Mathematical Sciences: Explorations in Brownian Motion, Intrinsic Ultracontractivity, Martingales and Lancunary Series

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Banuelos The investigator will study several problems which lie in the intersection of probability and various areas of analysis. Specifically, subject areas include extremal problems which are motivated from the study of Brownian motion in simply connected planar domains; a geometric characterization ofr intrinsic ultracontractivity for certain types of simply connected planar domains; martingale inequalities and a conjecture of T. Iwaniec for the Beurling-Ahlfors operator; and the zero Khintchin inequality for lacunary trigonometric series. This award will provide funding to further study the interrelationship between mathematical analysis and probability theory. Several problems will be investigated dealing with geometry, operators, series and Brownian motion. This should lead to a better understanding of these areas of mathematics, especially as they relate to one another.