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：高维平坦空间上的布朗运动和酉群上的布朗运动，后者给出了空间的随机连续旋转的模型。这样的过程将与另一类随机对象一起研究：大型随机矩阵的谱理论。随机矩阵理论是一个最近发展起来的研究领域，在过去的二十年里引起了人们的极大关注。它完美地结合了许多数学领域，并具有重要的外部应用，例如对多元统计和蜂窝通信网络的应用。使用随机分析的现代技术，首席研究员托德·坎普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、不断增长的领域，它融合了概率论、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随机矩阵模型的涨落和变形提供新的见解，并旨在给出与这类系统中的信息和信息有关的重要公开问题的更完整的解。本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，在几何学、应用数学和其他领域都很重要。布朗运动被用来模拟科学和工程中的许多过程：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。随机分析是关于布朗运动行为的系统理论。除了它的研究价值，这个数学研究项目的另一个同样核心的影响是通过它的教育目标，专注于创建一个新的暑期研究项目，CURE：协作本科生研究体验。它的目的是为主要是当地本科生的群体提供研究经验。CURE计划旨在通过情境学习和合作参与一线研究，将新手数学家引入实践社区，基于本提案研究部分更容易获得和计算的方面。它还将为研究生提供指导经验，他们将帮助协调研究工作；因此，CURE计划垂直整合到所有学术研究领域。数学已经成为一个日益合作的领域；CURE计划将向数十名参与者灌输研究合作的价值。通过促进多样性和发展年轻数学家的才华，这个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地位

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