Topics in Differential Geometry and Mathematical Physics

摘要DMS - 0204002。在这个提议中，PI的第一个程序是关于具有完全歪斜扭转的厄米或超厄米连接的几何。这些结果将适用于量子力学黑洞的研究。PI的第二个项目是研究零流形的复杂变形理论和厄米几何。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完整和辛结构的角度研究了这些空间上的几何。一位资深参与者提出了紧致四元数Kaehler流形的结构和有限第二同伦的非自旋流形的A-hat格。这两个问题都是通过分析椭圆属的刚性来解决的。通过椭圆属理论，两位作者将共同计算与全纯自同构的全纯接触结构或有理曲线不变量相关的Hilbert多项式。这里提出的项目是由理论物理学家和数学家在场论，复杂变形理论和不动点集的微分拓扑的知识。所有提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的项目将涉及一种数学家比物理学家更不了解的几何类型。这些结果将增加数学家对厄米几何的知识，并直接解决理论物理学家所关注的问题。资深参与者的项目解决了一个长期存在的猜想在全息领域。这个项目的核心工具是一个新的差分拓扑机制，用于研究关于群体行为的不动点集。研究人员的联合项目将结合他们的专业知识来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.