Non-Asymptotic Random Matrix Theory and Geometric Functional Analysis

这项研究旨在为概率分析和泛函分析这两个数学领域提供新的联系。调查的主要对象之一是随机矩阵，即随机数据的大型矩形阵列。PI努力理解这样的阵列的高概率保持的属性，以及这些属性对随机条目的性质和矩阵结构的依赖。这项研究将具有超越纯数学领域的潜在应用，因为随机矩阵被用于统计学、计算机算法和无线通信。研究的另一个方向是高维凸集的研究，重点是其复杂性和逼近性。这项研究还将应用于计算机科学，包括各种高维算法的速率估计。本研究的主要方向之一是随机矩阵的非渐近理论，这是一个新的快速发展的研究领域，分析大但固定大小的随机矩阵的谱特性，努力以高概率获得有效的界。PI旨在研究大型随机矩阵的不同集合的奇异值、特征值和特征向量。在这个方向上得到的结果将在随机矩阵理论中在证明随机矩阵的谱特征的极限定律方面有重要的应用。另一组问题来自几何泛函分析，这是一个研究高维凸体和赋范空间的数学领域。在这个方向上的进展将导致更好地理解这类物体的截面和投影的结构，以及用具有某些良好性质的物体来近似一般凸体的可能性。这两个方向都有一个与理论计算机科学相关的重要组成部分。

The research is intended to provide new connections between two areas of mathematics, probability and functional analysis. One of the main objects of investigation is a random matrix, a large rectangular array of random data. The PI strives to understand the properties of such arrays which hold with high probability and the dependence of those properties on the nature of random entries and the structure of the matrix. This study will have potential applications beyond the realm of pure mathematics, as random matrices are used in statistics, computer algorithms, and wireless communication. Another direction of the proposed research is the study of high dimensional convex sets with the emphasis on their complexity and approximation. This research will also have computer science applications including rate estimates for various high-dimensional algorithms.One of the main directions of this research is the non-asymptotic theory of random matrices, a new and rapidly developing area of research analyzing spectral characteristics of a random matrix of a large but fixed size and striving to obtain bounds valid with high probability. The PI intends to study singular values, eigenvalues, and eigenvectors of different ensembles of random matrices of a large size. The results obtained in this direction would have important applications within the random matrix theory in proving limit laws for the spectral characteristics of random matrices. Another group of problems comes from geometric functional analysis, an area of mathematics concerned with the study of high-dimensional convex bodies and normed spaces. A progress in this direction would lead to better understanding of the structure of sections and projections of such bodies, as well as possibility of approximation of a general convex body by a body with certain nice properties. Both directions have a significant component related to the theoretical computer science.