Georgia Institute of Technology College of Sciences

Given integers $1 < s < t$, what is the maximum size of a $K_s$-free subgraph that every $n$ vertex $K_t$-free graph is guaranteed to contain? This problem was posed by Hajnal, Erdős and Rogers in the 1960s as a way to generalize classical graph Ramsey numbers (which corresponds to the case $s=2$). We  prove almost optimal results in the case $t=s+1$ using recent constructions in Ramsey theory. We also consider the problem where we replace $K_s$ and $K_t$ by arbitrary graphs $H$ and $G$ and discover several interesting new phenomena.  This is joint work with Jacques Verstraete.

As is well known, many materials freeze at low temperatures. Microscopically,
  this means that their molecules form a phase where there is long range order
  in their positions. Despite their ubiquity, proving that these freezing
  transitions occur in realistic microscopic models has been a significant
  challenge, and it remains an open problem in continuum models at positive
  temperatures. In this talk, I will focus on lattice particle models, in which
  the positions of particles are discrete, and discuss a general criterion
  under which crystallization can be proved to occur. The class of models that
  the criterion applies to are those in which there is *no sliding*, that is,
  particles are largely locked in place when the density is large. The tool
  used in the proof is Pirogov-Sinai theory and cluster expansions. I will
  present the criterion in its general formulation, and discuss some concrete
  examples. This is joint work with Qidong He and Joel L. Lebowitz.

 For a positive integer , define  to be the smallest number such that the additive energy of any subset and any  is at most . In this talk, I will survey recent results on bounds for , explore the connections with (variants of) the Hausdorff-Young inequality in analysis and with the Balog-Szemeredi-Gowers theorem in additive combinatorics, and then discuss new results on the asymptotic behavior of  as .

The uniformization theorem states that every Riemann surface is a quotient of some subset of the complex projective line by a group of Mobius transformations. However, a number of closely related questions regarding the structure of uniformization maps remain open. For example, it is unclear how one might associate a uniformizing map to a given Riemann surface. In this talk we will discuss an approach to this question due to Gunning by attaching a projective line bundle to a Riemann surface and studying its analytic properties.