Vorträge in der Woche 10.06.2024 bis 16.06.2024

Montag, 10.06.2024: What is the tropicalization of a matrix?

Martin Ulirsch (Frankfurt)

Tropicalization is a process that associates to an algebro-geometric object a piecewise linear polyhedral shadow that captures its essential combinatorial structure. In this talk, I will give an overview of the numerous ways on how to extract tropical information from a matrix over a non-Archimedean field. Each perspective will give rise to inherently quite different phenomena. Central instances of this rich panorama include the tropical geometry of vector bundles, logarithmic concavity results for valuated (bi-)matroids (using techniques from combinatorial Hodge theory), and the geometry of affine buildings. This talk draws from joint work with Andreas Gross and Dmitry Zakharov; Andreas Gross, Inder Kaur, and Annette Werner; Felix Röhrle; Jeff Giansiracusa, Felipe Rincon, and Victoria Schleis; Luca Battistella, Kevin Kühn, Arne Kuhrs, and Alejandro Vargas; as well as with Desmond Coles.

Dienstag, 11.06.2024: Local and global fields II

Felix Röhrle

Mittwoch, 12.06.2024: Classification of real Riemann surfaces in the critical case.

Pietro Giavedoni (Berlin)

The problem of classification of real Riemann surfaces in the critical case can be formulated as the challenge to decide if a real Riemann surface - or equivalently, a projective, smooth and irreducible curve - is endowed with real points, based on any "real" period matrix of its. Solved only for genus two (by Comessatti), it has been open for more than one century nowadays. I will propose an exhaustive and effective solution valid for all genera.

Donnerstag, 13.06.2024: Pseudo-integrale halbzahlige Polygone

Martin Bohnert (Universität Tübingen)

Ein pseudo-integrales halbzahliges Polygon verhält sich ähnlich wie ein Gitterpolygon in dem Sinne, dass die Anzahlen der Gitterpunkte in seinen ganzzahligen Vielfachen als Werte eines Polynoms, seines Ehrhart-Polynoms, bestimmt werden können. Wir klassifizieren die Ehrhart-Polynome aller pseudo-integralen halbzahligen Polygone. Insbesondere erhalten wir, dass für jede positive ganze Zahl $i$ das Polynom $\frac{4i+5}{2}t^2+\frac{2i+7}{2}t+1 \in \mathbb{Q}[t]$ ein Ehrhart-Polynom eines rationalen Polygons ist und dass es daher pseudo-integrale halbzahlige Polygone mit $i$ inneren Gitterpunkten und $2i+7$ Randgitterpunkten gibt. Außerdem werden wir sehen, dass die pseudo-integralen halbzahligen Polygone mit genau einem inneren Gitterpunkt gerade die dualen Polygone der 30 LDP-Polygone mit Gorensteinindex 2 sind.

Donnerstag, 13.06.2024: Magic moire materials with a twist

Simon Becker (ETH Zürich)

I will review the mathematical analysis behind two classes of new exciting materials: Twisted bilayer graphene (TBG) and twisted semiconductors (TMDs) with an emphasis on their mathematical properties and differences. If time permits I will discuss results on these models under disorder. Joint work with Maciej Zworski Izak Oltman Martin Vogel and Mengxuan Yang.