Nowhere-zero flows in the linear lattice

09/17/2013 - 12:15pm
09/17/2013 - 1:10pm
Ghassan Sarkis (Pomona College)

A flow of a directed graph is an integer labeling of the graph's edges such that, at each vertex, the sum of the labels over the edges into that vertex equals the sum of the labels over the edges out of that vertex. For an undirected graph, the sum of the labels over the edges at each vertex has to equal zero. A flow is called nowhere-zero if all the labels are nonzero. Conjectures about flows focus on the existence of labelings with small absolute value. By viewing flows as vectors in the nullspace of the graph's incidence matrix, one can generalize the study to flows of bipartite graphs in which one set of vertices is labeled. I will report on work with Shahriar Shahriari and the Pomona College Undergraduate Research Circle that computes explicit flows between two levels of the lattice of subspaces of a finite-dimensional vector space over a finite field.

Mudd Science Library 126, Pomona College