Harmonic Maps, Loop Groups, and Integrable Systems

研究者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和其他人合作，确定这些空间的连通成分和基本群，在域是两个球体的情况下。该方法利用适当的群作用在这些空间上产生谐波映射的连续变形，并结合莫尔斯理论论证。他还将与F. E. Burstall等人合作，建立对域为两球面或环面情况下调和映射构造的共同理解，即有限单位数调和映射或有限类型调和映射。该方法利用了环群的几何特性，并结合了可积系统理论中的方法。研究者将继续他以前在全纯映射空间拓扑上的工作。这是调和映射空间研究的一个重要组成部分，也是一个独立的兴趣，与模空间拓扑的最新发展有关。这项工作的动机是对称在求解数学和物理方程中所起的作用。在过去的三十年里，数学家们逐渐认识到，一些描述现实世界的最重要的微分方程受制于非常广泛但“隐藏”（因此有些神秘）的对称性。即使这些方程描述的是一个有限维的过程，比如一个简单的机械系统，隐藏的对称性在程度上也可以是无限维的。这种对称性往往是解方程的关键。因此，这一领域的数学研究旨在预测某个特定方程是否存在隐藏的对称性，然后试图利用这些对称性来求解该方程。研究者的目标是对调和映射方程做这个。研究像这样的单个方程的对称性将最终导致具有隐藏对称性的方程的一般理论。这样的理论对于理解由这类方程控制的许多自然现象将是一个重大的进步。

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.