Gauge Theory, Harmonic Maps to Singular Spaces and Applications to Topology

DMS-0204191 PI：Georgios Daskalopoulos PI提出了三个项目。 首先，PI提出在完备Teichmueller空间中研究调和映射的正则性. 主要的两个应用是辛Lefschetz代数和超刚性格的映射类组。 在第二个项目中，PI提出了Riemann曲面上Yang-Millequations上Atiyah和Bott的代数的高维部分类似物的证明。 这些都是非trivialgeneralizations高维的PI以前的工作theconnection之间的杨米尔斯和理论的全纯向量丛。 最后，在第三个项目中，PI提出了一些关于从三维流形到树的调和映射的问题，推广了著名的从曲面到树的调和映射理论，第一个项目的动机是利用调和映射来解析地理解Kaimanovich-Masur和Farb-Masur关于映射类群中秩至少为2的格的超刚性的结果. 在这个过程中，PI意识到这些方法也应该在一级病例中发挥作用。 第二个项目表明，杨-米尔斯方程的分析与代数几何之间仍然有着非常密切的联系。 这些连接已被利用非常详细的几个人在黎曼曲面的情况下，但很少知道在高维. PI的工作在这个方向上提供了一些启示。

DMS-0204191 PI: Georgios DaskalopoulosThe PI proposes three projects. In the first, the PI proposes tostudy regularity of harmonic maps into the completion of Teichmuellerspace. The two main applications are on symplectic Lefschetz pencilsand on superrigidity of lattices in the mapping class group. In thesecond project the PI proposes a proof of higher dimensional partialanalogues of the conjectures of Atiyah and Bott on Yang-Millsequations over Riemann surfaces. These are nontrivialgeneralizations to higher dimensions of the PI's previous work on theconnection between Yang-Mills and the theory of holomorphic vectorbundles. Finally in the third project the PI proposes some questionsabout harmonic maps from three-dimensional manifolds to trees whichgeneralize the well-known theory of harmonic maps from surfaces.The motivation for the first project is to use harmonic maps in orderto understand analytically the results of Kaimanovich-Masur andFarb-Masur on the superrigidity of lattices of rank at least 2 in themapping class group. In the process the PI realized that the methodsshould also shed light in the rank one case. The second projectshows that there are still very close connections between theanalysis of the Yang-Mills equations and algebraic geometry. Theseconnections have been exploited in great detail by several people inthe case of Riemann surfaces but very little is known in higherdimensions. The PI's work sheds some light in this direction.