Harmonic Maps, Loop Groups, and Integrable Systems

9704443研究人员将研究从某些紧黎曼曲面到紧李群和对称空间的调和映射空间的拓扑和几何性质。他将继续与Y.Ohnita等人合作，在区域是两个球面的情况下，确定这些空间的连通分支和基本群。该方法利用这些空间上的适当群作用来产生调和映射的连续变形，并结合Morse理论论证。他还将与F.E.Burstall等人合作，在区域是两个球面或环面的情况下，即有限单位数的调和映射或有限类型的调和映射的情况下，建立对调和映射构造的共同理解。这里的方法使用了环群的几何，并结合了可积系统理论的方法。这位研究者将继续他之前在全纯映射空间拓扑方面的工作。这是调和映射空间研究的一个重要组成部分，也是独立的兴趣，与模空间拓扑学的最新发展有关。这项工作的动机是对称性在求解数学和物理方程中所起的作用。在过去的三十年里，数学家们逐渐意识到，一些描述现实世界的最重要的微分方程受到非常广泛但“隐藏”(因此有些神秘)的对称性的影响。即使方程描述了一个有限维的过程，例如一个简单的机械系统，隐藏的对称性也可以是无限维的。这样的对称性往往被证明是解方程的关键。因此，这一领域的数学研究旨在预测特定方程是否存在隐藏对称，然后尝试使用这些对称来求解该方程。投资触发器的目的是为调和映射方程做这件事。像这样研究单个方程的对称性最终将导致具有隐藏对称性的方程的一般理论。这样的理论将在理解由这类方程支配的许多自然现象方面取得重大进展。

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.