U.S.-Italy Cooperative Research on Quantum Groups and Solutions of Soliton Equations

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This award will support a four month visit by Professor Victor Kac of MIT to Pisa, Italy to carry out research with Professor Corrado DeConcini of the Scuola Normale in Pisa. The two mathmeticians intend to work on quantum groups at roots of unity and on quasiperiodic solutions of exceptional hierarchies of soliton equations. There has been a great deal of work recently on quantum groups Uq(g), where g is a finite-dimensional simple Lie algebra, for generic q. For q = root of 1 the properties of Uq(g) are dramatically different. The structure of the center of Uq(g) for this case is of particular interest to the researchers. In the area of quasiperiodic solutions of soliton equations, thanks to the Krichever constructions Riemann theta functions are seen as solutions of the Kadomtsev- Petviashvili (KP) equations. This result has been extended to show that Prym theta functions are also solutions of KP equations of type B. The proposed joint work will attempt to show that some further generalizations, for example the Prym- Tjurin theta functions, will also turn out to be solutions to other exceptional hierarchies of soliton equations.