Problems at the Interface of Analysis with Geometry and Physics

Proposal: DMS-9800783 Principal Investigator: Duong Phong Abstract: The influence of mathematical analysis has increased considerably over the last few decades. However, the field is at an important new juncture, where analytical methods have to be blended with techniques and insights from other fields, most particularly geometry, probability, and theoretical physics. This is a recurrent feature in the proposal. More specifically, the proposal addresses issues of bounds and stability for oscillatory integrals, regularity of Radon transforms and degenerate Fourier integral operators, asymptotic behavior of Green's functions, exact solutions of supersymmetric field and string theories, Hamiltonian theory of solitons and non-linear WKB methods. These are arguably among the most pressing problems in mathematics and theoretical physics today. Oscillatory integrals are, for example, a well-known feature of any physical scattering process, and progress along the lines suggested here can reduce greatly the extensive effort presently required for their numerical studies. On the other hand, the problems in the proposal concerning supersymmetric theories reflect very recent advances, indicating a deep and surprising correspondence between three seemingly different fields - namely, supersymmetric gauge theories, soliton equations, and Riemann surfaces. The origin of this correspondence is still mysterious, and it may well be essential to on-going attempts at understanding the laws of nature at their most fundamental level.