Harmonic Analysis on the Hamming Cube

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Many complex structures that arise in nature are perceived globally as continuous objects, although at the microscopic level they emerge from certain laws that appear to be discrete. What the relation is between the discrete and continuous "worlds", how one impacts the other, and what laws they obey are the fundamental questions that the PI intends to investigate. This project focuses on the Hamming cube, the simplest discrete object that consists of binary strings of given length and that can encode any complex information. As the complexity of the information increases, the lengths of the encoded strings increase exponentially. The goal of the project is to develop mathematics on the Hamming cube that would allow us to encode the structures of the continuous world into the discrete world, and vice versa. The new techniques developed could have surprising applications in complexity theory as well as classical and quantum algorithms. Many of these problems are accessible to students and this project aims to continue the PI's work with undergraduate and graduate students in this area. This project will consider a series of problems in the Gauss space and, most importantly, in its discrete counterpart, the Hamming cube. Unlike the classical case of the unit circle, many fundamental questions of Fourier analytic type are still open for the Hamming cube due to its unique discrete geometric structure. The first and main direction of the proposal is to resolve Weissler's conjecture, an old open problem in complex hypercontractivity theory on the Boolean cube that gives necessary and sufficient conditions for boundedness of the Hermite operator with complex time. Weissler's conjecture has important implications in several areas of mathematics including combinatorics, computer science, probability, isoperimetry, and approximation theory. The new techniques will be based on developing "two-point" inequalities. The second direction of this research is to develop methods of harmonic analysis to find good estimates on the norms of various linear operators acting on a class of functions on the Hamming cube whose Fourier spectra belong to given prescribed sets. The basic examples include Bernstein-Markov type inequalities and their reverse forms for functions on the Hamming cube that live on the low and high frequencies respectively. The third direction is to understand the universality phenomena of the Gaussian measure. In particular, the goal is to investigate uniqueness of the functional Ehrhard inequality, which is the sharp analog of Brunn-Minkowski inequality for the Gaussian measure. The analysis will be based on Monge-Ampere type partial differential equations and semigroup methods. The fourth direction is to develop the duality between martingale inequalities related to sharp dyadic square function estimates and the problems of isoperimetric type; that is, gradient estimates for functions on the Hamming cube. Our methods will use heat envelopes, "four-point" inequalities, and "inf-sup" Legendre transform. The fifth direction aims to obtain sharp forms of the classical triangle inequalities in the Lp spaces using the theory of developable surfaces and minimal concave functions.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.