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.
have important applications in very diverse areas of science such as physics, evolutionary biology, statistics, and theoretical computer science. For example, the eigenvalues of large random matrices are believed to model the behavior of the energy levels of heavy nuclei. Furthermore, there are remarkable connections between the distribution of the eigenvalues of large random matrices and the distribution of the zeros of Riemann zeta function, a fundamental object in Number Theory. The P.I. has studied the behavior of the eigenvalues of large random matrices with independent entries and obtained new important results about the fluctuation of the outliers in the spectrum of finite rank deformations of such matrices, the linear statistics of the eignevalues, and the spectra of the products of random matrices. In addition, the P.I. studied the entropy of the so-called beta ensembles of random matrices. The project outcomes have been published/ accepted for publication in the form of nine research papers in leading international journals in Probability and Mathematical Physics. The P.I. successfully mentored five Ph.D. students and four undergraduate students. During this research period, the P.I. successfully collaborated with mathematicians from France and Ukraine. The P.I. has given talks about his research at numerous international and domestic mathematical conferences.
