Strings, Solitons and Random Matrices

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Abstract Proposal: DMS-9802077 Principal Investigator: Mark Adler What is the connection of random matrices with integrable systems? Is this connection useful?The answer to these questions leads to a new and unifying approach to the theory of random matrices. Introducing an appropiate t-dependence in the probability distribution of the matrix ensemble, leads to vertex operator expressions for the n-point correlation functions (probabilities of n eigenvalues in infinitesmal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalues in a Borel set of R). This representation leads to PDE's for the Fredholm determinant. This covers the finite Hermitean ensemble,the symplectic and symmetric ensemble and even a coupled matrix ensemble. These are related to reductions of 2-Toda and 1-Toda and the KP hierarchy. We work in the general theory of nonlinear equations of engineering and mathematical physics.The type of equations we study figure into ultra-short transmissions in fiber optics,intense pulses in lasers, water wave transmission in narrow channels,slow nuclear reactions and the physical properties of bulk metals at high temperatures as well as the general excitation theory of heavy nuclei. We also study equations connected with string theory in quantum field theory, an exciting area of research in modern physics.