Topics in Differential Geometry and Mathematical Physics

抽象DMS-0204002本方案中的第一个程序是关于具有完全斜扭转的厄米特或超厄米特连接的几何。这一结果将适用于量子力学黑洞的研究。PI的第二个项目是研究复形变理论和零流形的厄米特几何。与经典模和扩张模的代数分析类似，他和他的合作者从特殊完整和辛结构的观点研究了这些空间上的几何。一位资深与会者建议分析紧致四元数Kaehler流形和具有有限第二同伦的非自旋流形的A-hat亏格的结构。这两个问题都是通过分析椭圆亏格的刚性来解决的。通过椭圆亏格理论，两个提出者将共同计算关于全纯自同构的与全纯接触结构或有理曲线不变量相关的希尔伯特多项式，所提出的项目是基于理论物理学家和数学家关于场论、复变形理论和不动点集微分拓扑的知识。所有拟议的项目都旨在解决与当前物理或数学问题相关的问题。PI的项目将解决一种比物理学家更少为数学家所知的几何类型。这些结果将增加数学家关于厄米几何的知识，并直接解决理论物理学家所关心的问题。这位资深参与者的项目解决了完整学领域一个长期存在的猜想。这个项目的中心工具是一个新的微分拓扑学，用于研究关于群作用的不动点集。研究人员的联合项目将结合他们的专业知识，提取代数几何中一类空间中心的拓扑信息。

ABSTRACT DMS - 0204002.The PI's first program in this proposal is concerned with the geometry of Hermitian or hyper-Hermitian connections with totally skew torsions. The results will be applicable to an investigation in quantum mechanical black holes. The PI's second program is to study the complex deformation theory and Hermitian geometry of nilmanifolds. Parallel to the algebraic analysis on the classical and extended moduli, he and his collaborators investigatge the geometry on these spaces from the viewpoint of special holonomy and symplectic structures. A senior participant proposes to analyse the structures of compact quaternionic Kaehler manifolds and the A-hat genus of non-spin manifolds with finite second homotopy. Both issues are tackled through an analysis of the rigidity of elliptic genus. Through the theory of elliptic genus, the two proposers will jointly calculate a Hilbert polynomial relevant to holomorphic contact structures or rational curves invariant with respect to a holomorphic automorphisms.The proposed projects here are enabled by theoretical physicists' and mathematicians' knowledge on field theory, complex deformation theory and differential topology of fixed point sets. All proposed projects are intended to address issues relevant to current physcial or mathematical issues. The PI's projects will address a type of geometry less known to mathematicians than to physicists. The results will increase mathematicians' knowledge on Hermitian geometry and directly addresses issues to theoretical physcists' concerns. The senior participant's project addresses a long-standing conjecture in the field of holonomy. The central tool in this project is a novel differntial topological mechinary for studying fixed point sets with respect to group actions. The investigators joint project will combine their expertise to extract topological information on a class ofspace central in algebraic geometry.