Georgia Institute of Technology College of Sciences

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

In this talk we will explore the interplay between tropical convexity and its classical counterpart. In particular, we will focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and of a ray in Rn/R1 and show that tropical convex hull and classical convex hull commute in R3/R1. Finally, we prove results on the dimension of tropical convex fans and give an upper bound on the dimension of the tropical convex hull of tropical curves under certain hypothesis. 

The talk will be held online via Bluejeans, use the following link to join the meeting.

Already for bivariate tropical polynomials, factorization is an NP-Complete problem.In this talk, we will introduce a rich class of tropical polynomials in n variables, which admit factorization and rational factorization into well-behaved factors. We present efficient algorithms of their factorizations with examples. Special families of these polynomials have appeared in economics,discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to open problems of interest in discrete geometry.

The talk will be held online via Bluejeans. Use the following link to join the meeting.

Merris and Watkins interpreted results of Littlewood to give generating functions for symmetric group characters induced from one-dimensional characters of Young subgroups.  Beginning with an n by n matrix X of formal variables, one obtains induced sign and trivial characters by expanding sums of products of certain determinants and permanents, respectively. We will look at a new analogous result which holds for hyperoctahedral group characters induced from the four one-dimensional characters of its Young subgroups.  This requires a 2n by 2n matrix of formal variables and four combinations of determinants and permanents.  This is joint work with Jongwon Kim.