Tuesdays at 4:10 in
September 15: David Offner (University of Illinois-Chicago) Discrete Aleksandrov solutions of the Monge-Ampere Equation
September 8: Gerard Awanou (University of Illinois-Chicago) Discrete Aleksandrov solutions of the Monge-Ampere Equation
September 15: David Offner
Title: Polychromatic Colorings of the Hypercube
Abstract: Given a graph G which is a subgraph of the n-dimensional hypercube Qn, an edge coloring of Qn with r ≥ 2 colors such that every copy of G contains every color is called G-polychromatic.
Denote by p(G) the maximum number of colors with which it is possible to G-polychromatically color
the edges of any hypercube. Originally introduced by Alon, Krech and Szab´o in 2007 as a way to
prove bounds for Tura´n type problems on the hypercube, polychromatic colorings have proven to be
worthy of study in their own right. This talk will survey what is currently known about polychromatic colorings and introduce some open questions. In particular, we will discuss the best known constructions that give good lower bounds on p(G) for many graphs G, and a lemma that follows from Ramsey’s Theorem that gives good upper bounds. Exact values for p(Qd) are known for all d, but there are many graphs G for which p(G) cannot be determined using current techniques. In addition,
there are many related open problems. For example, it is not known whether for all r there is a
graph G such that p(G) = r. In addition there are some natural generalizations and variations of
the problem that are only partially understood, and a number of questions about the relationship of
polychromatic numbers to Tura´n type problems on the hypercube.
September 8: Speaker: Gerard Awanou (UIC)
Title: DISCRETE ALEKSANDROV SOLUTIONS OF THE MONGE-AMPERE EQUATION
Abstract: The Monge-Amp`ere equation is a nonlinear partial differential equation which appears in a wide range of applications, e.g. geometric optics and material sciences. We present convergence results for finite difference discretizations with the weak solution in the sense of Aleksandrov. The numerical solution is computed as the minimizer of a convex functional of the gradient and under convexity and nonlinear constraints. For monotone schemes we obtain uniform convergence on compact subsets and for the standard finite difference discretization convergence of the discretization for approximate problems. The main tool used is approximation by smooth functions. Part of this talk is based on joint work with R. Awi and L. Matamba Messi.