Two combinatorial viewpoints on geometry
07.12.2022, 15:00 (unusual!)
– Campus Golm, Haus 9, Raum 2.22 + Zoom
Christian Haase, FU Berlin and Federico Ardila, San Francisco State University
15:00 Christian Haase: Lattice Point Semigroups — Toric and Non-Toric
15:45 Tee und Kaffee Pause
16:15 Federico Ardila: Combinatorial Intersection Theory: Three Examples
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Christian Haase, FU Berlin: Lattice Point Semigroups — Toric and Non-Toric
I will start with a combinatorialist's crash course on toric varieties.
These are algebraic varieties with deep connections to convex polytopes, lattice points and affine semigroups.
Newton-Okounkov theory is an attempt to play as much of the toric game as possible with non-toric varieties. The theory associates an affine semigroup with a projectively embedded variety and tries to draw conclusions from the asymptotic convex geometry of this semigroup. Many of these theorems assume that the semigroup is finitely generated, but checking finite generation seems to be hard. I will sketch how subtle the finite generation question becomes if one deviates from the purely toric path ever so slightly.
This contains joint work with Klaus Altmann, Alex Küronya, Karin Schaller & Lena Walter.
Federico Ardila, San Francisco State University: Combinatorial Intersection Theory: Three Examples
Intersection theory studies how subvarieties of an algebraic variety X intersect. Algebraically, this information is encoded in the Chow ring A(X). When X is the toric variety of a simplicial fan, Brion gave a presentation of A(X) in terms of generators and relations, and Fulton and Sturmfels gave a "fan displacement rule” to intersect classes in A(X), which holds more generally in tropical intersection theory. In these settings, intersection theoretic questions translate to algebraic combinatorial computations in one point of view, or to polyhedral combinatorial questions in the other. Both of these paths lead to interesting combinatorial problems, and in some cases, they are important ingredients in the proofs of long-standing conjectural inequalities.
This talk will survey three problems on matroids and root systems that arise in combinatorial intersection theory. It will feature joint work with Montse Cordero, Graham Denham, Chris Eur, June Huh, Carly Klivans, and Raúl Penaguião.