Geometric Inequalities in Fourier Analysis

本研究项目的目的是对流形上的几何分析有一个广泛的概念性的理解，它把流形上的傅里叶分析中的尖锐不等式作为编码流形的内在不变量、对称性和基本几何结构的几何信息的作用。我们的直接重点是在李群、对称空间、复流形和仿射几何（包括双曲空间、SL（2））的背景下，得到关于等周不等式、临界Sobolev嵌入、log Sobolev和Levy-Gromov泛函以及多线性分数次积分的新的和更清晰的结果，这些结果包含了Hardy-Littlewood-Sobolev不等式、Stein-Weiss积分、Birman-Schwinger核和Riesz势的框架。R）和Heisenberg群.精确嵌入估计是建立偏微分方程解的存在性和正则性以及控制流形上振荡行为的重要工具.该计划反映了一个自然的相互作用和刺激与问题的共形变形，流体动力学，几何概率，统计力学，弦理论和湍流。计算方法是更清楚地理解几何对称性复杂性的必要工具，描述物理现象的数学模型和描述动力学过程的微分方程分析需要丰富多样的分析框架.科学家们说“数学定律是我们宇宙结构的基础。“这项研究计划的目的是开发一个支持的数学结构，帮助我们理解连接不确定性，熵和无序，几何对称性，重整化和标度参数，时空模型的非均匀性，表面积和体积之间的等周比较，以及双曲几何的精细细节的关系。我们希望获得广泛而严格的理解数学工具，支持定量描述的基本规律的性质，并开发和探索数学结构，模型的物理现象，从流体动力学到弦理论，特别是通过使用模式的对称性和固有常数，编码几何信息的基础理论框架。

AbstractThe objective of this research program is to develop a broad conceptualunderstanding of geometric analysis on manifolds which takes as itsparadigm the role of sharp inequalities in Fourier analysis on themanifold as encoding geometric information about intrinsic invariants,symmetries and the underlying geometric structure of the manifold. Our direct focus is to obtain new and sharper results on isoperimetricinequalities, critical Sobolev embedding, log Sobolev and Levy-Gromovfunctionals, and multilinear fractional integrals that incorporate theframework of the Hardy-Littlewood-Sobolev inequality, Stein-Weissintegrals, Birman-Schwinger kernels and Riesz potentials, especially inthe setting of Lie groups, symmetric spaces, complex manifolds and affinegeometry including hyperbolic space, SL(2,R) and the Heisenberg group.Sharp embedding estimates are a critical tool to establish existence andregularity for solutions to pde's, and to control oscillatory behavior on a manifold. This program reflects a natural interplay and stimulus withproblems in conformal deformation, fluid dynamics, geometric probability,statistical mechanics, string theory and turbulence. Computational methodsare essential tools to obtain a clearer understanding of the complexity ofthe geometric symmetry.Mathematical models that characterize physical phenomena and the analysisof differential equations that describe dynamical processes require a richand diverse analytic framework. Scientists say that "mathematical lawsunderpin the fabric of our universe." The aim of this research program is to develop a supporting mathematical structure that helps us to understand relations that connect uncertainty, entropy and disorder, geometric symmetry, renormalization and scaling arguments, non-homogeneity of space-time models, isoperimetric comparison between surface area and volume, and the fine detail of hyperbolic geometry. We want to obtain broad and rigorous understanding ofmathematical tools that support a quantitative description of the fundamentallaws of nature, and to develop and explore mathematical structures that modelphysical phenomena ranging from fluid dynamics to string theory, especiallyby using patterns of symmetry and intrinsic constants that encode geometricinformation about the underlying theoretical framework.