Multilinear inequalities: Combinatorial and geometric aspects, and extremization

数学研究旨在发现适用于任何可能的现象的普遍数学定律，这些现象可以通过数学精确描述或建模。基本物理定律通常通过系统最小化或最大化（即极值化）某个量的原理来表达。例如，光线沿着最小化通过具有不同折射率的介质的传播时间的路径。系统最小化各种动作或能量。其他的物理定律都被编码在偏微分方程中。数学分析提供了工具，可以用来理解这两个微分方程，并极值化的许多类型的泛函。该项目关注的是开发用于分析函数和集合的技术，这些函数和集合使与经典（非相对论）物理空间的基本代数和几何结构密切相关的不等式极端化或接近极端化。近似极值的表征提供了对精确极值的更稳健的理解，其考虑了小的缺陷和扰动。在要研究的不等式是一些最基本和最广泛使用的数学分析不等式，包括不等式管理的傅立叶变换，卷积，并在欧几里得空间的总和。这些不等式中的每一个都是多线性的，涉及产品或其他成对的相互作用，或者具有潜在的多线性方面。多线性泛函位于线性和完全非线性现象之间的前沿。其他多线性不等式也将作为本项目的一部分进行研究，包括傅立叶变换的变体。我们将研究勒贝格空间中的基本多线性不等式和相关的范数。原型包括Riesz-Sobolev不等式（关于集合对称化的积分），Brunn-Minkowski不等式（关于集合和），Hausdorff-Young不等式（关于傅立叶变换）和Young不等式（关于卷积）。 在该项目的主要部分，注意力将集中在性质和数量属性的函数和集，几乎，但不完全，极端化这样的不平等。这将建立更强大的版本现有的精确极值的特征。建立极端化序列的紧凑性的方法，添加剂组合的考虑，而不是浓度的基础上，将开发。这些工具包括小的和集集的集，具有适度大，但控制大小的和集的集，和具有大的附加能量的集的特征。精细的不平等将被制定和建立，纳入第二项测量结构，而不是大小。 算术级数、区间、椭球、凸集和高斯函数将为这种测量提供背景。提高对仿射不变多线性不等式中近似群结构和近似极值之间相互作用的理解是一个主要目标。首席研究员还将离散多线性不等式应用于计算机科学。 严格的下限之间的通信内存层次结构的元素将被证明，并计算这些界限的方法将被设计。 其他的研究问题进行调查，包括反问题的非对角行为的伯格曼内核相关的高权力的积极复杂的线丛，不等式的多线性振荡积分算子有关的Hausdorff-杨不等式，并可能表征平等的一类多线性不等式以前分析的保持器，罗杰斯，Riesz，Sobolev，和Brascamp-Lieb-Luttinger。

Mathematical research aims to discover universal mathematical laws that apply to any possible phenomenon that can be accurately described or modeled through mathematics. Fundamental physical laws are often expressed through the principle that a system minimizes or maximizes (that is, extremizes) some quantity. Light rays, for instance, follow paths that minimize travel time through media with varying indices of refraction. Systems minimize various actions or energies. Other physical laws are encoded in partial differential equations. Mathematical analysis provides tools that can be used to understand both such differential equations, and the extremization of many types of functionals. This project is concerned with development of techniques for the analysis of functions and sets that extremize, or nearly extremize, inequalities that are closely tied to the underlying algebraic and geometric structures of classical (nonrelativistic) physical space. Characterization of near-extremizers provides a more robust understanding of exact extremizers, which takes into account small imperfections and perturbations. Among the inequalities to be studied are some of the most fundamental and widely used inequalities of mathematical analysis, including inequalities governing the Fourier transform, convolutions, and sums of sets in Euclidean space. Each of these inequalities is multilinear, involving products or other pairwise interactions, or has underlying multilinear aspects. Multilinear functionals lie on the frontier between linear and fully nonlinear phenomena. Other multilinear inequalities will also be investigated as part of this project, including variants of the Fourier transform. Fundamental multilinear inequalities in Lebesgue space and related norms will be investigated. Prototypes include inequalities of Riesz-Sobolev (concerning integrals over symmetrizations of sets), Brunn-Minkowski (concerning sums of sets), Hausdorff-Young (concerning the Fourier transform) and Young (concerning convolutions). In a major part of the project, attention will focus on the nature and quantitative properties of functions and sets that nearly, but not exactly, extremize such inequalities. This will establish more robust versions of existing characterizations of exact extremizers. Methodology for establishing compactness for extremizing sequences, based on additive combinatorial considerations rather than on concentration, will be developed. These tools include characterizations of sets with small sumsets, of sets with sumsets of moderately large but controlled size, and of sets with large additive energies. Refined inequalities will be formulated and established, incorporating second terms measuring structure, rather than size. Arithmetic progressions, intervals, ellipsoids, convex sets, and Gaussian functions will provide the context for such measurement. An improved understanding of the interplay between approximate group structure and near extremality in affine-invariant multilinear inequalities is a primary goal. The principal investigator will also apply discrete multilinear inequalities to computer science. Rigorous lower bounds for communication between elements of memory hierarchies will be proved, and methods for computing these bounds will be devised. Other research problems to be investigated include an inverse problem concerning the off-diagonal behavior of Bergman kernels associated to high powers of positive complex line bundles,inequalities for multilinear oscillatory integral operators related to the Hausdorff-Young inequality,and a possible characterization of equality in a class of multilinear inequalities previously analyzed by Holder, Rogers, Riesz, Sobolev, and Brascamp-Lieb-Luttinger.