Large Random Matrices

P.I.将研究随机矩阵理论和概率论中的问题。研究的重点是大型随机矩阵的特征值和特征向量的统计性质。特别是，P.I.打算研究随机矩阵的Wigner、样本协方差和相关集合中最大特征值的分布。P.I.建议为足够广泛的这类随机矩阵建立局部普适性结果。除了使用矩方法外，P.I.最近还发展了预解方法，使人们能够为特征值的局部线性统计建立递归关系系统。此外，P.I.还将与他的博士生Sean O‘Rourke、Pierre Dueck和David Renfrew合作，研究大部分光谱中本征值的高斯涨落，以及随机矩阵的变形Wigner系综的光谱特性。在过去的几十年里，随机矩阵理论已经成为数学和理论物理中最令人兴奋的领域之一，应用范围从量子力学(重核高激发能级的统计特性)、理论计算机科学(计算复杂性、误差统计分析、线性数值算法)、数学金融学和生物学。为了表明随机矩阵理论技术的应用范围，可以评论说，P.I.最近的作品被各种各样的研究人员引用，如哈佛医学院从事人口生物学工作的教员和资本基金管理集团(法国)从事金融数学工作的专家。此外，随机矩阵理论与现代数学的许多领域有着深刻的联系，包括著名的关于黎曼Zeta函数零点分布的黎曼假设。

The P.I. will work on problems in Random Matrix Theory and Probability Theory. The main emphasis of the research is on statistical properties of eigenvalues and eigenvectors of large random matrices. In particular, the P.I. intends to study the distribution of the largest eigenvalues in Wigner, sample covariance, and related ensembles of random matrices. The P.I. proposes to establish local universality results for a sufficiently wide class of such random matrices. In addition to employing the method of moments, the P.I. has been recently developing the resolvent method that allows one to establish the system of recursive relations for local linear statistics of the eigenvalues. Also, the P.I. will work with his Ph.D. students Sean O'Rourke, Pierre Dueck, and David Renfrew on the Gaussian fluctuation of the eigenvalues in the bulk of the spectrum and on the spectral properties of the deformed Wigner ensembles of random matrices.Over the last few decades, Random Matrix Theory has become one of the most exciting areas of mathematics and theoretical physics with applications ranging from Quantum Mechanics (statistical properties of highly excited energy levels of heavy nuclei), Theoretical Computer Science (computational complexity, statistical analysis of errors, linear numerical algorithms), Mathematical Finance, and Biology. To indicate the breadth of applications of Random Matrix Theory techniques, it can be commented that recent works of the P.I. have been cited by such diverse groups of researchers as faculty members of Harvard Medical School working in Population Biology and experts from the Capital Fund Management group (France) working in Financial Mathematics. In addition, Random Matrix Theory has deep connections to many areas of modern Mathematics, including the famous Riemann hypothesis about the distribution of the zeros of the Riemann zeta-function.