Colloquia — Summer 2003
Thursday, June 5, 2003
Semigroups and the Road Coloring Problem
Southern Illinois University, Carbondale
J. S. Ratti
Suppose one is given a strongly connected, aperiodic directed graph. Think of the vertices of the directed graph as buildings connected by unnamed one-way roads, and assume there is a person in each building. Under what conditions can one color (that is, name) the roads, so that the same set of instructions gets each person to the same building at the same time.
The problem above is known as the road coloring problem and has been open for almost thirty years. Recent efforts to solve the problem use algebraic methods, and the speaker will describe an approach using semigroup theory. Properties of the digraph will be given semigroup formulations and it will be shown how the structure of the minimal ideal of the “coloring semigroup” plays a critical role in the analysis of the problem. Recent results will be surveyed and a generalization of the problem to periodic graphs will be discussed.
Friday, May 16, 2003
On the existence of nontrivial solutions in some elliptic equations with slowly growing principal operators
Khoi Le Vy
University of Missouri — Rolla
We are interested in the existence of nontrivial solutions of mountain pass type for certain quasilinear elliptic equations. Our main tool is a version of the Mountain pass theorem for variational inequalities, without the Palais-Smale condition, in some appropriate Orlicz-Sobolev space.