Inequalities, Symmetry, Extremality, and Multilinear Interactions

数学分析为数学的其他部分（线性和非线性偏微分方程，数值分析，数论）提供了定量的工具。这些工具中最基本和最广泛应用的是傅立叶变换，它实现了量子物理学的波粒二象性，以及卷积的概念。 该项目汇集了五个主题。 (i)傅里叶变换和卷积的定量性质由各种数学不等式表示。(ii)傅里叶变换是某些对称性的表现，反过来又受对称性的支配。 (iii)非线性偏微分方程在自相互作用物理系统的数学描述中是普遍存在的，它常常涉及到多线性泛函。(iv)多线性泛函通常由组合问题，也就是复杂的计数问题所支配。(v)自然法则通常被表达为最小化或最大化原则。(For例如，广义相对论假设光沿着沿着的路径传播，在某种意义上，这些路径是时空点之间的最短路线。） 该项目侧重于数学分析中的一系列主题，这些主题相互作用。该项目的目标是发现和严格建立新的不等式，以及现有不等式的改进。该项目的目标是深入了解线性和多线性泛函，特别是那些表现欧几里德或其他阿贝尔群结构的泛函。 原型包括Riesz-Sobolev不等式、Brunn-Minkowski不等式、Young不等式和Hausdorff-Young不等式。 先验上界与非显式常数，和显式的最佳常数的不等式，将被调查。 特别注意将集中在性质和数量属性的功能和集，最大限度地提高这些功能，或几乎这样做。精细的不等式将开发，纳入额外的条款测量结构，而不是大小;算术级数，间隔，椭球和凸集是基本的结构对象。 涉及熵和相关量的扩展将被调查。 在变形下泛函的单调性将被利用，如果可用的话。 其中特别关注的焦点是Holder-Brascamp-Lieb（HBL）多线性不等式的稳定性，放松对称假设下Rogers-Brascamp-Lieb-Luttinger泛函的变体，HBL不等式的Schatten范数和混合范数版本，这个奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的知识优点和更广泛的影响审查标准。

Mathematical analysis provides quantitative tools that underpin other parts of mathematics (linear and nonlinear partial differential equations, numerical analysis, number theory). Among the most fundamental and widely applicable of these tools are the Fourier transform, which implements the wave-particle duality of quantum physics, and the concept of convolution. This project brings together five themes. (i) The quantitative natures of the Fourier transform and of convolution are expressed by various mathematical inequalities. (ii) The Fourier transform is a manifestation of certain symmetries, and in turn is governed by symmetries. (iii) Nonlinear partial differential equations, which are ubiquitous in mathematical description of self-interacting physical systems, often involve multilinear functionals. (iv) Multilinear functionals are often governed by combinatorial issues, that is, by sophisticated counting problems. (v) Natural laws are often expressed as minimization or maximization principles. (For instance, general relativity posits that light travels along paths that are, in a sense, shortest routes between spacetime points.) The project focuses on a circle of topics in mathematical analysis in which these themes interact. The goal is to discover and rigorously establish new inequalities, and refinements of existing inequalities.The goal of the project is insight into linear and multilinear functionals, especially those manifesting Euclidean or other Abelian group structure. Prototypes include the inequalities of Riesz-Sobolev, Brunn-Minkowski, Young, and Hausdorff-Young. Both a priori upper bounds with nonexplicit constants, and inequalities with explicit optimal constants, will be investigated. Special attention will focus on the nature and quantitative properties of functions and sets that maximize such functionals, or that nearly do so. Refined inequalities will be developed, incorporating extra terms measuring structure, rather than size; arithmetic progressions, intervals, ellipsoids, and convex sets are fundamental structural objects. Extensions involving entropy and related quantities will be investigated. Monotonicity of functionals under deformations will be exploited, where available. Among the specific foci of attention are stability of Holder-Brascamp-Lieb (HBL) multilinear inequalities, variants of the Rogers-Brascamp-Lieb-Luttinger functionals under relaxed symmetry hypotheses, Schatten norms and mixed-norm versions of HBL inequalities, and multilinear inequalities for compact and discrete Abelian groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.