Georgia Institute of Technology College of Sciences

Young tableaux arise in the enumerative geometry of linear series on curves in formulas for the Chow class and the holomorphic Euler characteristic of Brill--Noether varieties. I will discuss an intriguing tropical generalization of these two facts: the formulas for Chow class and Euler characteristic of Brill--Noether loci on a general curve occur in the first and last terms of the Ehrhart polynomial of the tropical Brill--Noether loci on a chain of loops. I will speculate on some generalizations and algebraic analogs of this calculation.

Chow rings and augmented Chow rings of matroids played important roles in the settlement of the Heron-Rota-Welsh conjecture and the Dowling-Wilson top-heavy conjecture. Their Hilbert series have been extensively studied since then. It was shown by Ferroni, Mathern, Steven, and Vecchi, and independently by Wang, that the Hilbert series of Chow rings of matroids are $\gamma$-positive using inductive arguement followed from the semismall decompositions of the Chow ring of matroids. However, they do not have an interpretation for the coefficients in the $\gamma$-expansion. Recently, Angarone, Nathanson, and Reiner further conjectured that Chow rings of matroids are equivariant $\gamma$-positive under the action of groups of matroid automorphisms. In this talk, I will give a proof of this conjecture without using semismall decomposition, showing that both Chow rings and augmented Chow rings of matroids are equivariant $\gamma$-positive. Moreover, we obtain explicit descriptions for the coefficients of the equivariant $\gamma$-expansions. Then we consider the special case of uniform matroids which extends Shareshian and Wachs Schur-$\gamma$-positivity of Frobenius characteristics of the cohomologies of the permutahedral and the stellahedral varieties.

Retinal images are often used to examine the vascular system in a non-invasive way. Studying the behavior of the vasculature on the retina allows for noninvasive diagnosis of several diseases as these vessels and their behavior are representative of the behavior of vessels throughout the human body. For early diagnosis and analysis of diseases, it is important to compare and analyze the complex vasculature in retinal images automatically.

During this talk, we will talk about a geodesic tracking approach that is better able to handle difficult structures, like high curvature and crossings. Additionally, we discuss how one can identify connected components in images that allow for small interruptions within the same component. Both methods takes place in the lifted space of positions and orientations SE(2), which allows us to differentiate between crossings and bifurcations.

The rank r matrix completion problem studies whether a matrix where some of the entries have been filled in with generic complex numbers can be completed to a matrix of rank at most r. This problem is governed by the bipartite rigidity matroid, which is a matroid studied in combinatorial rigidity theory. We show that the study of the bipartite rigidity matroid is related to the study of tensor codes, a topic in information theory, and use this relation to understand new cases of both problems. Joint work with Joshua Brakensiek, Manik Dhar, Jiyang Gao, and Sivakanth Gopi.