Georgia Institute of Technology College of Sciences

Upon quantization, hyperbolic Hamiltonian systems generically exhibit universal spectral properties effectively described by Random Matrix Theory. Semiclassically this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to single-particle systems. I will discuss an extension of this program to hyperbolic coupled map lattices with a large number of sites (i.e., particles). The crucial ingredient is a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such a symbolic dynamics can be constructed explicitly.

The knot concordance group consists of knots in the three-sphere modulo the equivalence relation of smooth concordance. We will discuss two concordance invariants coming from knot Floer homology: tau and epsilon.

Lately there was a growing interest in studying self-similarity and fractal properties of graphs, which is largely inspired by applications in biology, sociology and chemistry. Such studies often employ statistical physics methods that borrow some ideas from graph theory and general topology, but are not intended to approach the problems under consideration in a rigorous mathematical way. To the best of our knowledge, a rigorous combinatorial theory that defines and studies graph-theoretical analogues of topological fractals still has not been developed. In this paper we introduce and study discrete analogues of Lebesgue and Hausdorff dimensions for graphs. It turned out that they are closely related to well-known graph characteristics such as rank dimension and Prague (or Nesetril-Rodl) dimension. It allowed us to formally define fractal graphs and establish fractality of some graph classes. We show, how Hausdorff dimension of graphs is related to their Kolmogorov complexity. We also demonstrate fruitfulness of this interdisciplinary approach by discover a novel property of general compact metric spaces using ideas from hypergraphs theory and by proving an estimation for Prague dimension of almost all graphs using methods from algorithmic information theory.