Harmonic Maps, Loop Groups, and Integrable Systems

9704443 客人 研究人员将研究拓扑和几何性质 一类紧Riemann的调和映射空间 紧李群和对称空间。 他将 继续与Y合作。Ohnita和其他人确定 这些空间的连通分量和基本群 在域是两个球体的情况下。该方法使用 在这些空间上适当的群作用，以产生连续的 变形的调和映射，连同莫尔斯理论的论点。 他还将与F。E.伯斯托尔和其他人建立了 对调和映射的构造的共同理解 域是两个球面或一个环面的情况，即调和映射 有限型调和映射或有限型调和映射。 这里的方法使用了循环组的几何结构， 从可积系统的理论。研究者将继续 他以前的工作拓扑空间的全纯映射。 这是研究调和空间的一个基本要素 地图，也是独立的利益，与最近的 模空间拓扑学的发展。 这项工作的动机是对称性在解决 数学和物理方程。 在过去的三十年里，数学家们逐渐意识到，一些最重要的 描述真实的世界的微分方程服从于非常广泛但“隐藏的”（因此有些神秘的）对称性。 即使方程描述的是有限维的过程，比如一个简单的力学系统，隐藏的对称性也可以是无限维的。 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特定方程的隐藏对称性的存在，然后尝试使用这些对称性来求解方程。 投资大亨的目标是 这是调和映射方程。像这样研究单个方程的对称性，最终将导致一个具有隐藏对称性的方程的一般理论。 这样的理论将是一个重大的进步，在理解许多自然现象，这是由这种类型的方程。

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.