Speaker: Vlad Margarint (University of Colorado)

Title: SLE theory, Rough Paths and beyond

Abstract: In this talk, I will give an overview of some work carried out at the intersection of Rough Path Theory and Schramm-Loewner Evolutions (SLE) Theory. Specifically, I will cover a study of the Loewner Differential Equation using Rough Path techniques (and beyond). The Loewner Differential Equation describes the evolution of a family of conformal maps. We rephrase this in terms of Singular Rough Differential Equations. In this context, it is natural to study questions on the stability, and approximations of solutions of this equation. First, I will give an introduction to these two theories, and then I will present a result on the continuity of the dynamics and related objects in a natural parameter that appears in the problem. The first approach will be based on Rough Path Theory, and a second approach will be based on a constructive method of independent interest: the square-root interpolation of the Brownian driver of the Loewner Differential Equation. This is based on joint work with Dmitry Belyaev and Terry Lyons. If time permits, I will also touch on some further applications to the simulations of the SLE traces, that is based on joint works with James Foster and Jiaming Chen.