# Mathematics Colloquia and Seminars

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### Random matrices, integrable systems and vertex operators.

Mathematical Physics & Probability

 Speaker: Professor Pierre van Moerbeke, Mathematical Science Reseach Institute Location: 693 Kerr Start time: Tue, Mar 23 1999, 4:10PM

The purpose of this seminar is to show that, upon introducing appropriate time-parameters t, matrix models for general beta can be generated by certain vertex operators. It shows that these matrix models satisfy Virasoro constraints.

For beta=2, the matrix models are well-known to satisfy the Toda Lattice, besides the Virasoro constraints. For beta=1,4, they satisfy the Pfaff lattice. As a general recipe, combining the Lattice equations with the Virasoro constraints and setting t=0, leads to partial differential equations for the probabilities of finding the eigenvalues in an interval for the corresponding random matrix model.