Conformal Invariance and Restriction in Multiply Connected Domains and Riemann Surfaces

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The P.I.'s research programme is to study conformally invariant measures first in multiply connected domains and then on Riemann surfaces. In particular: Identify diffusions on an appropriate moduli space that give rise to conformally invariant measures on curves satisfying the restriction property; Measuring the restriction defect for other diffusions, and calculating or characterize intersection probabilities as functions of the moduli; Link the measure and its change under perturbations of the conformal structure to highest weight representations; Study conformally invariant quantities for domino tilings and the Gaussian free field in multiply connected domains. Self-avoiding planar curves arise in many natural contexts. For example, if a very shallow plate is filled with water and oil, we can consider the curve (or curves) separating the two liquids. Another example are coastlines, the borders of countries on maps, or rivers seen from an airplane. Typically, these curves are very irregular. But what exactly is ``typical,'' and how can the irregularity be quantified? Since all coastlines are different, it does not make sense to look for one self-avoiding curve as the mathematical model of a coastline. On the other hand, it does make sense to look for a probability distribution on the set of all self-avoiding curves ( the ``possible'' coastlines) so that if we sample from the set of these curves according to this distribution, then the statistics are in agreement with statistics of real coastlines. The P.I. proposes to study these probability distributions in the case when the geometry is complicated by holes (think of pebbles in the shallow plate with water and oil) or the curves are situated on a curved surface such as a sphere (the globe), a donut, or a pretzel.