Mathematical Sciences: Completely Integrable Hamiltonian Systems

Professor Li will investigate the relationship between eigenvalue algorithms and integrable Hamiltonian systems. Together with his collaborators, he has proved the complete integrablity of the Toda (QL) flows on orbits through generic symmetric matrices. Further research will focus on the Hamiltonian structure and the question of complete integrability for the LU flows, the Chloesky flows and the extension of the Toda flows to nonsymmetric matrices. Professor Li will also investigate two related classes of flows which are important derivates of the Toda flows and are linkeed to the QL family of algorithms, such as QZ and SVD. These flows are not isospectral, but one of them preserves singular values of matrices, while the other preserves eigenvalues of matrix pencils. This research is part of theoretical efforts aimed at a better understanding of computational methods related to matrix eigenvalue problems.