Next meeting

First meeting in the Summer Semester 2024


Past meetings

February 2, 2024 - Second meeting in the Winter Semester 2023/24

The talks will be given in a hybrid format. If you are close-by, please join us in Frankfurt in room 711 (groß), Robert-Mayer-Str. 10, for two in-person talks. Otherwise, we're hoping to see you on Zoom. The Zoom info will be sent out to the mailing list as usual.


14:30-15:30 Andreas Bernig (Universität Frankfurt): Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
15:30-16:00 Break
16:00-17:00 Manoel Zanoelo Jarra (Universität Groningen): Category of matroids with coefficients

Andreas Bernig: Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
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.

Manoel Zanoelo Jarra: Category of matroids with coefficients
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.


December 14, 2023 - First meeting in the Winter Semester 2023/24

14:30-15:30 Adam Afandi (Universität Münster): Stationary Descendents and the Discriminant Modular Form
15:30-16:00 Break
16:00-17:00 Ajith Urundolil-Kumaran (University of Cambridge): Refined tropical curve counting with descendants

Adam Afandi: Stationary Descendents and the Discriminant Modular Form
Abstract: By using the Gromov-Witten/Hurwitz correspondence, Okounkov and Pandharipande showed that certain generating functions of stationary descendent Gromov-Witten invariants of a smooth elliptic curve are quasimodular forms. In this talk, I will discuss the various ways one can express the discriminant modular form in terms of these generating functions. The motivation behind this calculation is to provide a new perspective on tackling a longstanding conjecture of Lehmer from the middle of the 20th century; Lehmer posited that the Ramanujan tau function (i.e. the Fourier coefficients of the discriminant modular form) never vanishes. The connection with Gromov-Witten invariants allows one to translate Lehmer's conjecture into a combinatorial problem involving characters of the symmetric group and shifted symmetric functions.

Ajith Urundolil-Kumaran: Refined tropical curve counting with descendants
Abstract: We introduce the enumerative geometry of curves in the algebraic torus (C*)^2. We show that a certain class of invariants associated with moduli spaces of curves in (C*)^2 can be calculated explicitly using a refined tropical correspondence theorem. If time permits we will explain how the proof relies on higher double ramification cycles and work of Buryak-Rossi on integrable systems on the moduli space of curves. This is joint work with Patrick Kennedy-Hunt and Qaasim Shafi.


Details of past events are available in the Archiv (Vorträge und Veranstaltungen)



Dr. Andreas Gross
E-mail: gross[at]math.uni-frankfurt.de

Prof. Dr. Martin Ulirsch
E-Mail: ulirsch[at]math.uni-frankfurt.de

FB 12 - Institut für Mathematik
Johann Wolfgang Goethe-Universität
Robert-Mayer-Str. 6-8
D-60325 Frankfurt am Main