## Aktuelle Vorträge und Veranstaltungen

### Mi, 20. Oktober 2021

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Remi Reboulet (Universite Grenoble Alpes): tba.

### Do, 28. Oktober 2021

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Talk 1: Overview (28.10.) [by default: Martin U.] The organizer will give an overview of the topic of the seminar.
• Talk 2: Matroid basics I – cryptomorphisms and examples (28.10)∗[Ingmar Metzler] Introduce the different cryptomorphic axiom systems for matroids as in [Wel76, Section 2.2](in particular their definition in terms of “independent sets” and Theorem 1-5 without proofs).Then give some central examples of matroids (uniform matroids, vectorial matroids, graphic and cographic matroids), as in [Wel76, Section 2.3] and [Kat16, Section 4]. Mention what it means for a matroid to be realizable [Kat16, Def. 3.3].

### Mi, 10. November 2021

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Yujie Xu (Harvard University): tba.

### Do, 11. November 2021

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Talk 3: Matroid basics II – exercise session (11.11.) [no speaker] Exercise session where participants will be split in breakout rooms and are asked to work in groups on a selection of elementary exercises to familiarize themselves with matroids.
• Talk 4: Matroid basics III – the lattice of flats (11.11.)∗[Arne Kuhrs] Introduce the lattice of flats, as e.g. in [Oxl92, Section 1.7] and explain how it forms yet another cryptomorphic definition of a simple matroid. The central result is [Oxl92, Theorem 1.7.5] and, ideally, a full proof should be given. You will need to introduce some terminology first: simple matroids, posets, and (geometric) lattices (also see [Bak18, Section 3.1] for a quick overview.

### Do, 25. November 2021

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Talk 5: Operations on matroids (25.11.)∗[Pedro Souza] Introduce the basic operations on matroids, as e.g. summarized in [Kat16, Section 5]. Themost important ones for the rest of the seminar are deletion/contraction, direct sums, and duality (that correspond to [Kat16, Section 5.1, 5.2, and 5.3]). Mention the further operations from [Kat16, Section 5.4-5.7] only if there is time to do so.
• Talk 6: The characteristic polynomial (25.11.)∗or∗∗[Lucie Devey] Introduce the characteristic polynomial of a matroid, as in [Kat16, Section 7.1 and 7.3]. Finish by stating the Rota-Welsh log-concavity conjecture from [Kat16, Section 7.5]. If there is time, you can include the motivic characterization of the characteristic polynomial in [Kat16, Section7.2] for motivation (for which you would have to know about the Grothendieck ring of varieties). The same material is also surveyed in [Bak18, Section 3].

### Do, 9. Dezember 2021

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Talk 7: A crash course on toric varieties (9.12.)∗∗[Felix Goebler] Give a crash course on the basic theory of toric varieties, in particular how to construct them from a rational polyhedral fan, as e.g. in [Kat16, Section 10.1]. Since this is only a 60-minute talk, it might be a good idea to mostly focus on examples. This talk is perfect for someone, who has already encountered toric varieties before and wants to share this story with everyone else. It requires no knowledge of the previous talks.
• Talk 8: Minkowski weights and the Chow ring of a toric variety (9.12)∗∗[LucaBattistella] Start by recalling the basic notions of intersection theory and explain how to think of of the Chow cohomology ring of a smooth and complete toric variety in terms of Minkowski weights, as outlined in [Kat16, Section 10.2]. If there is time, say something about how to prove this identification (see [FS97]). This talk is particularly suitable for a speaker who has already seen some basic notions of intersection theory and requires no knowledge of any of the previous talks but Talk 7.

### Do, 20. Januar 2022

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Talk 9: Bergman fans and the permutohedral variety (20.1.)∗∗[Stefan Rettenmayr] Introduce the Bergman fan of a matroid and explain how it allows us to think of a matroid as a Minkowski weight on the permutohedral fan. This can be carried out as in [Kat16, Section11]; but also [Huh18b] can help. The purpose of this talk is to connect Talks 2-6 with Talk 7-8.
• Talk 10: Intersection theory on the permutohedral variety (20.1.)∗∗∗[AndreasGross] Prove the log-concavity conjecture for realizable matroids using standard (but non-trivial)methods from algebraic geometry (as outlined in the proof of [Kat16, Theorem 12.2]). The wonderful compactification of a hyperplane arrangement will have a surprise appearance here. Also compare this to [Huh18b] and the original article [HK12]. This talk requires a fairly strong background in complex algebraic geometry.

### Mi, 2. Februar 2022

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Christin Bibby (Louisiane State University): tba.

### Do, 03. Februar 2022

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Talk 11: The Chow ring of a matroid (3.2.)∗∗∗[Alex Küronya] Define the Chow ring of an arbitrary matroid, as in [AHK18, Section 4 and 5], state the Hodge-Riemann bilinear relations and the Hard Lefschetz Theorem, and give an overview how to deduce the general log-concavity conjecture from these results (as in [AHK18, Section9]).
• Talk 12 by June Huh (Princeton): Kazhdan-Lusztig theory of matroids and its relation to Hodge theory (3.2)The foremost expert on the topic of our seminar will give a virtual talk on the big picture and the next chapter of this story.

### Do, 17. Februar 2022

Hodge theory of matroids. GAUS-Seminar – WiSe 2021/22. Organizer: Martin Ulirsch

• Last meeting: Discussion of topics for the next seminar (17.2)

Bereits gehaltene Vorträge im WiSe 2021/22:

### Mi, 06. Oktober 2021

• Bachelorabschlussvortrag (on zoom)
11:00 - 12:00: Tim Paulini (Univ. Frankfurt): Der Fox'sche Differentialkalkül

Den zoom link zu dieser Veranstaltung erhalten Sie bei Interesse per email von stix[at]math.uni-frankfurt.de

### Sommersemester 2021

Das Seminar findet bis auf weiteres über Zoom statt. Der Link wird am Morgen des Seminartermins via Zoom an die Mitglieder der Arbeitsgruppe geschickt. Falls Sie die Emails unserer Arbeitsgruppe nicht bekommen, melden Sie sich bitte mit einer kurzen Email an ulirsch[at]math.uni-frankfurt.de an.

The Seminar will be held via Zoom for the foreseeable future. The link will be sent to all members of our working group on the morning of the seminar. If you do not receive the emails of our working group, please register with a short email to ulirsch[at]math.uni-frankfurt.de.

* Das Seminar findet zusammen mit der Technischen Universität Darmstadt, der Ruprecht-Karls-Universität Heidelberg und der Johannes Gutenberg-Universität Mainz als DaFraHeiMai statt.

Bereits gehaltene Vorträge im SoSe 2021:

Mi, 21. April 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00 - 17:00: Noémie Combe (MPI Leipzig): The realm of Frobenius manifolds.
• This talk will focus on different facets of so-called Frobenius manifolds, a mathematical object that arose in the process of axiomatisation of Topological Field Theory (TFT). Until 2019 there were three main classes: 1. Quantum cohomology, in relation to Gromov--Witten invariants. 2. Saito manifold (unfolding spaces of singularities), in relation to Landau--Ginzburg models. 3. The moduli space of solutions to Maurer--Cartan equations appearing in the Barannikov--Kontsevich theory, related to Gerstenhaber--Batalanin--Vilkoviskiy algebras. In a result 2020, in a joint work with Yu. Manin, we have proved that there exists a very unexpected bridge between algebraic geometry (involving moduli spaces of curves, Gromov--Witten invariants, unfolding spaces of isolated singularities, known as the Saito manifold) and statistical manifolds, central objects for machine learning, information geometry and decision theory. In this talk we will discuss different aspects of Frobenius manifolds in particular the Saito manifold (unfoldings of isolated singularities) and consider relations to Grothendieck—Teichmuller theory.

Do, 22. April 2021, Oberseminar Algebra und Geometrie (on Zoom)

DaFraHeiMai* / GAUS-Seminar: The P=W conjecture

• 15:15-16:15 (Talk 1) Martin Möller:  "What is the P=W conjecture about?"
This talk is an overview over the topic serving as a guideline for the following talks.
• 16:15-16:45: Coffee break
• 16:45-17:45 (Talk 2) Felix Göbler: "The Betti moduli space"
In this talk we will be introduced to one of the several moduli spaces the conjecture is concerned with - the moduli space of (twisted) representations of the fundamental group into GL(n,C)

Fr, 30. April 2021, TGiF-Tropical Seminar - First meeting in the Summer Semester 2021 (on Zoom)

• 14:00-15:00 Felipe Rincon (Queen Mary University)
15:00-15:15: Break
• 15:15-16:15 Jeremy Usatine (Brown University)
16:15-16:30: Break
• 16:30-17:30 Shiyue Li (Brown University)

Mi, 05. Mai 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00 - 17:00: Marcin Lara (IMPAN/Warsaw): Fundamental groups of rigid spaces, geometric arcs and specialization morphism
We introduce a new category of coverings in rigid geometry, called geometric coverings, and show it is classified by a certain topological fundamental group. Geometric coverings generalize the class of étale coverings, introduced by de Jong, and its various natural modifications, and have certain desirable properties that were missing from those older notions: they are étale local and closed under taking infinite disjoint unions. The definition is based on the property of unique lifting of “geometric arcs”. On the way, we answer some questions from the foundational paper of de Jong.
In a separate project, for a formal scheme over a complete rank one valuation ring, we prove existence of a specialization morphism from the de Jong fundamental group of the rigid-analytic generic fiber to the pro-étale fundamental group of the special fiber.
This is joint work with Piotr Achinger and Alex Youcis.

Do, 06. Mai 2021, Oberseminar Algebra und Geometrie (on Zoom)

DaFraHeiMai* / GAUS-Seminar: The P=W conjecture

• 15:15-16:15 (Talk 3) Paul Kiefer:  "Hodge structures and mixed Hodge structures"
• 16:15-16:45: Coffee break
• 16:45-17:45 (Talk 4) Matti Würthen: "The mixed Hodge structure of the Betti moduli space"

Mi, 19. Mai 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00 - 17:00: Andrea Petracci (FU Berlin): On deformations of toric varieties and applications to moduli of Fano varieties.
• Abstract: Toric varieties are algebraic varieties whose geometry is encoded in certain discrete combinatorial objects, such as cones and polyhedra. After recalling the deformation theory of toric affine varieties (due to Klaus Altmann), in this talk I will show some applications to the local study of the recently constructed moduli space of K-polystable Fano varieties (i.e. Fano varieties admitting a Kähler-Einstein metric). In particular, I will explain that this moduli space is singular - this is joint work with Anne-Sophie Kaloghiros.

Do, 20. Mai 2021, Oberseminar Algebra und Geometrie (on Zoom)

DaFraHeiMai* / GAUS-Seminar: The P=W conjecture

• 15:15-16:15 (Talk 5) Felix Röhrle:  "The de Rham moduli space and Riemann-Hilbert correspondence"
• 16:15-16:45: Coffee break
• 16:45-17:45 (Talk 6) Johannes Schwab: "The Dolbeault moduli space and the abelian Hodge correspondence"

Fr, 28. Mai 2021, TGiF-Tropical Seminar - Second meeting in the Summer Semester 2021 (on Zoom)

• 14:00-15:00 Margarida Melo (Roma Tre University)
15:00-15:15: Break
• 15:15-16:15 Baldur Sigurðsson (UNAM Cuernavaca)
16:15-16:30: Break
• 16:30-17:30 Jenia Tevelev (UMass Amherst)

Mi, 02. Juni 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00 - 17:00: Michael Groechenig (Toronto): Hypertoric Hitchin systems and p-adic integration.
The first half of this talk will be devoted to a panoramic overview of p-adic integration for Hitchin systems. In particular, I will explain the main ideas that were used in joint work with Dimitri Wyss and Paul Ziegler to resolve the Hausel--Thaddeus conjecture.
In the second half we will turn to concrete examples. A construction due to Hausel and Proudfoot associates to a graph a complex-analytic integrable system. In joint work with Michael McBreen we introduce a formal-algebraic analogue of their spaces and compute the p-adic volumes of the fibres in graph-theoretic terms.

Do, 10. Juni 2021, Oberseminar Algebra und Geometrie (on Zoom)

DaFraHeiMai* / GAUS-Seminar: The P=W conjecture

• 15:15-16:15 (Talk 7) Riccardo Zufetti:  "The Hitchin map and spectral data"
• 16:15-16:45: Coffee break
• 16:45-17:45 (Talk 8) Jakob Stix: "Non-Abelian Hodge Correspondence"

Fr, 11. Juni 2021, Mini-Workshop on Toric Degenerations (on Zoom)

• 14:00-15:00 Victor Batyrev (Univeristät Tübingen)
15:00-15:15: Break
• 15:15-16:15: Lara Bossinger (UNAM Oaxaca)
16:15-16:30: Break
• 16:30-17:30 Chris Manon (University of Kentucky)

### Mi, 23. Juni 2021

• Bachelorabschlussvortrag (on Vidyo)

14:00 - 15:00: Marcel Feth (Goethe-Universität Frankfurt): Aufbau, Funktionsweise und Sicherheit digitaler Signaturen am Beispiel von RSA und ElGamal.
Die Arbeit behandelt den Aufbau der digitalen Signaturverfahren RSA und ElGamal unter Verwendung von Hashfunktionen und betrachtet mit der Faktorisierungsproblematik ganzer Zahlen und dem Problem des diskreten Logarithmus die den Verfahren zugrunde liegende Sicherheit.

Den Zugangscode erhalten Sie bei Interesse per e-mail an "colmar@math.uni-frankfurt.de".

### Mi, 23. Juni 2021

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Mattia Talpo (Pisa): Derived categories of parabolic sheaves with rational weights.

I will talk about some of my past work (joint with N. Sibilla and S. Scherotzke) describing semi-orthogonal decompositions for the derived category of parabolic sheaves with rational weights on certain log schemes. I will start by recalling the notion(s) of parabolic bundles and sheaves on a pair, their relationship with bundles and coherent sheaves on (finite and infinite) root stacks, and then I will explain how to apply known results about semi-orthogonal decompositions on root stacks.

### Do, 24. Juni 2021

• Oberseminar Algebra und Geometrie (on Zoom)
DaFraHeiMai* / GAUS-Seminar: The P=W conjecture
• 15:15-16:15 (Talk 9) Anton Güthge:  "The constructible derived category and intersection complexes"
• 16:15-16:45: Coffee break
• 16:45-17:45 (Talk 10) Can Yaylali: "Perverse sheaves and the topology of algebraic maps"

### Fr, 25. Juni 2021

TGiF-Tropical Seminar - Third meeting in the Summer Semester 2021 (on Zoom)

• 14:00-15:00 Hülya Argüz (Université de Versailles)
15:00-15:15: Break
• 15:15-16:15 Stefano Mereta (Swansea University)
16:15-16:30: Break
• 16:30-17:30 Eric Katz (Ohio State University)

### Mi, 30. Juni 2021

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Victoria Hoskins (Radboud University Nijmegen): The geometry and cohomology of moduli spaces of vector bundles and Higgs bundles.

Moduli spaces are geometric solutions to classification problems in algebraic geometry. One of the most classical examples is moduli of vector bundles and Higgs bundles on a Riemann surface, which has very rich geometry and has connections with representation theory and mathematical physics. I will describe the geometry of these moduli spaces and survey some results on their various cohomological invariants. Finally I will present some joint work with Simon Pepin Lehalleur on the motives of these moduli spaces, which unify different cohomological invariants and also encode Chow groups describing subvarieties of these moduli spaces.

### Mi, 07. Juli 2021

• Masterabschlussvortrag (on Vidyo)
16:00 - 17:00: Melda Görür (Univ. Frankfurt): Tropische Realisierungen von gewichteten Fächern.

In meiner Arbeit untersuche ich tropische Realisierungen eines gegebenen gewichteten Fächers. Hierzu verwende ich insbesondere Kenntnisse aus der torischen und tropischen Geometrie. Außerdem beschäftige ich mich mit tropischen Realisierungen des multiplizitätsfreien gewichteten Matroidfächers, der mithilfe eines Matroids definiert wird und durch Ardila und Klivans eingeführt worden ist.

### Do, 08. Juli 2021

• Oberseminar Algebra und Geometrie (on Zoom)
DaFraHeiMai* / GAUS-Seminar: The P=W conjecture
• 15:15-16:15 (Talk 11) Martin Ulirsch:  "P=W for tautological classes"
• 16:15-16:45: Coffee break
• 16:45-17:45 (Talk 12) Luca Battistella: "P=W for G = GL(2, C)"

### Di, 13. Juli, Do, 15. Juli, Fr. 16. Juli 2021

• Ruth Moufang Lectures by Jennifer Balakrishnan (Boston University): Rational points on curves: from Diophantus to the present day
• 13. Juli, 15:00 - 15:45 Uhr Andrea Blunck (University of Hamburg): Ruth Moufang and her work on projective planes
• 13. Juli, 16:00 - 17:00 Uhr Jennifer Balakrishnan: Questions about rational points on curves
• 15. Juli, 16:00 - 17:00 Uhr Jennifer Balakrishnan: Mordell’s conjecture and the last ninety-nine years
• 16. Juli, 16:00 - 17:00 Uhr Jennifer Balakrishnan: Explicit methods for rational points on curves

### Mi, 14. Juli 2021

• Bachelorabschlussvortrag
14:00 - 15:00: Julian Schneider: Amenable Gruppen und das Banach-Tarski-Paradoxon.

### Mi, 14. Juli 2021

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Fabio Bernasconi (University of Utah): Log liftability for del Pezzo surfaces and applications to singularities in positive characteristic.

In a recent work with Arvidsson and Lacini, we proved a liftability result to characteristic zero for singular del Pezzo surfaces over perfect fields of characteristic $p>5$. I will explain exactly what notion of liftability we use for singular surfaces (and pairs), and I will sketch some parts of the proof.
From this, I will explain the importance of this result for positive characteristic birational geometry: we prove a Kawamata-Viehweg vanishing theorem for such surfaces that I successively used, in a work with Kollár, to deduce properties of singularities of klt threefolds and new liftability results for threefolds Mori fibre spaces in positive characteristic.

### Mi, 04. August 2021

• Oberseminar Algebra und Geometrie (on Zoom)
16:00 - 17:00: Xuesen Na (University of Maryland): Limiting configuration of SU(1,2) Higgs bundles
The moduli space of Higgs bundles, or the space of solutions of Hitchin equations has been a focus of intensive studies in algebraic geometry, symplectic geometry and topology. Recently the asymptotics near the ends of the moduli space has been investigated by studying behavior of solutions for (E,t\Phi) as $t\to\infty$ by Mazzeo et al (2014), Mochizuki (2016) and Fredrickson (2018) for some cases of SL(n,C) Higgs bundles.
In this talk I will present a new result of the limiting behavior of solutions SU(1,2) Hitchin equation, as a first step of extending the study to the G-Higgs bundle with G a real rank-one Lie group. The proof relies on construction of approximate solutions by gluing local models on disks to decoupled solutions which converge to limiting configuration after appropriate scaling. A by-product of the study is an explicit description of spectral data of generic SU(1,2) Higgs bundle by Hecke transformations.