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.
______________________________________________________________________________________________________________