Georgia Institute of Technology College of Sciences

Pseudoholomorphic curves are pivotal in establishing uniqueness and finiteness results in the classification of symplectic manifolds. In a series of works, Wendl used punctured pseudoholomorphic foliations to classify symplectic fillings of contact three-manifolds supported by planar open books, turning it into a problem about monodromy factorizations. In a joint work with Hyunki Min and Agniva Roy, we build on the works of Lisi--Van Horn-Morris--Wendl in using spinal open books to further delve into the classification problem of symplectic fillings of higher genus open books. In particular, we provide the local model of the mysterious "exotic fibers" in a generalized version of Lefschetz fibrations, which captures a natural type of singularity at infinity. We will give some applications to classifying symplectic fillings via this new phenomenon.

In this talk I will present both mathematical and numerical analysis as well as experiments to study a few basic computational issues in using neural network to approximate functions: (1) the stability and accuracy, (2) the learning dynamics and computation cost, and (3) structured and balanced approximation. These issues are investigated for both approximation and optimization in asymptotic and non-asymptotic regimes.

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.