Mathematical Sciences: Geometric Inequalities in Fourier Analysis

这项研究将集中在几何方法的数学问题的调和分析。流形的几何和对称结构决定了自然算子的先验解析不等式，这些算子刻画了流形上函数空间的分析。其中包括傅里叶变换、格林函数和拉普拉斯-贝尔特拉米算子。这项工作试图通过研究李群和黎曼流形固有的微分算子、奇异积分和变分问题来发展和理解关于李群和黎曼流形的分析。主要方向是(1)重整化变元在控制振荡现象中的作用和(2)高阶多线性张量结构微分算子的几何分析。这项研究的中心是流形的几何结构和利用共形不变性和几何对称化的算子的调和分析之间的相互作用。变分问题的尖锐常数提供了丰富的几何和概率信息来源。从共形弦理论和统计力学中的精确模型计算中，利用临界现象的共形对称性，可以洞察到数学框架。计算分析，特别是计算和计算机图形学，被用来为几何结构过于复杂而无法进行初等分析的问题提供直观和策略。

This research will focus on a geometric approach to mathematical problems of harmonic analysis. The geometry and symmetry structure of a manifold determine a priori analytic inequalities for the natural operators that characterize analysis on function spaces over the manifold. These include the Fourier transform, Green's functions and the Laplace-Beltrami operator. The work seeks to develop and understand analysis on Lie groups and Riemannian manifolds through the study of differential operators, singular integrals and variational problems intrinsic to the manifold. Principal directions are (1) the role of renormalization arguments in controlling the behavior of oscillatory phenomena and (2) geometric analysis of higher-order multilinear differential operators with tensor structure. Central to this research is the interplay between geometric structure of manifolds and harmonic analysis of the operators using conformal invariance and geometric symmetrization. Sharp constants for variational problems provide a rich source of geometric and probabilistic information. Insight into the mathematical framework is gained from exact model calculations in conformal string theory and statistical mechanics using the conformal symmetry of critical phenomena. Computation analysis, especially computation and computer graphics, is used to provide intuition and strategy for problems where the geometric structure is too complex for elementary analysis.