Anti-Concentration, Random Matrices, and Random Functions

反集中、随机矩阵和随机函数

• 批准号: 1902825
• 负责人: Van Vu
• 金额: $ 22.5万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-05-15 至 2022-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902825&HistoricalAwards=false
• 关键词: Anti; Concentration; Random; Matrices; Functions

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.

该项目的目标是研究离散随机变量的分布问题，如随机多项式、随机和、随机矩阵等。这些问题与数学的其他部分以及统计学和机器学习中的应用有许多联系。PI的目标是继续他的反集中理论的发展。这个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。他还将研究Erdos和Moser关于非阿贝尔情况下避和集的一个古老而自然的问题，继他最近与T. Tao关于阿贝尔情况的工作之后。在提案的另一部分，PI提出研究几个关于随机矩阵和随机函数的问题。这些问题中的许多都是非常基本和容易表述的，但长期以来一直是数学界面临的主要挑战，最近才取得进展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Random matrix products: Universality and least singular values

随机矩阵乘积：普适性和最小奇异值

• DOI: 10.1214/19-aop1396
• 发表时间: 2020
• 期刊: The Annals of Probability
• 作者: Kopel, Phil;O’Rourke, Sean;Vu, Van
• 通讯作者: Vu, Van

Roots of random functions: A framework for local universality

随机函数的根：局部普遍性的框架

• DOI: 10.1353/ajm.2022.0000
• 发表时间: 2022
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: Nguyen, Oanh;Vu, Van
• 通讯作者: Vu, Van

Central limit theorems for the real zeros of Weyl polynomials

• DOI: 10.1353/ajm.2020.0034
• 发表时间: 2017-07
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: Yen Q. Do;V. Vu

Random polynomials: Central limit theorems for the real roots

• DOI: 10.1215/00127094-2020-0089
• 发表时间: 2019-04
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Oanh Nguyen;V. Vu

Recent progress in combinatorial random matrix theory

组合随机矩阵理论的最新进展

• DOI: 10.1214/20-ps346
• 发表时间: 2021
• 期刊: Probability Surveys
• 影响因子: 1.6
• 作者: Vu, Van H.