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

博耶9423752 早些时候，提出者发现了一类3-Sasakian流形;在这个过程中给出了明确的例子，简单连接强非齐次爱因斯坦流形的正标量曲率。这里提出的问题是分类，拓扑和可微，上述紧单连通3-Sasakian流形。他们还研究了从黎曼球面到复旗流形的基全纯映射空间的拓扑，这是由西格尔的瞬子模工作所激发的;给出了一个稳定性定理。这导致了他们的解决方案（博耶和曼恩与Hurtubise和Milgram合作）的阿蒂亚-琼斯猜想。他们建议扩展稳定性（和合理性）定理的范围，以包括某些几乎齐次的目标空间，而不是旗流形。 所提出的研究是在微分拓扑学和微分几何的接口;它可能奠定了规范理论的部分数学基础，作为各种统一场论的理论模型，在数学和理论物理。

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.