Inverse Problems and Their Applications

该项目集中在两个相关的研究领域：加法组合学中的反问题和随机矩阵理论中的奇异性问题。就逆问题而言，PI将专注于逆Littlewood-Offord问题的各种设置。他的主要目标之一是获得一个清晰的画面，为什么一个多线性形式的独立随机变量是高度集中在一个短的时间间隔。对于奇异性问题，他建议研究一些相关项的随机矩阵（如对称矩阵和双随机矩阵）的最小奇异值。在数学中的许多重要计数问题中，理解底层结构已被证明是基本的。由于我们的Littlewood-Offord逆问题将随机游动的浓度与算术级数等加法结构联系起来，因此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，建立相关条目的广泛随机矩阵的圆形定律和椭圆定律。这些定律支持了物理学家和概率学家多年前观察到的著名的普适性现象：关于随机矩阵特征值分布的许多事实似乎是普适的，它们不依赖于条目的分布。除了研究，PI将开发本科和研究生水平的课程，涵盖组合学和概率的重要主题。这些课程中的一些主题将与本项目中的问题有关。PI将继续撰写研究论文，组织研讨会，并进行广泛的演讲，从研究水平到入门级，以鼓励学生学习数学。

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.