[PAST EVENT] Mathematics Colloquium and EXTREEMS-QED Lecture: Xiangqian Zhou (Wright State University)

March 25, 2016
2pm - 3pm
Jones Hall, Room 301
200 Ukrop Way
Williamsburg, VA 23185Map this location
Abstract: A matroid $M$ is a pair $(E, \mathcal{I})$ where $E$ is a finite set and $\mathcal{I}$ is a non-empty collection of subsets of $E$ such that (1) $\mathcal{I}$ is closed under taking subsets; and (2) if $I , J \in \mathcal{I}$ with $|J| > |I|$, then there exists $x \in J \del I$ such that $I \cup \{x\} \in \mathcal{I}$. Members of $\{I}$ are called independent sets of $M$. Clearly the columns of a matrix over a field $F$ would naturally form a matroid. Such a matroid is called an $F$-representable matroid. Matroids that are $F$-representable over every field $F$ are called regular matroids.

A matrix is totally-unimodular if every square sub-matrix has determinant 0, 1 or -1. It turns out that a matroid is regular if it arises from a totally-unimodular matrix. Using matroid theory, Seymour gave a precise characterization of totally-unimodular matrices.

A matrix is dyadic if every square sub-matrix has determinant 0, $\pm$ powers of 2s. We study matroids that arise from a dyadic matrix. In this talk I will introduce the concept of matroids, give s short survey of Seymour's result, and then talk about recent progress on the study of dyadic matrices.

Gexin Yu