Peter Lin, University of Washington

PDL C-36

Abstract: Extremal length is a powerful, yet easy-to-estimate, conformal invariant. In the first part of the talk, we show how the definition of this invariant immediately leads to concrete applications such as estimating the hitting probabilities of brownian motion or random walks. We do not assume any knowledge of conformal maps or complex analysis.

In the second part of the talk we focus on the application of this technique to the problem of conformal welding. The conformal welding problem asks: given an equivalence relation on the boundary of a (possibly disconnected) riemann surface, how can we put a conformal structure on the resulting quotient space that is compatible with the existing structure? This construction naturally arises in places such as complex dynamics, random geometry, and teichmuller theory.