Georgia Institute of Technology College of Sciences

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham;  The Georgia Institute of Technology;  Emory University;  The University of Tennessee Knoxville.  The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. Catherine Williams (Columbia U);  Hugh Bray (Duke U);  Simon Brendle (Stanford U);  Spyros Alexakis (U of Toronto);  Alessio Figalli (U of Texas at Austin).   There will also be an evening public lecture by plenary speaker Hugh Bray (Duke U) entitled From Black Holes and the Big Bang to Dark Energy and Dark Matter: Successes of Einstein's Theory of Relativity.

The matching sequence of a graph is the sequence whose $k$th term counts the number of matchings of size $k$. The independent set (or stable set) sequence does the same for independent sets. Except in very special cases, the terms of these sequences cannot be calculated explicitly, and one must be content to ask questions about global behavior. Examples of such questions include: is the sequence unimodal? is it log-concave? where are the roots of its generating function? For the matching sequence, these questions are answered fairly completely by a theorem of Heilmann and Lieb. For the independent set sequence, the situation is less clear. There are some positive results, one startling negative result, and a number of basic open questions. In this talk I will review the known results, and describe some recent progress on the questions.