Georgia Institute of Technology College of Sciences

Matroid theory has seen fruitful developments arising from different algebro-geometric approaches, such as the K-theory of Grassmannians and Chow rings of wonderful compactifications. However, these developments have remained somewhat disjoint. We introduce "tautological bundles of matroids" as a new geometric framework for studying matroids. We show that it unifies, recovers, and extends much of these recent developments including log-concavity statements, as well as answering some open conjectures. This is an on-going work with Andrew Berget, Hunter Spink, and Dennis Tseng.

BlueJeans link: https://bluejeans.com/569437095

We write $F \rightarrow (H,G)$ for graphs $F$, $G$, and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let the induced Ramsey number $IR(H,G)$ be the smallest number of vertices in a graph $F$ such that $F \rightarrow (H,G)$. Deuber showed in 1975 that $IR(H,G)$ is well-defined for any graphs $H$ and $G$. Still, the determination of $IR(H,G)$ remains a challenge for most graphs. A striking contrast between induced and non-induced Ramsey numbers was demonstrated by Fox and Sudakov in 2008 by showing that $IR(H,G)$ is superlinear in $n$ when $H$ is a matching on $n$ edges and $G$ is a star on $n$ edges.

In this talk, I will address the case when $G= K_{1,n}$, a star on $n$ edges, for large $n$, and $H$ is a fixed graph. We prove that $$ (\chi(H)-1) n \leq IR(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$, sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

This is a joint work with Izolda Gorgol.

Live cell imaging and single particle tracking techniques have become increasingly popular amongst the mathematical biology community. We study endocytosis, the cellular internalization and transport of bioparticles. This transport is carried out in membrane-bound vesicles through the use of motor proteins. Lysosomes, known for endocytosis, phagocytic destruction, and autophagy, move about the cell along microtubules. Single particle tracking methods utilize stochastic models to simulate intracellular transport and give rise to rigorous analysis of the resulting properties, specifically related to transitioning between inactive to active states. This confidence in the stochastic modeling of particle tracking is useful not only for particle-containing lysosomes, but also broad questions of cellular transport studied with single particle tracking.

Meeting Link: https://gatech.bluejeans.com/348270750