CAREER: Non-Asymptotic Random Matrix Theory and Connections

随机矩阵在数学、科学和工程中无处不在。例如，它们可以为数学和统计物理中的物理系统建模，统计学中的样本协方差矩阵，数值分析中的基准，信号处理中的算法工具；它们还在几何泛函分析中的概率方法中发挥作用。该项目旨在开发随机矩阵理论的新方法，以期解决广泛领域的基本问题，包括经典和组合随机矩阵理论、数值分析、马尔可夫链蒙特卡罗、统计物理、偏差理论和精确算法。该项目的教育部分包括为所有级别的学生提供培训、研究和指导机会。一个关键要素将是针对研究生和职业生涯早期数学家的跨学科讲习班，重点是参与者的专业发展和促进研究合作。该研究计划大致由三个部分组成。第一部分是关于随机矩阵的极端奇异值的非渐近研究，其动机是基本问题，包括离散随机矩阵的奇异性问题和矩阵的Spencer猜想。第二部分涉及非渐近随机矩阵理论在统计和统计力学中几个经典马尔可夫链的混合时间的应用。第三部分重点研究反集中现象及其在组合学和计算机科学中的应用。这些方面的进展预计将导致新的概率和组合技术的发展。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Random matrices are ubiquitous throughout mathematics, science, and engineering. For instance, they can model physical systems in mathematical and statistical physics, sample covariance matrices in statistics, benchmarks in numerical analysis, algorithmic tools in signal processing; they also play a role in the probabilistic method in geometric functional analysis. This project aims to develop novel methods in random matrix theory with a view towards resolving fundamental problems in a wide range of areas, including classical and combinatorial random matrix theory, numerical analysis, Markov chain Monte Carlo, statistical physics, discrepancy theory, and exact algorithms. The educational component of this project includes training, research, and mentoring opportunities for students at all levels. A key element will be interdisciplinary workshops targeted at graduate students and early-career mathematicians, with a focus on professional development of participants and fostering research collaborations. The research program is broadly composed of three components. The first component concerns the non-asymptotic study of the extreme singular values of random matrices, motivated by fundamental problems including the singularity problem for discrete random matrices and the matrix Spencer conjecture. The second component involves the application of non-asymptotic random matrix theory to the mixing time of several classical Markov chains in statistics and statistical mechanics. The third component focuses on the study of the anti-concentration phenomenon and its applications to combinatorics and computer science. Progress in each of these components is anticipated to lead to the development of new probabilistic and combinatorial techniques.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.