Banuelos 9700585 The Principle Investigator will investigate several concrete open problems in different areas of analysis where probabilistic ideas and techniques have already had considerable success and where he believes further progress is possible. These include 1) applications of martingale inequalities to investigate the operator norm of the Beurling- Ahlfors operators in two and several dimensions, 2) the use of Brownian motions to investigate sharp bounds for the fundamental frequency and fundamental gap of the Laplacian in Euclidean domains, and 3) the use of asymptotic expansions for Wiener functionals to investigate the asymptotic expansion in the trace of Schrodinger operators and how the signs of the coefficients in such expansion depend on the sign of the potential. Probability Theory has its roots in various fields of applied sciences and hence it should not be surprising that modern stochastic analysis draws many of its technical tools from several distinct areas of mathematics. What is often far from obvious is that probabilistic ideas and techniques can be effectively used to investigate problems and applications which on the surface do not seem to be related to probability at all. The problems discussed above fall in this category. The Beurling-Ahlfors operators are (singular) integral operators which describe regularity (smoothness) properties of solutions to various nonlinear equations arising from, among other topics, elasticity. The computation of their operator norms is a fundamental problem with many applications. On the surface this operator appears to be very far from probability. The Principal Investigator, in collaboration with G. Wang and A. J. Lindeman, has successfully used the theory of martingales (fair games) to study this problem. The first part of the project describes various new probabilistic approaches for further work in this direction. Bounds on the fundamental frequency and fundament al gap of the Laplacian operator are basic quantities in the theory of vibrating membranes. The trace of Schrodinger operators plays a fundamental role in various problems in mathematical physics related to, among other things, scattering theory. Here, too, the connection to probability is not transparent. In the second part of the project the PI proposes to use the theory of Brownian motion and stochastic calculus to investigate several open problems in this areas. It is expected that these investigations will lead, as it has happened many times in the past, to new and surprising applications of probability to analysis, geometry, partial differential equations, and the application of these subjects to mathematical physics. Several students will most likely participate in these investigations.
