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Organizers: Laura Anderson, Michael Dobbins, Vaidy Sivaraman, and Thomas Zaslavsky.
There are several different topological representations of non-orientable matroids. In this talk, inspired by Swartz's work, I will show an explicit homotopy sphere arrangement that is a regular CW-complex whose intersection lattice is the geometric lattice of the corresponding matroid. I will also look at enumerative properties, including how the flag f-vector formula of Billera, Ehrenborg, and Readdy for oriented matroids applies to arbitrary matroids.
A finite-field Kakeya set is a subset of (Fq)n (the n-dimensional vector space over a finite field of order q) which contains a line in each direction. I will first show how to use polynomials to obtain bounds on the size of a Kakeya set in finite vector spaces. Then I will show how to modify this polynomial technique to obtain better bounds on the size of these Kakeya sets. Time permitting, I will discuss bounds on related incidence problems.
I will discuss domino tilings of three-dimensional manifolds. I will focus on the connected components of the space of tilings of such manifolds under local moves. Using topological techniques, I introduce two parameters of tilings: the flux and the twist. The main result characterizes, in terms of these two parameters, when two tilings are connected by local moves. I will not assume any familiarity with the theory of tilings.
A cycle with two blocks c(k,l) is an oriented cycle consisting of two internally disjoint directed paths of lengths at least k and l, respectively, from a vertex to another one. In 2007, Addario-Berry, Havet, and Thomassé asked if every strongly connected digraph D containing no c(k,l) has chromatic number at most k+l−1. In this talk, I show that such a digraph D has chromatic number at most O((k+l)2), improving the previous upper bound O((k+l)4) of Cohen, Havet, Lochet, and Nisse.
This is joint work with Seog-Jin Kim, Jie Ma, and Boram Park.