Harmonic Maps, Loop Groups, and Integrable Systems

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9704443 Guest The investigator will study topological and geometrical properties of the space of harmonic maps from certain compact Riemann surfaces to compact Lie groups and symmetric spaces. He will continue his collaboration with Y. Ohnita and others to determine the connected components and the fundamental groups of these spaces, in the case where the domain is a two sphere. The method uses suitable group actions on these spaces to produce continuous deformations of harmonic maps, together with Morse theoretic arguments. He will also collaborate with F. E. Burstall and others to establish a common understanding of the construction of harmonic maps in the cases where the domain is a two sphere or a torus, i.e. harmonic maps of finite uniton number or harmonic maps of finite type, respectively. The method here uses the geometry of loop groups, together with methods from the theory of integrable systems. The investigator will continue his previous work on the topology of spaces of holomorphic maps. This is an essential ingredient in the study of spaces of harmonic maps, and also of independent interest, being related to recent developments in the topology of moduli spaces. This work is motivated by the role played by symmetry in solving equations of mathematics and physics. Over the past thirty years, mathematicians have come to realize that some of the most important differential equations which describe the real world are subject to very extensive but "hidden" (and therefore somewhat mysterious) symmetries. Even when the equations describe a finite-dimensional process, such as a simple mechanical system, hidden symmetries can be infinite dimensional in extent. Such symmetries often turn out to be the key to solving the equation. As a consequence, mathematical research in this area aims to predict the existence of hidden symmetries for a particular equation and then attempts to use these symmetries to solve the equation. The inves tigator aims to do this for the harmonic map equation. Studying the symmetries of individual equations like this will lead eventually to a general theory of equations with hidden symmetries. Such a theory would be a significant advance in understanding the many natural phenomena which are governed by equations of this type.