CAREER: Beyond Independence: Random Matrices and Applications

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。本课题致力于随机矩阵理论及其应用的研究。随机矩阵自然而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。例如，随机矩阵最初被引入物理学以研究重原子核。这个项目的目的是了解在控制理论、统计遗传学和神经网络研究等几个不同领域中出现的某些随机矩阵模型的性质。这项研究为更深入地了解这些领域的应用打开了大门，并有可能为未来随机矩阵理论和相关领域的基础研究开辟道路。该项目还包括一些将研究和教学相结合的教育组成部分。这些组成部分包括为对高等数学感兴趣的高中生开设暑期学校；参加旨在让高中生和科学家在非正式环境中通过互动演示见面和互动的有组织的活动；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依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，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的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。第二个主题受到统计遗传学中的公开问题的启发，涉及由相依随机样本构造的样本协方差矩阵的谱性质。第三个主题的灵感来自于神经网络理论研究的前沿成果，涉及随机矩阵乘积。这一跨学科研究计划与学生研究和探索项目相结合，为从高中到研究生水平的学生设计。该奖项反映了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。

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.