Liouville Quantum Gravity and Its Applications

This project will study Liouville quantum gravity (LQG), a theory of random surfaces. These random surfaces arise in various areas of theoretical physics. In bosonic string theory, they can be used to model the surfaces swept out by a string moving through space. They can also be viewed as models of gravity in two dimensions. LQG surfaces have many interesting and surprising geometric properties and are connected to a variety of other mathematical objects, including various types of random curves, random permutations, and random graphs. This project will address some of the most important open problems in the subject. The awardee will also engage in outreach, mentorship, and dissemination activities, including mentoring undergraduate and graduate participation in the research, writing expository articles, organizing conferences, and giving talks and mini-courses at conferences, seminars, and summer schools. One goal of the project is to build a better understanding of distances between points in LQG surfaces. Another goal is to understand LQG surfaces in the so-called supercritical phase (corresponding to central charge between 1 and 25), which is much less well understood than the subcritical phase, but which is expected to be more interesting from a string theory perspective. The awardee also plans to study the extent to which one can make sense of the solutions to partial differential equations on LQG surfaces, and to further explore the applications of LQG to combinatorial objects, including planar maps, permutations, and meanders. 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.