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

Boyer，9423752早些时候，作者发现了一类3-Sasakian流形；在这个过程中，给出了单连通强非齐次正标量曲率爱因斯坦流形的例子。这里提出的问题是对上述紧致单连通3-Sasakian流形进行拓扑分类和可微分类。他们还研究了从Riemann球面到复旗流形的基全纯映射空间的拓扑，得到了一个稳定性定理。这导致了他们(Boyer和Mann与Hurtubise和Milgram合作)对Atiyah-Jones猜想的解决方案。他们建议扩展稳定性(和合理性)定理的范围，以包括除旗形流形之外的某些几乎齐次的目标空间。所提出的研究是在微分拓扑学和微分几何的交界处进行的，它可能为数学和理论物理中的部分规范理论奠定数学基础，这些规范理论是各种统一场论的理论模型。

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.