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.
