Disordered systems, dense graphs, normal approximation and applications

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It is proposed to study several problems in probability.One class of problems concerns the structure of high-dimensional Gaussian fields, with applications to the study of disordered systems, including spin glasses and polymers. A second class of problems involves dense random graphs, including unsolved questions about large deviations of subgraph counts and the properties of graphs with a given degree sequence. Finally, a third class of problems centers around a continuation of the proposer's earlier work of normal approximation in modern problems. Sensitivity to small perturbations in physical systems is broadly known as chaos. The proposer wishes to study the phenomenon of chaos in disordered systems, which are simplified physical models of materials that exhibit so-called `glassy' properties. The key focus of this proposal is on certain kinds of magnetic materials known as spin glasses. These have been studied by physicists and mathematicians for nearly forty years, but various aspects of their behavior are still shrouded in mystery, one of them being chaos. The proposer believes that he has new mathematical tools to understand this elusive phenomenon, which he has already successfully applied to certain polymer models in the past.