Variational Estimates, Green’s Mappings and Brownian Motion

This project investigates a list of hard-analysis problems rooted in complex analysis (Greens mappings) exploiting tools from harmonic analysis, geometric measure theory and Brownian motion. Harmonic analysis is a branch of mathematics designed for investigating physical phenomena of oscillatory nature. Its applications to other disciplines of mathematics include beautiful results in additive combinatorics, probability theory, differential equations, potential theory, geometric measure theory. The recent, explosive growth of the field is due (in part) to its role played in Wavelet analysis. Brownian motion, found a wide range of applications outside mathematics, e.g., in theoretical physics, social sciences and economy. Within mathematical analysis, the Brownian motion process is well suited to the analysis of harmonic functions, and more generally to solutions of elliptic partial differential equations such as Laplacians associated to complete complex manifolds. In this project, we use the tools of martingale theory and harmonic analysis to obtain probabilistic counterparts for stochastic integrals of Bourgains variational estimates, and investigate the Anderson Conjecture for Greens mappings on Greens fundamental domains