an alternative approach to the work of Li-Wu for Hilbert schemes of points on simple degenerations. We construct stacks which we prove to be equivalent to the underlying stack of some choices of logarithmic Hilbert schemes produced by Maulik-Ranganathan.

The standard conjecture of Hodge type for abelian fourfolds
2023-01-19 14:00 - 16:00 | Mainz, Hilbertraum (05-432)
Guiseppe Ancona (Strasbourg)
Let S be a surface, V be the Q-vector space of divisors on S modulo numerical equivalence and d be the dimension of V . The intersection product defines a non degenerate quadratic form on V . The Hodge index theorem says that it is of signature (1, d − 1). In the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is a consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable, thanks to p-adic Hodge Theory. Moreover, using classical product formulas on quadratic forms, the p-adic result will give non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.

Integral p-adic cohomology for open and singular varieties
2023-01-20 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Veronika Ertl (Universität Regensburg)
Abstract: In this talk I will explain a joint result with Johannes Sprang and Atsushi Shiho.
Under certain conditions of resolutions of singularities in positive characteristic,
we construct a “good” integral p-adic cohomology theory for open and singular varieties,
by using a version of Voevodsky’s h-topology.
I will explain the construction and clarify in which sense our cohomology is a “good” p-adic cohomology theory.
I will also touch on the question why a similar approach does not work in full generality without resolutions of singularities.

Slope of Siegel modular forms: some geometric applications
2023-01-24 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
We study the slope of modular forms on the Siegel space. We will recover known divisors of minimal slope for $gleq5$ and we discuss the Kodaira dimension of the moduli space of principally polarized abelian varieties $A_g$ (and eventually of the generalized Kuga’s varieties). Moreover we illustrate the cone of moving divisors on $A_g$. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel cusp forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on $A_g$ for $gleq5$.
Riccardo Salvati Manni (Sapienza University of Rome)
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).

Automorphisms of categories of schemes
2023-01-27 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Remy van Dobben de Bruyn (Universität Utrecht)
Abstract: Given two schemes S and S’, we show that any equivalence between Sch/S and Sch/S’ comes from a unique isomorphism between S and S’. In particular, the category of schemes does not have any nontrivial automorphisms. This eliminates all Noetherian and finite type hypotheses from a result of Mochizuki, and answers a series of questions of Brandenburg. The methods are analogous to those in anabelian geometry (but easier), and this talk also serves as an introduction to those ideas for non-experts.

Motives of moduli of bundles on curves
2023-01-27 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Seminar Arithmetic Geometry
Simon Pepin Lehalleur (Radboud-Universiteit Nijmegen)
Abstract: (Joint with Victoria Hoskins, Nijmegen) Moduli spaces and stacks of bundles on smooth projective curves are basic objects of algebraic geometry and part of the geometric set-up of the Langlands program for function fields. Their cohomology is well understood in some ways and still very mysterious in others. After some recollections on Voevodsky
motives, I will present several results about the motives of the moduli stack of vector bundles and the moduli spaces of semistable Higgs bundles, expressing them in terms of the motives of powers of the base curve. I will then explain how this can combined with constructions of Maulik-Shen to prove a motivic version of the “topological mirror symmetry” connecting moduli spaces of SL_n and PGL_n-Higgs bundles.
Zoom (Meeting-ID: 635 7328 0984, Password: [REDACTED] six digit prime)
