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.

Schedule:

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

Details:
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.

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