Asymptotic Geometric Analysis, Random Matrices, and Applications

该项目集中于渐近几何分析(AGA)的几个相关方向。这个领域涉及有限维对象的几何和线性性质，例如凸集和赋范空间，特别是当维度或一些其他相关的自由参数适当大或趋于无穷大时出现的特征行为。高维系统在数学和应用科学中非常常见，因此，理解高维现象变得越来越重要。在过去的十年中，随着新的强大的技术的发展，AGA有了巨大的增长，主要是概率性质的。由于AGA的一般框架、方法及其对相关领域的影响，AGA可以位于许多数学分支的“十字路口”：泛函分析、凸几何和离散几何，以及几个概率领域。AGA中的许多现象都与随机矩阵的奇异值行为密切相关。随机矩阵奇异值的分布问题在纯数学、应用数学、统计学、计算机科学、电子工程等领域有着重要的应用。经典随机矩阵理论对相应的极限分布的研究由来已久。与此形成鲜明对比的是，我们的兴趣集中在非限制性制度上。我们考虑高维随机矩阵，并寻求以压倒性概率成立的最大奇异值和最小奇异值的渐近锐界。该项目将为AGA的几个方向带来重大贡献。这将导致在快速发展的随机矩阵的前沿渐近无极限理论中发展新的认识、新的技术和新的结果。它还将导致随机图的邻接矩阵理论的发展，以及解决AGA其他方向的公开问题。该项目还将用于培养研究生和博士后研究员。

The project concentrates on several related directions of Asymptotic Geometric Analysis (AGA). This field is concerned with geometric and linear properties of finite dimensional objects, such as convex sets and normed spaces, especially with the characteristic behavior that emerges when the dimension, or a number of other relevant free parameters, is suitably large or tends to infinity. High-dimensional systems are very frequent in mathematics and applied sciences, hence, understanding high-dimensional phenomena is becoming increasingly important. The last decade has seen a tremendous growth of AGA, with the development of new powerful techniques, mainly of probabilistic nature. By virtue of AGA's general framework, methods, and its impact on related fields, AGA can be situated at the "crossroads" of many branches of mathematics: functional analysis, convex and discrete geometry, and several areas of probability. Many phenomena in AGA are closely related to the behavior of singular values of random matrices. Questions on distributions of singular values of random matrices are of major importance due to many applications in pure and applied mathematics, statistics, computer sciences, electrical engineering, among others. Classical random matrix theory extensively studied corresponding limiting distributions already for a long time. In sharp contrast, our interest concentrates on the non-limiting regime. We consider a high dimensional random matrix and seek asymptotically sharp bounds for the largest and smallest singular values which hold with an overwhelming probability. This project will bring significant contributions to several directions of AGA. It will lead to development of new understanding, new techniques, and new results in the fast growing cutting edge asymptotic non-limiting theory of random matrices. It will also lead to the development of the theory of adjacency matrices of random graphs as well as to solving open problems in other directions of AGA. The project will also serve to train graduate students and postdoctoral fellows.