Harmonic Analysis on the Hamming Cube

自然界中出现的许多复杂结构在全球范围内被视为连续的物体，尽管在微观层面上，它们来自某些似乎是离散的定律。离散的和连续的“世界”之间是什么关系，一个世界如何影响另一个世界，以及它们遵循什么规律，是PI打算调查的根本问题。这个项目的重点是汉明立方体，这是一种最简单的离散对象，由给定长度的二进制字符串组成，可以编码任何复杂的信息。随着信息复杂性的增加，编码字符串的长度呈指数增长。该项目的目标是在汉明立方体上开发数学，允许我们将连续世界的结构编码到离散世界中，反之亦然。开发的新技术可能会在复杂性理论以及经典和量子算法中有令人惊讶的应用。其中许多问题是学生可以接触到的，该项目旨在继续PI在这一领域与本科生和研究生的工作。这个项目将考虑高斯空间中的一系列问题，最重要的是，在其离散的对应空间中，汉明立方体。与单位圆的经典情形不同，由于海明立方独特的离散几何结构，许多傅立叶解析型的基本问题仍然是悬而未决的。第一个也是主要的方向是解决Weissler猜想。Weissler猜想是布尔立方体上复超收缩理论中的一个老问题，它给出了复时间Hermite算子有界的充要条件。魏斯勒猜想在组合数学、计算机科学、概率论、等周学和近似论等数学领域都有重要的意义。新技术将建立在发展“两点”不平等的基础上。这项研究的第二个方向是发展调和分析方法，以求对作用于汉明立方上的一类函数的各种线性算子的范数进行良好的估计，该函数的傅立叶谱属于给定的规定集。基本的例子包括分别存在于低频和高频上的汉明立方上的函数的Bernstein-Markov型不等式及其逆形式。第三个方向是理解高斯测量的普适性现象。特别地，我们的目的是研究泛函Ehrhard不等式的唯一性，它是关于高斯测度的Brunn-Minkowski不等式的尖锐类比。分析将基于Monge-Ampere型偏微分方程组和半群方法。第四个方向是发展与尖锐并矢方函数估计有关的鞅不等式与等周型问题之间的对偶性，即Hamming立方体上函数的梯度估计。我们的方法将使用热封、“四点”不等式和“inf-sup”勒让德变换。第五个方向旨在利用可展曲面和极小凹函数理论获得LP空间中经典三角不等式的尖锐形式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.