Geometric Inequalities in Fourier Analysis

这个研究项目的目的是发展对流形上几何分析的广泛的概念理解，以流形上的傅立叶分析中的尖锐不等式作为编码关于流形的内在不变量、对称性和基本几何结构的几何信息的模式。我们的直接目标是得到关于等周不等式、临界Sobolev嵌入、Log Sobolev和Levy-Gromov泛函以及包含Hardy-Littlewood-Sobolev不等式、Stein-Weissals积分、Bman-Schwinger核和Riesz势框架的多线性分数次积分的新的和更尖锐的结果，特别是在李群、对称空间、复流形和仿射几何包括双曲空间、SL(2，R)和Heisenberg群的背景下。尖锐嵌入估计是建立Pde解的存在性和正则性以及控制流形上的振荡行为的关键工具。该程序反映了与保形变形、流体动力学、几何概率、统计力学、弦理论和湍流等问题的自然相互作用和刺激。计算方法是更清楚地了解几何对称的复杂性的基本工具。描述物理现象的数学模型和描述动力学过程的微分方程的分析需要丰富多样的分析框架。科学家们说，“数学定律支撑着我们宇宙的结构。”这个研究计划的目的是建立一个支持性的数学结构，帮助我们理解联系不确定性、熵和无序、几何对称性、重整化和标度变元、时空模型的非均质性、表面积和体积之间的等周比较以及双曲几何的精细细节之间的关系。我们希望获得对数学工具的广泛和严格的理解，这些工具支持对基本自然定律的定量描述，并开发和探索模拟从流体动力学到弦理论的各种物理现象的数学结构，特别是通过使用对称模式和内部常量来编码关于基本理论框架的几何信息。

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.