Geometric, Optimizational and Spectral Problems in Large Random Structures

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The research supported by this award focuses on large random structures that have strong motivation from a number of physical disciplines. The PI attempts to improve understanding on a range of fundamental models and phenomena in mathematical physics and computer science, including the insulator-conductor transition, the limit of computing power in solving realistic problems, and some two-dimensional random surface models of importance to conformal field theory. In addition, the PI is actively collaborating on application-oriented projects, and one of the specific goals is to help consolidating theoretical foundations for some statistical learning procedures. Finally, the PI intends to provide research opportunities for graduate students and postdoctoral researchers in probability theory.The PI will study limiting behavior for stochastic models with emphasis on geometric, optimizational and spectral aspects as well as interactions among them. The stochastic processes under consideration cover a wide range of spectrum and are of fundamental importance in respective research communities. For instance, random geometry of Gaussian free field has been a phenomenal research topic in two-dimensional probability, and the closely related Liouville quantum gravity surfaces can be thought of as the canonical models of random two-dimensional Riemannian manifolds; random constraint satisfaction problems have strong connections with spin glass theory and complexity theory, and it is of major interest to understand the connections between the phase transitions of the solution spaces and the algorithmic transitions; random field model has been a classic example in understanding how presence of disorder affects the behavior of statistical physics models; Anderson localization for random Schroedinger operators is motivated by understanding the conductor-insulator transition. In order to make progress on these problems, the PI will attempt to bring in new insights from physics, develop new mathematical tools and identify new connections among different mathematical branches.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.