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
