CAREER: Free Probability and Connections to Random Matrices, Stochastic Analysis, and PDEs

这个数学研究项目将集中于两种布朗运动的随机分析：高维平面空间和酉群，后者给出了空间随机连续旋转的模型。这类过程将与另一类随机对象一起研究：大随机矩阵的谱理论。随机矩阵理论是近二十年来发展起来的一个新兴研究领域。它完美地结合了许多数学领域，并具有重要的外部应用，例如多元统计和蜂窝通信网络。利用当代随机分析技术，首席研究员Todd Kemp将探索大型随机矩阵的行为，实现为具体的无限维对象。这项研究使用了自由概率的工具：这是一个强大的、不断发展的领域，融合了概率论、复分析、算子代数和组合学的思想。所要解决的问题将为自由概率中的随机矩阵模型的波动和变形提供新的见解，并旨在为此类系统中与熵和信息有关的重要开放问题提供更完整的解决方案。这个数学研究项目是在自由概率论和布朗运动的一般领域。布朗运动的概念是分析的核心，在几何、应用数学等领域都很重要。布朗运动被用来为科学和工程中的许多过程建模：从股票价格的波动到计算机或生物网络中排队系统的大规模行为，再到量子系统的核心行为。随机分析是布朗运动行为的系统理论。除了它的研究价值之外，这个数学研究项目的另一个同样重要的影响是通过它的教育目标，重点是创建一个新的夏季研究项目，CURE：合作本科生研究经验。它旨在为主要由本地本科生组成的群体提供研究经验。CURE计划旨在通过在一线研究中的情境学习和合作参与，将新手数学家引入实践社区，该计划基于本提案的研究组成部分的更易于获取和计算方面。它还将为研究生提供指导经验，以协助协调研究工作；因此，CURE项目垂直整合了整个学术研究领域。数学已经成为一个协作性越来越强的领域；CURE项目将向数十名参与者灌输研究合作的价值。通过促进多样性和培养年轻数学家的才能，该项目将为下一代研究人员增加自由概率、随机矩阵和随机分析的形象

This mathematics research project will focus on the stochastic analysis of two kinds of Brownian motions: on high-dimensional flat spaces, and on unitary groups, the latter giving a model for random continuous rotations of space. Such processes will be studied in conjunction with another class of stochastic objects: the spectral theory of large random matrices. Random matrix theory is a recently developed research area that has garnered much attention in the last two decades. It beautifully combines many fields of mathematics and has important outside applications, for example to multivariate statistics and cellular communication networks. Using contemporary techniques from stochastic analysis, the principal investigator Todd Kemp will explore the behavior of large random matrices, realized as concrete infinite-dimensional objects. The research uses the tools of free probability: a robust, growing field which incorporates ideas from probability theory, complex analysis, operator algebras, and combinatorics. The problems to be addressed will give new insight into the fluctuations and deformations of random matrix models in free probability and are designed to yield more complete solutions to important open problems relating to entropy and information in such systems.This mathematics research project is in the general area of free probability theory and Brownian motions. The notion of Brownian motion is central to analysis, and is important in geometry, applied mathematics, and beyond. Brownian motion is used to model many processes throughout science and engineering: from the fluctuations of stock prices to the large-scale behavior of queueing systems in computer or biological networks, to the core behavior of quantum systems. Stochastic analysis is the systematic theory of the behavior of Brownian motion. Besides its research value, another, equally central impact of this mathematics research project is through its educational goals, focusing on the creation of a new summer research program, CURE: Collaborative Undergraduate Research Experience. It is designed to give research experience to groups of primarily local undergraduate students. The CURE program seeks to introduce novice mathematicians to the community of practice, through situated learning and cooperative engagement in front-line research, based on the more accessible and computational aspects of the research component of this proposal. It will also provide mentoring experience to graduate students who will assist in coordinating the research effort; thus the CURE program is vertically integrated across the full spectrum of academic research. Mathematics has become an increasingly collaborative field; the CURE program will instill the value of research collaboration in its dozens of participants. By promoting diversity and developing the talent of young mathematicians, this project will increase the profile of free probability, random matrices, and stochastic analysis for the next generation of researchers