Random Surfaces and Related Questions

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This research concerns the study of random surfaces, random trees and random curves. These objects are relevant to many different fields of mathematics (such as combinatorics, complex analysis, graph theory, and random matrix theory) and to many different areas of physics (such as string theory, gauge theory, particle physics and two-dimensional statistical mechanics). The PI will help mentor and support both undergraduate and graduate students, as well as postdoctoral researchers.The specific objects under study include Schramm-Loewner evolution, continuum random trees, and several random surface models, including Liouville quantum gravity, the Brownian map, the peanosphere, and Liouville conformal field theory. These random surface models are all in some sense equivalent, but proving and understanding their equivalence has been a major undertaking within both mathematics and physics, and many open problems remain. The specific goals include a better understanding of growth models and quantum Loewner evolution, a better understanding of the relationship between surface sums and gauge theory, a better understanding of high genus surfaces, and a more detailed understanding of the associated lattice models and random planar maps. Overall, the goal is to provide a stronger mathematical understanding of some of the most fundamental models for physical phenomena.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.