Comparing root subgroup and Verblunsky coordinates
The group of homeomorphisms of a circle has a natural semigroup compactification (`increasing functions on a circle', or essentially probability measures on the circle). There are multiple ways to parameterize (smoothness classes of) this space, each with its own advantage. In this talk I will compare two parameterizations, so called root subgroup coordinates, which is tied to composition for homeomorphisms, and Verblunsky coefficients, which is tied to composition of loops in SU(1,1). In some way which remains mysterious, these two coordinates are intertwined by inversion, at least qualitatively. My own interest in these coordinates is related to the speculation that natural measures, such as Werner's measure on self-avoiding loops in the plane, may have tractable expressions in one or the other of these coordinates.
This talk is in person but we will also stream it via Zoom: https://arizona.zoom.us/j/85470607369