Noncommutative functional analysis, operator algebras and operator spaces

非交换泛函分析、算子代数和算子空间

• 批准号: 0800674
• 负责人: David Blecher
• 金额: $ 7.97万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800674&HistoricalAwards=false
• 关键词: Noncommutative; functional; analysis; operator; algebras

This research project is directed on fundamental problems of the theory of random matrices and random polynomials and their applications, and on related problems in statistical physics. The cornerstone of the problems is different conjectures of universality, which state that as the size of a random matrix (or the degree of a random polynomial) approaches infinity, the correlations between properly scaled eigenvalues (or zeros) approach a universal limit. In the current project the PI continues his studies of the universality in random matrix models, random polynomials, and statistical physics. This includes: (i) The Riemann-Hilbert (RH) approach to double scaling limits in random matrix models. (ii) RH approach to random matrices with external source. (iii) Semiclassical asymptotics and RH approach to multi-matrix models. (iv) RH approach to the six-vertex model of statistical physics. (v) Scaling limits and universality in non-Gaussian ensembles of random polynomials and random algebraic varieties.The project has an interdisciplinary character and it lies on the frontier between physics and mathematics. The problems of scaling and universality are central in many areas of modern science: theory of critical phenomena and phase transitions, statistical physics and quantum field theory, theory of quantum chaos, nonlinear dynamics, etc. This project is directed on development of powerful mathematical methods to the problems of scaling and universality in the theory of random matrices, random polynomials, and related topics. It involves different areas of mathematics: analysis, theory of integrable systems, probability theory, semiclassical asymptotics for systems of differential equations, complex analysis, etc. The research project under consideration has direct applications to various physical problems: combinatorial asymptotics related to quantum gravity, exactly solvable models of statistical physics, spin systems on random surfaces, theory of critical phenomena and phase transitions, quantum chaos. Possible further applications include theory of knots and links and related problems in molecular biology, growth models, statistical data analysis, and others.

该研究项目针对随机矩阵和随机多项式及其应用理论的基本问题，以及统计物理学中的相关问题。问题的基石是普遍性的不同猜想，它表明，作为随机矩阵的大小（或随机多项式的程度）接近无穷大，而正确缩放的特征值​​（或零）之间的相关性接近了普遍的极限。 在当前项目中，PI继续他对随机矩阵模型，随机多项式和统计物理学的研究继续研究。这包括：（i）Riemann-Hilbert（RH）在随机矩阵模型中进行双缩放限制的方法。 （ii）与外部源的随机矩阵的RH方法。 （iii）多矩阵模型的半经典渐近学和RH方法。 （iv）统计物理学六个vertex模型的RH方法。 （v）随机多项式和随机代数品种的非高斯集合中的缩放限制和普遍性。该项目具有跨学科的特征，它位于物理和数学之间的边界。在现代科学的许多领域中，缩放和普遍性的问题是核心：批判现象和相变理论，统计物理学和量子场理论，量子混乱，非线性动力学理论等。该项目针对强大的数学方法开发，以实现随机矩阵，随机矩阵，随机矩阵，随机型，多种元素和相关典型的理论。它涉及数学的不同领域：分析，可集成系统的理论，概率理论，用于微分方程系统的半经典渐近学，复杂的分析等。所考虑的研究项目直接适用于各种物理问题：与量子重力相关的组合渐近剂，与统计物理系统的可溶模型，统计型系统的可溶模型，量子置换量和阶段的跨度，理论，阶段，阶段和阶段。可能的进一步应用包括分子生物学，增长模型，统计数据分析等中的结和联系理论以及相关问题。