GT ADG-Systèmes DynamiquesThe Loewner equation and Hölder geometry
Kyle Kinneberg (UCLA)
Tuesday 18 March 2014 14:00 - Orléans - Salle de Séminaire
The Loewner equation provides a one-to-one correspondence between continuous functions on $\R$ and a certain class of shrinking domains in the upper half-plane. Originally introduced to study coefficient problems for univalent maps, the Loewner equation has received renewed interest for its use in probability: driving the equation with (rescaled) Brownian motion produces conformally invariant processes (SLE) that describe scaling limits of several discrete planar processes. An important question, in both the deterministic and stochastic settings, is the relationship between analytic properties of the real-valued driving term and the geometric properties of the domain it generates. In this talk, we will review some of the work that has been done on this question, most of which investigates geometric properties that are induced by certain classes of driving terms. Then, we will discuss some recent work on the converse question: which geometric properties give good analytic control on the driving term? Here, we are motivated by the fact that SLE domains are, almost surely, Hölder domains.