Georgia Institute of Technology College of Sciences

The symmetric configurations for the equilibrium shape of a fluid interfaceare given by the geometric differential equation mean curvature isproportional to height. The equations are explored numerically tohighlight the differences in classically treated capillary tubes andsessile drops, and what has recently emerged as annular capillary surfaces. Asymptotic results are presented.

We study an old problem of Linial and Wilf to find the graphs with n vertices and m edges which maximize the number of proper q-colorings of their vertices. In a breakthrough paper, Loh, Pikhurko and Sudakov asymptotically reduced the problem to an optimization problem. We prove the following structural result which tells us how the optimal solutionlooks like: for any instance, each solution of the optimization problem corresponds to either a complete multipartite graph or a graph obtained from a complete multipartite graph by removing certain edges. We then apply this result on optimal graphs to general instances, including a conjecture of Lazebnik from 1989 which asserts that for any q>=s>= 2, the Turan graph T_s(n) has the maximum number of q-colorings among all graphs with the same number of vertices and edges. We disprove this conjecture by providing infinity many counterexamples in the interval s+7 <= q <= O(s^{3/2}). On the positive side, we show that when q= \Omega(s^2) the Turan graph indeed achieves the maximum number of q-colorings. Joint work with Humberto Naves.

This talk is a sequel to the speaker's previous lecture given in the January 31st Combinatorics Seminar, but attendance at the first talk is not assumed. We begin by carefully reviewing our generalized cycle-cocyle reversal system for partial graph orientations. A self contained description of Baker and Norin's Riemann-Roch formula for graphs is given using their original chip-firing language. We then explain how to reinterpret and reprove this theorem using partial graph orientations. In passing, the Baker-Norin rank of a partial orientation is shown to be one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation. We conclude with an overview of how these results extend to the continuous setting of metric graphs (abstract tropical curves).