Applications of integrable systems in geometry and topology

得到了调和映射和调和映射空间的几何和拓扑的一些结果，特别是在定义域为Riemann曲面，目标空间为紧李群或对称空间的情况下.来宾使用了一个推广的维尔斯特拉斯表示的最小表面研究调和映射从二维球（或更一般地说，调和映射的有限uniton数，从任何黎曼曲面）的酉群。Uhlenbeck，Segal，Dorfmeister-Pedit-Wu，Burstall-Guest的早期结果被发展成为描述这类地图的有效工具。特别是，得到了一个明确的标准形，这是用来研究所有这样的地图的空间。主要应用是描述从二维球面到酉群的调和映射空间的连通分支。Ohnita基于Hitchin在规范理论中的早期工作，使用了一种不同的方法，获得了一个研究调和映射空间几何（特别是前辛几何）的框架。调和映射方程可以被视为一个可积系统，上述工作为其他可积系统提供了启示。从这一观点出发，研究了可积系统的另外两个例子，得到了初步的结果。第一个例子，研究的客人，是理论的量子微分方程。建立了与调和地图的比较，为今后这方面的工作奠定了基础。关于对称空间的量子上同调的结果也由Ohnita和Nishimori得到，关于旗流形的量子上同调的结果由Guest和Otofuji得到。第二个例子，研究Burstall和Calderbank，是可积系统方面的共形和麦比乌斯几何，并提出了一种新的方法。

Results were obtained on the geometry and topology of harmonic maps and spaces of harmonic maps, especially in the case where the domain is a Riemann surface and the target space is a compact Lie group or symmetric space. Guest used a generalization of the Weierstrass representation for minimal surfaces to study harmonic maps from the two-dimensional sphere (or, more generally, harmonic maps of finite uniton number, from any Riemann surface) to the unitary group. Earlier results of Uhlenbeck, Segal, Dorfmeister-Pedit-Wu, Burstall-Guest were developed into an effective tool for describing such maps. In particular, an explicit canonical form was obtained, and this was used to study the space of all such maps. The main application was a description of the connected components of the space of harmonic maps from the two-dimensional sphere to the unitary group. Ohnita used a different approach, based on earlier work of Hitchin in gauge theory, to obtain a framework for studying the geometry (in particular, the pre-symplectic geometry) of spaces of harmonic maps.The harmonic map equation can be regarded as an integrable system, and the above work sheds light on other integrable systems. Two other examples of integrable systems were studied from this point of view, and preliminary results obtained. The first example, studied by Guest, was the theory of quantum differential equations. Parallels with harmonic maps were established, forming the basis for future work in this direction. Results on quantum cohomology of symmetric spaces were obtained also by Ohnita and Nishimori, and on quantum cohomology of flag manifolds by Guest and Otofuji. The second example, studied by Burstall and Calderbank, was the integrable systems aspect of conformal and Mobius geometry, and a new approach was initiated.