Inequalities, Symmetry, Extremality, and Multilinear Interactions

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

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.