On the trace of Loewner chains

Abstract: The theory of Loewner chains developed by C. Loewner in an attempt to Bieberbach's conjecture reappeared with the seminal work of O. Schramm leading to the development of Schramm-Loewner-Evolutions. The theory gives a one to one correspondence between growing family of compact sets in a simply connected domain and continuous functions on its boundary. We will investigate further into this correspondence by studying the so called the trace of Loewner chains. This asks us to study boundary behaviour of associated conformal maps. To this end, we will study the so called reverse Loewner differential equation near the boundary and prove results about the trace in random and deterministic settings. The content of this talk is based on separate joints works one with Peter Friz and one with Yiling Wang.