UW Combinatorics and Geometry Seminar: Special episode: Graduate Student Lightning Talks

Talk titles and abstracts:

Natasha Crepeau: Constructing stability conditions from triangulations of Lawrence polytopes

Abstract: A stability condition of a graph $G$is a collection of generalized break divisors of size $|\mathcal{ST}(G)|$. In this talk, we'll construct stability conditions of $G$from triangulations of the Lawrence polytope of the matroid dual to the graphic matroid $\mathcal{M}(G)$, and discuss why these stability conditions are important in the study of compactified Jacobians.

Lauren Nowak: Tropical oriented matroids & tropical (pseudo)hyperplane arrangements

Abstract: Representability is a core aspect of studying matroids. In this expository talk, we will discuss what a tropical oriented matroid is, whether it can be realized as a tropical (pseudo)hyperplane arrangement, and how these questions connect to mixed subdivisions of dilated simplices.

Joe Rogge: Geometry and Combinatorics of P-matrices

Abstract: P-matrices are a broad class of "positive" matrices, generalizing well-known matrix classes such as positive definite matrices, totally positive matrices, and M-matrices (which themselves generalize graph Laplacians). We compute the dimension of the space of P-matrices as a semi-algebraic set, get some insight into the topology of this space, and describe the combinatorics of low order P-matrices.

Cameron Wright: A Categorical Approach to Matroids

Abstract: Following the work of Baker and Bowler, several classes of matroids can be described as matroids over (perfect) idylls $F$, which are a special kind of tract. Such classes include matroids, oriented matroids, regular matroids, and valuated matroids. Generalizing strong maps to $F$-matroids, we obtain notions of short exact sequences and rudimentary homological algebra. We describe this setup and some consequences, considering in particular connections with quiver representations and tropical vector bundles. Based on Joint work with Jaiung Jun and Alex Sistko.

Michael Zeng: K-Theory of Spanning Line Configurations

Abstract: The space of spanning line configurations is$X_{n,k}$a generalization of the flag variety first introduced by Brendan Pawlowski and Brendon Rhoades. It provides geometric foundations for the combinatorics of ordered set partitions aka. Fubini words. Moreover, its cohomology ring is $R_{n,k}$a generalization of the classical coinvariant algebra. The $K_0$ring of a variety is an enrichment of the ordinary cohomology ring by recording how vector bundles assemble and decompose. In this talk, we outline a computation of the $K_0$ring of $X_{n,k}$using Fulton and Lascoux's degeneracy loci formula in terms of double Grothendieck polynomials. We show that $K_0(X_{n,k})$is again isomorphic to $R_{n,k}$, providing a new instance of an exceptional isomorphism between cohomology and $K_0$.