Topics in Differential Geometry and Mathematical Physics

摘要DMS -0204002. PI的第一个计划是关于具有全斜挠率的厄米特或超厄米特连接的几何。这些结果将适用于量子力学黑洞的研究。PI的第二个计划是研究复变形理论和厄米几何的nilmanifold。在对经典模和扩展模进行代数分析的同时，他和他的合作者从特殊的完整性和辛结构的角度对这些空间进行了几何分析。一位资深参与者建议分析紧凑四元数Kaehler流形的结构和具有有限第二同伦的非自旋流形的A-hat亏格。这两个问题都是通过椭圆亏格的刚性分析来解决的。通过椭圆亏格理论，两个提议者将共同计算一个与全纯接触结构或关于全纯自同构不变量的有理曲线相关的Hilbert多项式。这里提出的项目利用了理论物理学家和数学家在场论、复变形理论和不动点集微分拓扑学方面的知识。所有建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的项目将解决一种数学家比物理学家更不了解的几何学。这些结果将增加数学家对厄米几何的认识，并直接解决理论物理学家关心的问题。高级参与者的项目解决了在完整领域的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机制，用于研究关于群作用的不动点集。研究人员的联合项目将联合收割机他们的专业知识，以提取拓扑信息的一类空间中心的代数几何。

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.