CAREER: Random Matrices and Related Topics

随机矩阵理论主要研究不同类型随机矩阵的特征值的分布和渐近分布。随机矩阵是不同学科的科学家非常活跃的研究领域，包括数学家、统计学家和物理学家。在这个方案中，将研究Toeplitz、Hankel和马尔可夫矩阵的几个公开问题。除了在这一主题上提高数学知识，随机矩阵理论在量子物理和统计学中也有自然的应用，最近，在金融市场也发现了新的应用。与本项目的研究相结合的是教育活动。这些计划包括发展新课程以协助学生利用电脑软件理解理论、建立国际教学和研究合作关系，以及为幼年至幼年的幼童举办外展活动，以提高他们对数学的欣赏能力。这项计划的研究和教育活动将有助寻找不同学科之间的联系，包括数学、物理和统计学等。可能会产生一些新的理论。本科生和研究生通过建议的教育计划更容易地学习理论。一些K-12的学生和他们的老师将对数学和科学有更好的理解。建立国际学术交流机制。

The primary concern of Random Matrix Theory is the distributions and asymptotic distributions of eigenvalues of different kinds of random matrices. Random matrices are a very active area of research by scientists of various disciplines, including mathematicians, statisticians, and physicists. In this proposal, several open problems on Toeplitz, Hankel and Markov matrices will be studied. In addition to advancing mathematical knowledge on the subject, Random Matrix Theory has natural applications in Quantum Physics and Statistics, and recently, new applications in the financial market have also been found. Integrated with the research in this project are educational activities. They include development of new courses to help students understand theories with aids of computer software, building an international co-operation on teaching and research, and outreach activities to K-12 students to help increase their appreciation of mathematics from an early age.The research and educational activities of this project will help find connections among different disciplines of the science, including, among others, mathematics, physics and statistics. Some new theories may be possibly created. Undergraduate and graduate students learn theories with easier efforts through proposed plan of education. Some K-12 students and their teachers will get better understanding of mathematics and science. International academic exchanges will be established.