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.

Abstracterhrhardtthis奖是根据2009年的《美国复苏与重新投资法》（公法111-5）资助的。PI计划调查操作者理论中某些具体问题，主要围绕着Toeplitz的操作员和矩阵以及其修改和概括（Toeplitz+Hankel，wienerereererecrestors，Wiener-Hopf-hankel-eerer-Hankel-elereTors）。这些对象的决定因素的渐近学尤其令人感兴趣，并且与一些仅部分解决的30年猜想有关，而解决后，这导致了新的开放问题。研究此类决定因素的渐近学的工具包括渐近光谱理论，BANACH代数技术（非共同的Gelfand理论），数值分析的某些方面（稳定性理论）以及Wiener-HOPF因素化理论。如果一个人想研究例如toeplitz矩阵的渐近特征值分布（这是一个天真的，但只是部分解决的问题），则决定因素的渐近学是兴趣。另一个应用是随机矩阵理论中的某些问题。第一种问题涉及各种随机矩阵集合的线性统计信息在某些缩放范围内。第二类问题与所谓的差距概率的渐近学有关，对于各种类型的随机矩阵而言。事实证明，线性统计数据的分布函数以及差距概率都可以根据某些操作员的决定因素来表示，其中基础操作员依赖于所考虑的矩阵集合。该研究项目的重点是使用此链接，并应用操作者理论的技术来解决随机矩阵理论中的这些问题。该提案试图连接数学的两个领域，操作员理论和随机矩阵理论。操作员理论研究某些数学对象的分析特性，并具有许多可用的硬性和强大工具。随机矩阵理论是一个领域，已经与数学的其他分支有许多连接。它的目标是，其目标是用固有的随机行为建模和解释复杂的系统。这种系统出现在统计物理学的模型中，也出现在例如无线通信中。随机矩阵理论中存在非常``基本''的问题，这些问题尚未或最近才找到他们的解决方案。事实证明，其中一些问题可以根据某些决定因素的渐近学提出。操作者理论及其复杂的工具可以被视为能够解决和解决这些问题。该提议的目的是研究源自随机矩阵理论的操作者理论中的那些问题。希望这将解决那里的一些基本问题。另一方面，随机矩阵理论的影响可能会导致新的发展，并大量扩展操作者理论中现有工具和方法，这些工具和方法与渐近性决定性问题以及光谱理论有关。