Georgia Institute of Technology College of Sciences

The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

We consider the question of coloring the first n integers with two colors in such a way as to avoid copies of a given arithmetic configuration (such as 3 term arithmetic progressions, or solutions to x+y=z+w). We know from results of Van der Waerden and others that avoiding such configurations completely is a hopeless task if n is sufficiently large, so instead we look at the question of finding colorings with comparatively few monochromatic copies of the configuration. At the very least, can we do significantly better than just closing our eyes and coloring randomly? I will discuss some partial answers, experimental results, and conjectured answers to these questions for certain configurations based on joint work with Steven Butler and Ron Graham.

This is part two of a lecture series investigating questions in contact geometry from the perspective of Riemannian geometry. Interesting questions in Riemannian geometry arising from contact geometry have a long and rich history, but there have been few applications of Riemannian geometry to contact topology. In these talks I will discuss basic connections between Riemannian and contact geometry and some applications of these connections. I will also discuss the "contact sphere theorem" that Rafal Komendarczyk, Patrick Massot and I recently proved as well as other results.

Despite its long history, the theory of ellipticpartial differential equations in non-smooth media is abundant with openproblems. We will discuss the main achievements in the theory, recentdevelopments, surprising paradoxes related to the behavior of solutions nearthe boundary, and some fundamental questions which still remain open.