Holomorphic Vector Bundles, Harmonic Maps, and the Topology of Moduli Spaces

摘要：DMS-0805797PI: Richard A. wentworth该基金资助了几个几何分析方面的研究项目。第一部分继续研究高维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Harder-Narasimhan滤波相关的代数奇点的比较。第二个项目分析了黎曼曲面上相干系统模空间的拓扑结构。相干系统与各种几何对象有关，如高阶Brill-Noether轨迹和到格拉斯曼和其他齐次变体的全纯映射。奇异空间条件下莫尔斯理论的新方法为这些计算提供了一个框架。一个正在进行的项目研究能量最小化映射到Teichmueller空间的Weil-Petersson完成。这种映射与黎曼流形的基本群与紧致取向曲面的映射类群的同态有关。谐波映射的正则性是主要问题，因为这与刚性问题有关。第四个项目包括与表面群的表现形式相关的进一步主题。PI将继续研究由与面群表示相关的等变调和映射的能量定义的Teichmueller空间上的泛函。一个特别的目标将是为这个泛函的极小值的适当性和唯一性发展新的准则。结果将对表示模空间上映射类群作用的动力学产生影响。复杂球的等距组表示与球形cr结构有关，关于它们如何与双曲结构相关，存在许多悬而未决的问题。并提出建立新的存在性和刚度结果。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。对称也是物理系统的自然和基本部分，这些对称的动力学携带着重要的信息。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——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，因此对数学家和物理学家都非常重要。

Abstract: DMS-0805797PI: Richard A. WentworthThe grant addresses several research projects in geometric analysis. The first part continues work on the Yang-Mills flow on higher dimensional Kaehler manifolds. A special focus is a comparison of the analytic singularities that occur along the flow with the algebraic singularities associated to Harder-Narasimhan filtrations of holomorphic vector bundles. The second project analyzes the topology of moduli spaces of coherent systems on Riemann surfaces. Coherent systems are related to a variety of geometric objects such as higher rank Brill-Noether loci and holomorphic maps to Grassmannians and other homogeneous varieties. New methods of Morse theory in the setting of singular spaces provide a framework for these computations. An ongoing project studies energy minimizing maps to the Weil-Petersson completion of Teichmueller space. Such maps are associated to homomorphisms of fundamental groups of Riemannian manifolds to the mapping class group of a compact oriented surface. Regularity of harmonic maps is the main issue, as this is related to rigidity questions. The fourth project consists of further topics related to representation varieties of surface groups. The PI will continue to investigate the functional on Teichmueller space defined by the energy of equivariant harmonic maps associated to surface group representations. A particular aim will be to develop new criteria for the properness and uniqueness of minima of this functional. Results will have implications for the dynamics of the mapping class group action on the moduli space of representations. Representations into the isometry group of the complex ball are related to spherical CR-structures, and there are many open questions as to how these relate to hyperbolic structures. The project also proposes to establish new existence and rigidity results. 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. Symmetries are also a natural and fundamental part of physical systems, and the dynamics of these symmetries carries important information. 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 and are therefore are of great importance to both mathematicians and physicists.