Georgia Institute of Technology College of Sciences

We consider one or more volumes of a liquid or semi-molten material sitting on a substrate, while the vapor above is assumed to have the same medium in suspension. There may be both evaporation and condensation to move mass from one cell to another. We explore possible equilibrium states of such configurations. Our examples include a single sessile drop (or cell) on the plate, connected clusters of cells of the material on the plate, as well as a periodic configuration of connected cells on the plate. The shape of the configurations will depend on the type of energy that we take into consideration, and in settings with a vertical gravitational potential energy the clusters are shown to exhibit a preferred granular scale. The majority of our results are in a lower dimensional setting, however, some results will be presented in 3-D.

Mapping classes are the natural topological symmetries of surfaces. Their study is often restricted to the orientation-preserving ones, which form a normal subgroup of index two in the group of all mapping classes. In this talk, we discuss orientation-reversing mapping classes. In particular, we show that Lehmer's question from 1933 on Mahler measures of integer polynomials can be reformulated purely in terms of a comparison between orientation-preserving and orientation-reversing mapping classes.

Let P be a system of unique shortest paths through a graph with real edge weights (i.e. a finite metric). An obvious fact is that P is "consistent," meaning that no two of these paths can intersect each other, split apart, and then intersect again later. But is that all? Can any consistent path system be realized as unique shortest paths in some graph? Or are there more forbidden combinatorial intersection patterns out there to be found? In this talk, we will characterize exactly which path systems can or can't be realized as unique shortest paths in some graph by giving a complete list of new forbidden intersection patterns along these lines. Our characterization theorem is based on a new connection between graph metrics and certain boundary operators used in some recent graph homology theories. This connection also leads to a principled topological understanding of some of the popular algebraic tricks currently used in the literature on shortest paths. We will also discuss some applications in theoretical computer science.