Topics in Differential Geometry and Mathematical Physics

University Of California-Riverside, Riverside CA

Investigators

Abstract

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.