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提出研究调和映射在teichmueller空间补全中的正则性。两个主要的应用是关于辛Lefschetz铅笔和映射类群中格的超刚性。在第二个项目中，PI在黎曼曲面上的Yang-Millsequations上提出了对Atiyah和Bott猜想的高维部分类似物的证明。这些是PI之前关于Yang-Mills和全纯向量束之间的联系的工作在高维上的非平凡推广。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，这些问题推广了众所周知的曲面调和映射理论。第一个项目的动机是使用调和映射，以便解析地理解kaimanoovich - masur和farb - masur关于映射类群中至少2阶格的超刚性的结果。在这个过程中，私家侦探意识到这些方法也应该对排名第一的案件有所帮助。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非常密切的联系。在黎曼曲面的情况下，这些联系已经被一些人非常详细地利用了，但在高维中却知之甚少。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.