Inverse Problems and Their Applications

该项目集中在两个相关的研究领域：加性组合中的逆问题和随机矩阵理论中的奇点问题。就反问题而言，PI将重点关注littlewood - offford反问题的各种设置。他的主要目标之一是清楚地了解为什么独立随机变量的多线性形式高度集中在短间隔上。对于奇异性问题，他提出研究一些相关项的随机矩阵的最小奇异值，如对称矩阵和双随机矩阵。在数学中许多重要的计数问题中，理解底层结构已被证明是基本的。由于我们的littlewood - offford逆问题将随机游走的集中与等差数列等附加结构联系起来，预计对这些问题的系统研究将产生许多复杂的应用。我们项目的一个关键部分是将这些应用之一发展到随机矩阵理论中，为广泛的相关项随机矩阵建立圆律和椭圆律。这些定律支持物理学家和概率论家多年前观察到的著名的普世性现象：关于随机矩阵特征值分布的许多事实似乎是普世性的，它们不依赖于条目的分布。除了研究之外，PI还将开发涵盖组合学和概率论重要主题的本科和研究生课程。这些课程中的一些主题将与本项目中的问题相关。为了鼓励学生学习数学，PI将继续撰写研究论文，组织研讨会，并从研究水平到介绍性的广泛讲座。

The project concentrates on two related areas of research: inverse problems in additive combinatorics and singularity problem in random matrix theory. As far as the inverse problems are concerned, the PI will be focusing on various settings of the inverse Littlewood-Offord problems. One of his main goals is to obtain a clear picture as to why a multilinear form of independent random variables is highly concentrated on a short interval. For the singularity problem, he proposes to study the least singular value for a number of random matrices of correlated entries such as symmetric matrices and doubly stochastic matrices. In numerous important counting problems in mathematics, understanding the underlying structure has been shown to be fundamental. As our Littlewood-Offord inverse problems connect the concentration of random walks to additive structures such as arithmetic progressions, it is expected that a systematic study of these problems would yield many sophisticated applications. A key part of our project is to develop one of these applications into random matrix theory, establishing the circular law and elliptic law for a broad range of random matrices of correlated entries. These laws support the well-known universality phenomenon observed by physicists and probabilists many years ago: many facts about the distribution of eigenvalues of random matrices seem to be universal, they do not depend on the distribution of the entries.In addition to research, the PI will develop undergraduate and graduate level courses which cover important topics in combinatorics and probability. Some of the topics in these courses will be related to the problems in this project. The PI will continue to write research papers, organize seminars, and give a wide range of talks, from research level to introductory one in order to encourage students into studying mathematics.