Random curves and surfaces with conformal symmetries

This research concerns the study of random conformal geometry, particularly random surfaces and random curves. These objects are of major interest for the modelling of physical phenomena and they provide firm mathematical ground for fundamental theories of physics. They arise in a wide variety of settings and applications due to their natural and intrinsic formulation. The PI will mentor students at all levels and early-career researchers, will organize seminars and workshops suitable for junior researchers, she will give talks aimed at a broad audience including junior researchers and non-probabilists. Two particular objects of interest are the random fractal curves known as Schramm-Loewner evolutions (SLE) and the random fractal surfaces known as Liouville quantum gravity (LQG) surfaces. The discovery of SLE in 1999 revolutionized the understanding of critical phenomena in statistical mechanics, while LQG, whose study goes back to the physics literature in the 80s, has intimate relationships with SLE and discrete surfaces known as planar maps. The PI will study fundamental properties of SLE and LQG, will define and study generalizations of them which are of major interest for applications and will prove scaling limit results. 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.