Georgia Institute of Technology College of Sciences

The first meeting of our new professional development seminar for postdocs and other interested individuals (such as advanced graduate students). A discussion of the triumvirate of faculty positions: research, teaching, and service.

We study various binomial and monomial ideals related to the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe minimal polyhedral cellular free resolutions for these ideals. We will show that the resolutions of all these ideals are closely related and that their Betti tables coincide. As corollaries we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related in the theory of chip-firing games on graphs -- including Merino's proof of Biggs' conjecture and Baker-Shokrieh's characterization of reduced divisors in terms of potential theory -- also follow immediately from our general techniques and results.

Studying the ferromagnetic Ising model with zero applied field reduces to sampling even subgraphs X of G with probability proportional to \lambda^{|E(X)|}. In this paper we present a class of Markov chains for sampling even subgraphs, which contains the classical single-site dynamics M_G and generalizes it to nonlocal chains. The idea is based on the fact that even subgraphs form a vector space over F_2 generated by a cycle basis of G. Given any cycle basis C of a graph G, we define a Markov chain M(C) whose transitions are defined by symmetric difference with an element of C. We characterize cycle bases into two types: long and short. We show that for any long cycle basis C of any graph G, M(C) requires exponential time to mix when \lambda is small. All fundamental cycle bases of the grid in 2 and 3 dimensions are of this type. Moreover, on the 2-dimensional grid, short bases appear to behave like M_G. In particular, if G has periodic boundary conditions, all short bases yield Markov chains that require exponential time to mix for small enough \lambda. This is joint work with Isabel Beichl, Noah Streib, and Francis Sullivan.

In this talk I will describe an Lp theory for outer measures, which could be used to connect two themes of Lennart Carleson's work: Carleson measures and time frequency analysis. This is joint work with Christoph Thiele.