Topics in Operator Theory with Applications to Random Matrices

算子理论及其在随机矩阵中的应用主题

• 批准号: 0901434
• 负责人: Torsten Ehrhardt
• 金额: $ 16.74万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901434&HistoricalAwards=false
• 关键词: Topics; Operator; Theory; Applications; Random

AbstractErhrhardtThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI plans to investigate certain concrete problems in Operator Theory, mostly centered around Toeplitz operators and matrices as well as their modifications and generalizations (Toeplitz+Hankel, Wiener-Hopf-Hankel operators). The asymptotics of the determinants of these objects is of particular interest and is related to some 30-year conjectures, which have only been partially solved, and when solved have led to new open problems. The tools for studying the asymptotics of such determinants include asymptotic spectral theory, Banach algebra techniques (non-commutative Gelfand theory), certain aspects of Numerical Analysis (Stability Theory), and also Wiener-Hopf factorization theory. The asymptotics of determinants is interest if one wants to study the asymptotic eigenvalue distribution of, for instance, Toeplitz matrices (which is one naive, but only partially solved problem). Another application are certain problems in Random Matrix Theory. There a first type of problem concerns the linear statistics for various random matrix ensembles in certain scaling limits. A second class of problem is related to the asymptotics of so-called gap probablities, again for various types of random matrices. It has turned out that both the distribution function for the linear statistics as well as the gap probabilities can be expressed in terms of the determinants of certain operators, where the underlying operator depends on the matrix ensemble under consideration. The focus of this research project is to use this link and apply the techniques from operator theory to solve those problems in Random Matrix Theory. The proposal tries to connect two areas of Mathematics, Operator Theory and Random Matrix Theory. Operator Theory studies the analytical properties of certain mathematical objects and has a lot of hard and powerful tools available. Random Matrix Theory is a field which has already many connections to other branches of Mathematics. It has also an applied aspect in that its goal is to model and explain complex systems with an inherent random behavior. Such systems arise in models of Statistical Physics, but also in, for instance, Wireless Communication. There are very ``basic'' problems in Random Matrix Theory, which have not yet or only recently found their solution. It has turned out that some of these questions can be formulated in terms of the asymptotics of certain determinants. Operator Theory along with its sophisticated tools can be considered as being capable of tackling and solving these questions. It is the goal of the proposal to study in particular those problems in Operator Theory that arise from Random Matrix Theory. This will hopefully solve some of the basic questions there. On the other, the influence from Random Matrix Theory will likely lead to the development of new and the considerable extension of existing tools and methods in Operator Theory which are connected to asymptotic determinant problems as well as spectral theory.

本奖项由2009年美国复苏与再投资法案（公法111-5）资助。PI计划研究算子理论中的某些具体问题，主要集中在Toeplitz算子和矩阵以及它们的修改和推广（Toeplitz+Hankel， Wiener-Hopf-Hankel算子）。这些对象的行列式的渐近性是特别有趣的，并且与一些30年来的猜想有关，这些猜想只得到部分解决，当解决时又导致了新的开放问题。研究这些行列式的渐近性的工具包括渐近谱理论、巴拿赫代数技术（非交换Gelfand理论）、数值分析的某些方面（稳定性理论）以及Wiener-Hopf分解理论。行列式的渐近性对于研究例如Toeplitz矩阵的渐近特征值分布（这是一个朴素的，但只是部分解决的问题）是很有意义的。另一个应用是随机矩阵理论中的某些问题。第一类问题是关于各种随机矩阵集合在一定尺度极限下的线性统计量。第二类问题与所谓的间隙概率的渐近性有关，同样适用于各种类型的随机矩阵。结果表明，线性统计量的分布函数和间隙概率都可以用某些算子的行列式来表示，其中底层算子取决于所考虑的矩阵集合。本课题的研究重点是利用这一环节，运用算子理论中的技术来解决随机矩阵理论中的这些问题。该建议试图将数学的两个领域，算子理论和随机矩阵理论联系起来。算子理论研究某些数学对象的解析性质，并且有许多强大的工具可用。随机矩阵理论是一个与其他数学分支有许多联系的领域。它也有一个应用方面，因为它的目标是建模和解释具有固有随机行为的复杂系统。这样的系统出现在统计物理模型中，但也出现在，例如，无线通信中。随机矩阵理论中有一些非常“基本”的问题，这些问题还没有或只是最近才找到解决方案。事实证明，这些问题中的一些可以用某些行列式的渐近性来表述。算符理论及其复杂的工具可以被认为有能力处理和解决这些问题。本文的目标是研究算子理论中由随机矩阵理论引起的问题。希望这能解决一些基本问题。另一方面，随机矩阵理论的影响可能会导致算子理论中与渐近行列式问题以及谱理论相关的新工具和方法的发展和相当大的扩展。