Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

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This project is focused on three areas of mathematical analysis: (1) periods of hyperelliptic Riemann surfaces and Hamiltonian mechanics, (2) matrix problems, and (3) long-time behavior of integrable systems. The first topic involves the investigation of a new connection between the arithmetic- geometric mean and the computation of hyperelliptic integrals on the one hand, and Hamiltonian mechanics on the other. The second topic involves the analysis of high accuracy algorithms to compute the singular values of a matrix. This part of the project is also concerned with the problem of computing the eigenvalues and eigenvectors of products of matrices. A central role in the analysis is played by integrable Hamiltonian systems which interpolate the iterates of the singular value / eigenvalue algorithms at integer times. The third topic involves an analysis of the long-time behavior of integrable systems, specifically of the Toda shock problem. Attention will be focused on the case of low shock speed, the more general case for higher shock speeds, the stability problem in which a doubly infinite Toda lattice in equilibrium is perturbed locally, the question of the rate and the manner in which the eigenvalues fill the "phantom" band, and the analysis of the long-time behavior of the nonlinear wave equations in the Gel'fand-Dikii hierarchy. The work in this project deals with spectral problems and inverse spectral problems. The project is concerned with applications to Hamiltonian mechanics, with the development of algorithms which improve the accuracy of the computed singular values of a matrix, and with the study of the long-time behavior of integrable systems.