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方面的结果，特别是当区域是Riemann曲面，目标空间是紧李群或对称空间时。Guest使用极小曲面的魏尔斯特拉斯表示的推广来研究从二维球面到酉群的调和映射(或者，更一般地，从任意黎曼曲面的有限酉数的调和映射)到酉群。乌伦贝克、西格尔、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。特别地，得到了一个显式标准形，并用它研究了所有这类映射的空间。它的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用一种不同的方法，在Hitchin早期规范理论工作的基础上，得到了研究调和映射空间的几何(特别是预辛几何)的框架。调和映射方程可以看作是一个可积系统，上述工作对其他可积系统也有启发意义。从这个角度研究了另外两个可积系统的例子，得到了初步的结果。Guest研究的第一个例子是量子微分方程组理论。建立了调和映射的平行关系，为今后在这一方向上的工作奠定了基础。Ohnita和Nishimori也得到了对称空间的量子上同调，Guest和Otofuji得到了FLAG流形的量子上同调.Burstall和Calderbank研究的第二个例子是共形几何和Mobius几何的可积系方面，并开创了一种新的方法。

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.