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

这个数学研究项目将集中在两种布朗运动的随机分析：高维平坦空间和酉群，后者给出了空间随机连续旋转的模型。 这样的过程将研究与另一类随机对象：大随机矩阵的谱理论。 随机矩阵理论是最近发展起来的一个研究领域，在过去的二十年里受到了广泛的关注。 它完美地结合了许多数学领域，并具有重要的外部应用，例如多元统计和蜂窝通信网络。 利用当代的随机分析技术， 首席研究员托德肯普将探讨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：一个强大的，不断发展的领域，它结合了概率论，复分析，算子代数和组合学的思想。 这些问题将使我们对自由概率中随机矩阵模型的涨落和变形有新的认识，并旨在为这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。布朗运动的概念是分析的核心，在几何、应用数学等领域也很重要。 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的大规模行为，再到量子系统的核心行为。 随机分析是研究布朗运动行为的系统理论。 除了它的研究价值，另一个同样重要的影响，这个数学研究项目是通过其教育目标，专注于创建一个新的夏季研究计划，治愈：合作本科研究经验。 它旨在为主要是本地本科生的群体提供研究经验。 CURE计划旨在将新手数学家引入实践社区，通过情境学习和合作参与一线研究，基于本提案研究部分的更容易获得和计算方面。 它还将为研究生提供指导经验，帮助协调研究工作;因此，CURE计划在学术研究的全方位垂直整合。 数学已经成为一个越来越多的合作领域; 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概况