Pittsburgh Combinatorial Algebraic Geometry Mini-Workshop
University of Pittsburgh

April 20 - 21, 2024

Location: Room 703 or 704 Thackeray Hall (7th floor of the math department building)

Workshop poster

Schedule:

Saturday

9:30am-10:30am Austin Alderete
11am-12pm Christopher Manon
2pm-3pm Jenya Soprunova
3:30pm-4:30pm Dustin Ross

Sunday

9:30am-10:30am Kiumars Kaveh
11am-12pm Ivan Soprunov
 Abstracts: Jenya Soprunova The volume polynomial of lattice polygons Let P and Q be lattice polygons in the plane. Then their normalized volumes V(P) and V(Q) and their normalized mixed volume V(P,Q) satisfy the classical Minkowski inequality V(P)V(Q)<= V(P,Q)^2. We show that for any triple of nonnegative integers a,b,c that satisfy the inequality ab<= c^2 there exist lattice polygons P and Q in R^2 such that V(P)=a, V(Q)=b, and V(P,Q)=c. This shows that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons, which solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers. This is joint work with Ivan Soprunov. Ivan Soprunov The volume polynomial of zonoids and the absolute value of the Grassmannian Abstract: The volume polynomial is an n-variate homogeneous polynomial of degree d associated with a collection of n convex bodies in R^d. It originates in Brunn-Minkowski theory and has appeared in various fields including convex and algebraic geometry, combinatorics, and statistics. I will present a new framework that connects the volume polynomial of zonoids with combinatorics of the absolute value of the Grassmannian, which is defined as the image of the real Grassmannian under the coordinate-wise absolute value map. This allows us to derive new coefficient inequalities for the volume polynomial of n zonotopes in dimension d when (n,d)=(4,2), (6,2), and (6,3). This is joint work with Gennadiy Averkov, Katherina von Dichter, and Simon Richard. Dustin Ross New perspectives on tropical intersection theory Abstract: Tropical intersection theory aims to create analogues of intersection-theoretic tools from algebraic geometry in a piecewise-linear setting. In this talk, I’ll describe a few aspects of tropical intersection theory and discuss how these ideas can be used to build bridges between algebraic geometry, combinatorics, and convex geometry. Kiumars Kaveh Toric vector bundles and systems of Laurent polynomial equations Abstract: This is a report on the work in progress with Askold Khovanskii and Hunter Spink. We give a generalization of the famous Bernstein-Kushnirenko-Khovanskii theorem on number of solutions of a system of Laurent polynomial equations to vector-valued Laurent polynomial equations. The answer is in terms of mixed volume of certain virtual polytopes. This is directly related to (representable) matroids and torus equivariant vector bundles on toric varieties. The constructions also make sense for non-representable matroids and give Alexandrov-Fenchel type inequalities for certain virtual polytopes constructed from matroid data. Austin Alderete An Introduction to Matroids, Modularity, and Matroid Extensions Abstract: We review the background material necessary to define and study matroidal vector bundles. By analogy to the representable case, we introduce flats and the notion of a modular pair. Given a matroid M with a non-modular pair of flats, it is natural to ask whether there is some extension of M which repairs the modular defect. We describe the well-known Vamos matroid along with a non-modular pair of flats for which no rank-preserving extension repairs the defect along the image of the chains. In particular, given a pair of flags of flats, it may not be possible to extend to a matroid for which the image of the pair has an adapted basis. Christopher Manon Toric matroid bundles