Inequalities, Symmetry, Extremality, and Multilinear Interactions

数学分析提供了支撑数学其他部分（线性和非线性偏微分方程、数值分析、数论）的定量工具。这些工具中最基本和最广泛应用的是傅里叶变换，它实现了量子物理中的波粒二象性，以及卷积的概念。这个项目有五个主题。(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。非线性偏微分方程在自相互作用物理系统的数学描述中普遍存在，通常涉及多线性泛函。多线性泛函通常是由组合问题，即复杂的计数问题支配的。自然法则通常表示为最小化或最大化原则。（例如，广义相对论假定光沿着在某种意义上是时空点之间最短的路径传播。）该项目侧重于数学分析中的一个主题圈，其中这些主题相互作用。其目标是发现并严格建立新的不平等，并改进现有的不平等。该项目的目标是深入了解线性和多线性泛函，特别是那些表现出欧几里德或其他阿贝尔群结构的泛函。原型包括Riesz-Sobolev、Brunn-Minkowski、Young和Hausdorff-Young的不等式。我们将研究具有非显式常数的先验上界和具有显式最优常数的不等式。特别注意将集中在函数和集的性质和数量性质，使这些泛函最大化，或接近这样做。精细化的不等式将得到发展，纳入额外的术语来衡量结构，而不是规模；等差数列、区间、椭球体和凸集是基本的结构对象。涉及熵和相关量的扩展将被研究。在可能的情况下，将利用变形下泛函的单调性。其中特别关注的焦点是Holder-Brascamp-Lieb （HBL）多线性不等式的稳定性，松弛对称假设下Rogers-Brascamp-Lieb-Luttinger泛函的变体，HBL不等式的Schatten范数和混合范数版本，以及紧和离散阿别群的多线性不等式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.