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

职业：自由概率以及与随机矩阵、随机分析和偏微分方程的联系

• 批准号: 1254807
• 负责人: Todd Kemp
• 金额: $ 55万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1254807&HistoricalAwards=false
• 关键词: CAREER; Free; Probability; Connections; Random

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

该数学研究项目将重点介绍两种布朗动作的随机分析：在高维平面上和单一组上，后者为随机连续旋转空间提供了模型。 此类过程将与另一类随机对象结合研究：大型随机矩阵的光谱理论。 随机矩阵理论是一个最近开发的研究领域，在过去的二十年中引起了广泛关注。 它完美地结合了许多数学领域，并具有重要的外部应用程序，例如多变量统计和蜂窝通信网络。 使用随机分析的当代技术，主要研究者托德·肯普（Todd Kemp）将探索大型随机矩阵的行为，该矩阵被认为是混凝土无限二维对象。该研究使用了自由概率的工具：一个健壮的成长领域，结合了概率理论，复杂分析，操作员代数和组合学的思想。 待解决的问题将为自由概率的随机矩阵模型的波动和变形提供新的见解，并旨在为与此类系统中的熵和信息有关的重要开放问题产生更完整的解决方案。此数学研究项目位于自由概率理论和布朗尼运动的一般领域。布朗运动的概念对于分析至关重要，在几何，应用数学及其他方面很重要。 布朗运动用于对整个科学和工程的许多过程进行建模：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。 除了其研究价值外，该数学研究项目的另一个同样核心的影响是通过其教育目标，重点是创建新的夏季研究计划《 CURE：协作本科生研究经验》。 它旨在为主要是本地本科生的小组提供研究经验。 CURE计划旨在基于本提案的研究组成部分的更易于访问和计算方面，将新手数学家通过位置学习和合作参与介绍给实践社区。 它还将为将协调研究工作的研究生提供指导经验；因此，在整个学术研究中垂直整合了CURE计划。 数学已成为一个越来越多的协作领域。 CURE计划将在数十名参与者中灌输研究合作的价值。 通过促进多样性并发展年轻数学家的才能，该项目将增加自由概率，随机矩阵和随机分析的概况