Mathematical Sciences: Oscillatory and Singular Integrals in Analysis, Geometry, and Physics

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Phong DMS-9505399 Phong will investigate core problems on oscillatory and singular integrals, Radon transforms, Green's functions, and analyticity of generalized functions. He will study coalescent versuq uniform estimates for oscillatory integrals, the singular integral equations for large N field theories, and the problem of analytic continuation for divergent integrals on the moduli space of Riemann surfaces. He will also investigate potential connections with the cut-off phenomenon in probability theory, molecular collisions in chemistry, string theory, and two-dimensional models of quark confinement. The mathematics underlying this research is harmonic analysis, which has as its origins the attempt to approximate an arbitrary function by a linear combination of very regular, much in the way that a sound wave is approximated by a sum of its pure tones.