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

Phong DMS-9505399 Phong将研究振荡积分和奇异积分、Radon变换、格林函数和广义函数的解析性等核心问题。他将研究振荡积分的联合一致估计，大N场论的奇异积分方程组，以及黎曼曲面的模空间上发散积分的解析延拓问题。他还将研究与概率论中的截断现象、化学中的分子碰撞、弦理论和夸克限制的二维模型之间的潜在联系。这项研究背后的数学基础是调和分析，它的起源是试图通过非常规则的线性组合来逼近任意函数，就像声波通过其纯音的和来逼近一样。

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.