Problems in Geometric Analysis: Harmonic Maps and Holomorphic Vector Bundles

第一个项目涉及能量最小化映射到Weil-Petersson完成的Teichmueller空间。所提出的研究将为高维域建立足够的正则性，以便调和映射理论可以用来回答刚性问题。该提案的第二部分继续研究高维Kaehler流形上的Yang-Mills流。一个特别的焦点是沿流发生的解析奇点与与全纯向量束的hard - narasimhan滤波相关的代数奇点的比较。第三个项目包括与表面群体的表现形式相关的主题。包括对与面群表示相关的调和映射所定义的Teichmueller空间上的能量泛函的性质的研究。这个方向的结果将对表示模空间上映射类群作用的动力学产生影响。数学研究的一个重要分支是流形的几何、解析和代数性质之间的关系。流形是曲线和面的高维推广，它们出现在纯数学和应用数学的各种情况中。本提案中的研究项目将进一步加深我们对其中一些对象的理解。所研究的方程式——能量最小化图和杨-米尔斯流——起源于对物理世界的数学描述。因此，它们对数学家和物理学家都非常重要。

The first project deals with energy minimizing maps to the Weil-Petersson completion of Teichmueller space. The proposed research will establish sufficient regularity for higher dimensional domains so that harmonic map theory may be used to answer rigidity questions. The second part of the proposal continues work on the Yang-Mills flow on higher dimensional Kaehler manifolds. A special focus is a comparison of the analytic singularities which occur along the flow with the algebraic singularities associated to Harder-Narasimhan filtrations of holomorphic vector bundles. The third project consists of topics related to representation varieties of surface groups. Included is a study of properties of the energy functional on Teichmueller space defined by harmonic maps associated to surface group representations. Results in this direction will have implications for the dynamics of the mapping class group action on the moduli space of representations.A significant branch of mathematical inquiry has been the relationship between the geometric, analytic, and algebraic properties of manifolds. Manifolds are higher dimensional generalizations of curves and surfaces, and they appear in a variety of situations in pure and applied mathematics. The research projects in this proposal will further our understanding of some of these objects. The equations studied -- energy minimizing maps and the Yang-Mills flow -- have their origins in the mathematical description of the physical world. They are therefore of great importance to both mathematicians and physicists.