Geometry and Topology of Manifolds with Exceptional Holonomy

微分几何中的一个重要主题是研究具有特殊完整性的黎曼流形。Berger对可能的“素”完整群进行了分类，其中包括两种例外情况：具有完整群G_2的7-流形和具有完整群Spin的8-流形（7）。这些都是实现在所有首先证明了布莱恩特在1985年，并存在封闭流形与特殊holonomy证明了乔伊斯在1995年。自那时以来，人们对例外完整流形的认识不断取得进展，在例外完整闭流形的背景下，G_2方近年来在认识上取得了较大的进展;例如，自同构类型，以及示例的校准几何和规范理论。该项目将研究Spin（7）-流形的类似问题，特别是Joyce的加权射影空间构造。最初，在头12个月，重点将放在拓扑问题上：例如，计算不变量，调查适用的分类结果以及与有理同伦理论的关系。这将建立熟悉的地区和特定的建设，同时仍然有可能导致一些可持续的结果在短期内。这项工作的主要来源是Joyce的“紧致流形与特殊完整性”，而拓扑背景材料将包括Hatcher的“代数拓扑”，以及Milnor和Stasheff的“特征类”。更多关于校准几何的分析问题（例如，通过校准子流形搜索纤维化）和规范理论将被考虑在较长的时间内。为了进一步了解这些问题，必要的背景材料，如Sobolev空间和其他功能分析工具的理解，以及objectsincluding K3曲面和凯莱子流形，将通过TCC课程和独立的阅读。

An important theme in differential geometry is the study of Riemannian manifolds with special holonomy. Thepossible "prime" holonomy groups were classified by Berger, and include two exceptional cases: 7-manifolds withholonomy G_2 and 8-manifolds with holonomy Spin(7). That these are realised at all was first proved by Bryant in1985, and the existence of closed manifolds with exceptional holonomy was proved by Joyce in 1995. Since thenthere has been increasing progress on understanding exceptional holonomy manifolds.In the context of closed manifolds with exceptional holonomy, the G_2 side has in recent years seen more progressin understanding; for example, the diffeomorphism types, and the calibrated geometry and gauge theory of examples.The project will study similar problems for Spin(7)-manifolds, in particular from Joyce's weighted projective spaceconstruction.Initially, during the first 12 months, the focus will be on topological questions: for example, computing invariants,investigating applicable classification results and the relation to rational homotopy theory. This will build familiaritywith the area and with the particular constructions, while still potentially leading to some publishable results in theshort term. A primary source for this work is Joyce's 'Compact Manifolds with Special Holonomy', while thetopological background material will include Hatcher's 'Algebraic Topology', and Milnor and Stasheff's 'CharacteristicClasses'.More analytical problems concerning calibrated geometry (e.g. searching for fibrations by calibrated submanifolds)and gauge theory will be considered in the longer term. In order to progress to these problems, the necessarybackground material, such as understanding of Sobolev spaces and other functional analysis tools, as well as objectsincluding K3 surfaces and Cayley submanifolds, will be developed through TCC courses and independent reading.