In this talk, I will give an overview of the Schramm-Loewner Evolutions (SLE) theory and present some new results on this topic. In the first part, I will introduce the basic definitions and make an overview of some of the important results of the theory. In the second part, I aim to cover new results on SLE theory based on the analysis of some Stochastic Differential Equations (SDEs) that appear naturally in this context.
The first one appears when extending the conformal maps to the boundary and can be thought of as a singular Rough Differential Equation (RDE), as in Rough Path Theory. In the study of RDEs, questions such as continuity of the solutions, the uniqueness/non-uniqueness of solutions depending on the behavior of parameters of the equation, appear naturally. We adapt these type of questions to the study of the backward Loewner differential equation in the upper half-plane, and the conformal welding homeomorphism. This view will allow us to obtain some new structural information about the SLE traces in the regime where they have double points.
The second one appears when we study a coordinate change of the Loewner equation in which we obtain via a random time change, a stochastic dynamics on a specific line in the upper half-plane, that is depending only on the argument of the points. I will show how in this framework one obtains some closed-form expressions that exhibit a change in behaviour exactly at the critical values of a natural parameter in the SLE theory.
Prior joining NYU Shanghai as a Fellow of Mathematics, Vlad did his Bachelor's Degree in Mathematics in Romania at the Faculty of Mathematics of the University of Bucharest, followed by his Master's Degree in Mathematics at ETH Zurich, and his PhD in Mathematics under the supervision of Prof. Dmitry Belyaev and Prof. Terry Lyons at the University of Oxford.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai