Anti-Concentration, Random Matrices, and Random Functions

该项目的目标是研究有关离散随机变量的分布问题，如随机多项式，随机和，随机矩阵。这些问题与数学的其他部分有许多联系，以及在统计和机器学习中的应用。PI的目标是继续发展反集中理论。这一理论在数学的各个领域都有应用，包括概率论、数值线性代数和复杂性理论。他还将调查一个古老的和自然的问题，Erdos和Moser关于和避免集在非阿贝尔设置，继他最近的工作与T。关于阿贝尔的案子。在提案的另一部分，PI建议研究有关随机矩阵和随机函数的几个问题。这些问题中有许多是非常基本和容易陈述的，但长期以来一直是数学界的主要挑战，最近才取得了进展。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The goal of the project is to study problems concerning distribution of discrete random variables, such as random polynomials, random sums, and random matrices. These problems have many connections to other parts of mathematics, as well as applications in statistics and machine learning. The PI aims to continue his development of a theory of anti-concentration. This theory has found applications in various areas of mathematics, including probability, numercial linear algebra, and complexity theory. He will also investigate an old and natural problem of Erdos and Moser concerning sum-avoiding sets in the non-abelian setting, following his recent work with T. Tao concerning the abelian case. In another part of the proposal, the PI proposes to study several problems concerning random matrices and random functions. Many of these problems are very basic and easy to state, but have been major challenges to the mathematics community for a long time, and progress on them has been made very recently.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.