The Riemann mapping theorem states that any two simple connected proper domains in the complex plane are conformally isomorphic, i.e. there exists an angle preserving bijection between them. In general, there is no explicit description of the conformal isomorphism in terms of the conformal domains. Despite this, Loewner developed a way to connect the dynamics of a family continually evolving simply connected domains to the dynamics of their conformal isomorphisms to the the upper half plane. We will do a brief review/introduction on conformal maps before diving deeper into Loewner evolution. Time permitting, we will discuss Schramm's groundbreaking idea connecting Loewner evolution to statistical mechanics.
Conformal maps, the Riemann mapping theorem, and (Schramm--)Loewner evolution