CAREER: Beyond Independence: Random Matrices and Applications

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。本项目致力于随机矩阵理论及其应用的研究。随机矩阵自然地出现在许多不同的领域，包括统计学、数据科学、计算机科学和物理学。例如，随机矩阵最初是在物理学中被引入来研究重原子的原子核。这个项目的目的是了解某些随机矩阵模型的性质，这些模型出现在几个不同的领域，包括控制论、统计遗传学和神经网络的研究。本研究为深入了解这些领域的应用打开了大门，并有可能为随机矩阵理论和相关领域的未来基础研究创造途径。该项目还包括一些整合研究和教学的教育组件。这些组成部分包括面向对高等数学感兴趣的高中生的暑期学院；参与有组织的活动，旨在让高中生和科学家在非正式的环境中通过互动演示进行会面和互动；以及本科生和研究生的指导。本项目的主要研究目标是了解具有相关项的随机矩阵的特征值和特征向量的行为。研究计划分为三个主题，它们在背景、应用和工具上都有很大的不同。第一个主题涉及随机网络和图研究中出现的矩阵的特征值和特征向量，包括同步问题和网络控制理论中出现的矩阵。受统计遗传学开放问题的启发，第二个主题涉及由依赖随机样本构建的样本协方差矩阵的光谱特性。第三个主题受到神经网络理论研究的前沿结果的启发，涉及随机矩阵乘积。这个跨学科的研究项目与学生的研究和探索项目相结合，为从高中到研究生阶段的学生设计。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。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project is dedicated to the study of Random Matrix Theory and its applications. Random matrices arise naturally in many diverse fields including statistics, data science, computer science, and physics. For example, random matrices were originally introduced in physics to study the nuclei of heavy atoms. The aim of this project is to understand the properties of certain random matrix models that arise in several diverse domains including control theory, statistical genetics, and the study of neural networks. This research opens the door to a deeper understanding of applications in these domains and has the potential to create avenues of future fundamental research in Random Matrix Theory and related fields. This project also features a number of educational components that integrate research and teaching. These components include a summer academy for high school students interested in advanced mathematics; participation in organized activities designed to allow high school students and scientists to meet and interact in an informal setting using interactive demonstrations; and undergraduate and graduate student mentoring. The overarching research goal of this project is to understand the behavior of the eigenvalues and eigenvectors of random matrices with dependent entries. The research program is divided into three themes, which are quite disparate in background, application, and tools. The first theme concerns the eigenvalues and eigenvectors of matrices arising in the study of random networks and graphs, including matrices that appear in synchronization problems and network control theory. Motivated by open questions in statistical genetics, the second theme concerns the spectral properties of sample covariance matrices constructed from dependent random samples. The third theme is inspired by cutting-edge results in the theoretical study of neural networks and involves random matrix products. This interdisciplinary research program is integrated with student research and exploration projects, designed for students ranging from the high school to graduate level.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.