Mathematical Sciences: Quaternionic Geometry, Einstein Manifolds, and the Topology of Moduli Spaces

早些时候，提出者发现了一类3- sasaki流形；在此过程中给出了正标量曲率的单连通强非齐次爱因斯坦流形的明确例子。这里提出的问题是对上述紧单连通3- sasaki流形进行拓扑和可微分类。他们还研究了从黎曼球到复标志流形的基全纯映射空间的拓扑结构，受到Segal的瞬时模工作的启发；给出了一个稳定性定理。这导致了他们的解决方案（Boyer和Mann与Hurtubise和Milgram合作）的Atiyah-Jones猜想。他们提出扩展稳定性（和合理性）定理的范围，使其包括除标志流形以外的某些几乎齐次的目标空间。本文的研究方向是微分拓扑与微分几何的结合；它可以为数学和理论物理中作为各种统一场理论的理论模型的规范理论的一部分奠定数学基础。

Boyer 9423752 Earlier the proposers discovered a class of 3-Sasakian manifolds; in the process gave explicit examples of simply connected strongly inhomogeneous Einstein manifolds of positive scalar curvature. The proposed problem here is to classify, topologically and differentiably, the aforementioned compact simply connected 3-Sasakian manifolds. They also studied the topology of the space of based holomorphic maps from the Riemann sphere to a complex flag manifold, motivated by the instanton moduli work of Segal; gave a stability theorem. This led to their solutions (Boyer and Mann in collaboration with Hurtubise and Milgram) of the Atiyah-Jones conjecture. They propose to extend the scope of the stability (and rationality) theorem to include certain almost homogeneous target spaces other than the flag manifolds. The proposed research is in the interface of differential topology and differential geometry; it may lay a mathematical foundation for parts of gauge theories, which serve as theoretical models for various unified field theories, in mathematical and theoretical physics.