## Aktuelle Vorträge und Veranstaltungen

### Sommersemester 2019

Mi, 17. Juli 2019 Bachelor-Abschlussprüfung

• Hannah Laus
Schubertkalkül auf Grassmannschen
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (kl.), Robert-Mayerstr. 10

Do, 18. Juli 2019 Bachelor-Abschlussprüfung

• Jakub Nezam
Dressian und Matroid-Unterteilungen
Zeit und Ort: 14:30-16:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Do, 18. Juli 2019 Oberseminar Algebra und Geometrie

• Dmitry Zakharov (Central Michigan University)
Covers of algebraic curves and graphs with a finite group action

A classical subject of algebraic geometry is the study of curves with a group action. For example, a hyperelliptic curve is a curve with a Z/2Z-action such that the quotient is the projective line. Covers of a given curve X with structure group G are classified by monodromy representations from the fundamental group of X to G.
Tropical geometry studies degenerations of algebraic objects by means of certain polyhedral counterparts, which record the combinatorial structure of the degenerations. The tropical counterpart of an algebraic curve is a metric graph with vertex weights and is called a tropical curve. In my talk, I will describe a theory of G-covers of tropical curves, where G is a finite group. A G-cover of a tropical curve is a map of graphs together with an action of G on the source graph commuting with the cover. The principal complication is that the action is not required to be free, so the cover may have non-trivial stabilizers that vary along the graph.
I will focus mostly on the case when the structure group is abelian, which corresponds, in the number-theoretic setting, to class field theory. I will show that, when G is abelian, G-covers of a tropical curve T with given stabilizers are classified by a cohomology group that generalizes the first simplicial cohomology group of T with coefficients in G. I will also describe the relationship between cyclic covers of the curve T and the torsion in its Jacobian.
Joint work with Yoav Len and Martin Ulirsch.
Zeit und Ort: 16:00-18:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

### Wintersemester 2019/20:

Mi, 23. Oktober 2019 Oberseminar Algebra und Geometrie

• Dr. Giulio Bresciani (FU Berlin)
tba.
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (kl.), Robert-Mayerstr. 10

Bereits gehaltene Vorträge im Sommersemester 2019:

Mi, 03. April 2019 Oberseminar Algebra und Geometrie

Galois action on Riemann surfaces and their associated solenoids

Let S be a compact Riemann surface uniformised by a Fuchsian group Γ. For any element σ ∈ G := Gal(C/Q) the natural Galois action of G on the coefficients of the algebraic equation corresponding to S yields a new Riemann surface S^σ with uniformising group Γ^σ.
Little seems to be known about the relationship between Γ and Γ^σ as subgroups of PSL_2(R). In this talk I will attempt to show that some invariants of this Galois action can be found by studying the action of G on the solenoid associated to S. I will apply these results to present explicit non-Galois-conjugate (arithmetic) Fuchsian groups.
Zeit und Ort: 11:00-12:00 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

Mi, 24. April 2019 Frankfurter Seminar

• Karl-Theodor Sturm (Universität Bonn)

Ginkgo-Seminar ab 15.15 - 16.00 Uhr
Tee 16.15 - 16.45
Zeit und Ort: ab 16:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Do, 25. April 2019 Darmstadt-Frankfurt Seminar [Programm]
- Forschungsseminar Uniformity of rational points on curves -

• Timo Henkel, Berkovich analytic spaces (Talk 1)
• Paul Kiefer, Skeletons of curves I (Talk 2)

Zeit und Ort: ab 15:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Do, 2. Mai 2019 Bachelor Seminar

• David Zimmermann (Bachelor Vortrag im Abschlussseminar)
Arithmetik von Hurwitz-Quaternionen
Zeit und Ort: 15:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Mi, 15. Mai 2019 Frankfurter Seminar

• Andrea Colesanti (Universität Florenz)

Ginkgo-Seminar ab 15.15 - 16.00 Uhr
Tee 16.15 - 16.45
Zeit und Ort: ab 16:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Mi, 22. Mai 2019 Oberseminar Algebra und Geometrie

• Alexander Betts (MPIM Bonn)
Non-abelian Kummer maps for curves

The Q -pro-unipotent non-abelian Kummer map associated to a curve X is a certain function controlling the existence of Galois-invariant paths between points of X, and plays an important role in the non-abelian Chabauty method for finding rational points. In this talk, I will report on a project with Netan Dogra, in which we compute these maps explicitly when the base field is p-adic - by relating the problem to complex-analytic curve families over the punctured disc, we give a description of this map in terms of harmonic analysis on the reduction graph of X. As a result, we are able to prove injectivity results for these maps.
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

Do, 23. Mai 2019 Darmstadt-Frankfurt Seminar [Programm
- Forschungsseminar Uniformity of rational points on curves -

• Theresa Kumpitsch, Divisor theory on metric graphs (Talk 5)
• Sara Lamboglia, Jacobians of metric graphs (Talk 6)

Zeit und Ort: ab 15:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Mi, 05. Juni 2019 Frankfurter Seminar

• Michel Brion (Universität Grenoble)

Ginkgo-Seminar ab 15.15 - 16.00 Uhr
Tee 16.15 - 16.45
Zeit und Ort: ab 16:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Do, 06. Juni 2019 Oberseminar Algebra und Geometrie

• Lucia Mocz (Universität Bonn)
A New Northcott Property for Faltings Height

The Faltings height is a useful invariant in arithmetic geometry. In particular, it plays a key role in Faltings' proof of the Tate conjecture for abelian varieties, in which it is crucially used that the Faltings height satisfies a specific Northcott property. Here we demonstrate a different Northcott property: namely, assuming the Colmez Conjecture and the Artin Conjecture, we show that there are finitely many CM abelian varieties over the complex numbers of a fixed dimension which have bounded Faltings height, along with an unconditional statement within isogeny classes.
Zeit und Ort: 15:00-16:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10
• Dr. Hanieh Keneshlou (Max-Planck-Institut Leipzig)
Moduli of 6-gonal genus 11 curves with several pencils

Considering a smooth d-gonal curve C of genus g, one may naturally ask about the existing possible number of pencils of degree d on C. Motivated by some questions of Michael Kemeny, in this talk we will focus on this question for hexagonal curve of genus 11. We describe a unirational irreducible component of the schemes of 6-gonal genus 11 curves possessing certain numbers of pencils.
Zeit und Ort: 16:30-17:30 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Fr, 07. Juni 2019 TGiF-Tropical Geometry in Frankfurt - First meeting in Summer Semester

Schedule

• Margarida Melo (Università degli studi Roma Tre):
Combinatorics and moduli of line bundles on stable curves.

The moduli space of line bundles on smooth curves of given genus, the so called universal Jacobian, has a number of different compactifications over the moduli space of stable curves. These compactificatons have very interesting combinatorial properties, which can be used to describe their geometry. In the talk I will explain different features and applications of these interesting objects, focusing on properties which have a natural tropical counterpart.

• Farbod Shokrieh (University of Copenhagen):
Heights and moments of abelian varieties

We give a formula which, for a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron-Tate height of $(A,\lambda)$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a role.
(Based on joint works with Robin de Jong.)

• Philipp Jell (Universität Regensburg):
The tropical Hodge conjecture for divisors

The Hodge conjecture is one of the big open questions in algebraic geometry. Mikhalkin and Zharkov formulated a tropical analogue of this conjecture. In joint work with Johannes Rau and Kristin Shaw, we established this conjecture for divisors. I will introduce the notions that are necessary to state the tropical Hodge conjecture and then sketch the proof and further directions of research.

Mi, 12. Juni 2019 Oberseminar Algebra und Geometrie

• Dr. Samouil Molcho (Hebrew University of Jerusalem)
The Logarithmic Picard Group and its Tropicalization
Abstract: The Jacobian of a family C ---> S of smooth curves is an Abelian variety, that is, a proper smooth group scheme over S. On the other hand, when the family of smooth curves is allowed to degenerate to a nodal curve, there is in general no way to extend the Jacobian to a proper, smooth group scheme over the limiting nodal curve as well. However, in the setting of logarithmic geometry, such a degeneration of the Jacobian does exist: it is the logarithmic Picard scheme. In this talk I will define the logarithmic Picard group, discuss its main properties, and describe its structure. I will focus in particular on one of the essential pieces of this structure, which is the tropicalization of the logarithmic Picard scheme. This tropicalization, the tropical Picard scheme, can be understood as a moduli space on an associated tropical curve, and is closely related to the tropical Jacobian of a tropical curve. I will describe the tropical Picard scheme in detail and provide the necessary background in log geometry.
Zeit und Ort: 16:00-17:30 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

Mo, 24. Juni 2019 Oberseminar Algebra und Geometrie

• Prof. Dr. Katherine E. Stange (University of Colorado, Boulder)
An illustration in number theory: abelian sandpiles and Schmidt arrangements
Abstract:  This talk ranges over some number theoretical topics I've come across recently which have interesting visual aspects.  I'll discuss a question in abelian sandpiles and relate it to Schmidt arrangements, which are pictures, from the world of hyperbolic geometry, that illustrate properties of certain rings of integers.  The talk is largely colloquium-style and accessible to a general mathematical audience.
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

Mi, 26. Juni 2019 Oberseminar Algebra und Geometrie

• Prof. Dr. Jonathan Wise (University of Colorado, Boulder) Complete moduli of line bundles and divisors
Abstract:  The Jacobian of a smooth curve is an abelian variety, but if the curve is allowed to degenerate to acquire nodes, it may not be possible to degenerate the Jacobian simultaneously without sacrificing its smoothness, properness, or group structure. Similarly, the Abel-Jacobi morphism from a curve to its Jacobian fails to extend to extend to nodal curves. I will discuss how logarithmic geometry can correct these problems.
• Zeit und Ort: 16:00-17:30 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

Do, 27. Juni 2019 Darmstadt-Frankfurt Seminar [Programm]
- Forschungsseminar Uniformity of rational points on curves -

• Martin Luedtke, The method of Chabauty-Coleman I: An overview (Talk 9)
• Torsten Wedhorn, The method of Chabauty-Coleman II: Theories of p-adic integration (Talk 10)

Zeit und Ort: ab 15:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Fr, 05. Juli 2019 TGiF-Tropical Geometry in Frankfurt - Second meeting in Summer Semester

Schedule

• Madeline Brandt (University of California at Berkeley)
Matroids and their Dressians

Abstract: In this talk we will explore Dressians of matroids. Dressians have many lives: they parametrize tropical linear spaces, their points induce regular matroid subdivisions of the matroid polytope, they parametrize valuations of a given matroid, and they are a tropical prevariety formed from certain Plücker equations. We show that initial matroids correspond to cells in regular matroid subdivisions of matroid polytopes, and we characterize matroids that do not admit any proper matroid subdivisions. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. If time permits, we will also discuss an ongoing project extending these ideas to flag matroids.
Zeit und Ort: 13:15-14:15 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

• Dhruv Ranganathan (University of Cambridge)
Tropical curves, stable maps, and singularities in genus one

Abstract: In the early days of tropical geometry, Speyer identified an extremely subtle combinatorial condition that distinguished tropical elliptic space curves from arbitrary balanced genus one graphs. Just before this, Vakil and Zinger gave a very explicit desingularization of the moduli space of elliptic curves in projective space, with remarkable applications. Just after this, Smyth constructed new compactifications of moduli spaces of pointed elliptic curves, using worse-than-nodal singularities, as part of the Hasset-Keel program. A decade on, we understand these three results as part of a single story involving logarithmic structures and their tropicalizations. I will discuss this picture and how the unified framework extends all three results. This is joint work with Keli Santos-Parker and Jonathan Wise.
Zeit und Ort: 14:45-15:45 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

• Yoav Len (Georgia Institute of Technology)
Algebraic and Tropical Prym varieties

Abstract: My talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian vari- eties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concern- ing Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results con- cerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci. This is joint work with Martin Ulirsch.
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

Mi, 10. Juli 2019 Frankfurter Seminar

• Mihyun Kang (TU Graz)

Ginkgo-Seminar ab 15.15 - 16.00 Uhr
Tee 16.15 - 16.45
Zeit und Ort: ab 16:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10