CAREER: An integrated probabilistic approach to discrete and continuous extremal problems via information theory

University Of Delaware, Newark DE

Investigators

Abstract

Mathematics abounds with extremal problems problems where the goal is to minimize some functional applied to a class of objects under some constraint, identify the extremal objects, and investigate the stability of extrema. Relevant examples range from some in the continuous world (isoperimetric phenomena in convex geometry, functional analytic inequalities), to some in the discrete world (structural phenomena in additive combinatorics), and some in both (maximum entropy problems in statistics, limit theorems in probability). A natural language for all of these problem classes is probability, and, although not obvious, information theory. The project will develop new formulations of extremal problems from each of these fields in terms of information-theoretic inequalities, and then use a variety of tools from analysis, probability, convex geometry, combinatorics, and information theory, to make progress on them. The unifying nature of the perspective adopted will bridge discrete and continuous problems using a common set of tools, and enable significant cross-fertilization. Furthermore, some of the information-theoretic inequalities developed, combined with statistical decision theory, will be applied to novel statistical challenges involving multiple players that arise in engineering, economics, and biology (specifically, theoretical foundations for the problems of data pricing and distributed inference). The project will use information-theoretic thinking to make advances on challenging mathematical problems from the three seemingly disparate fields of convex geometry, arithmetic combinatorics, and probability. Apart from the intrinsic significance of these areas within mathematics, they have much practical significance - convex geometry finds applications in medical tomography, arithmetic combinatorics in computer science, and probability is ubiquitous as the foundation of statistical inference. The interpretability and unifying nature of the proposed research, and the diversity of tools it uses, create wonderful opportunities for student motivation. Newly developed courses and a resource website on information theoretic approaches to extremal problems will exploit these opportunities. The investigator will disseminate key findings through survey articles, organize an interdisciplinary workshop, and communicate the excitement of research through non-academic public lectures to attract promising students to the mathematical sciences. The applied component of the research would also have broad impact, by contributing to how data collectors and vendors come up with pricing mechanisms (e.g., for pricing of advertisements by search engines), and by improving the way networks of sensors collect and use data for various applications (e.g., for disaster recovery coordination or smart kindergartens).