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的黎曼流形。Berger对可能的“素”完整群进行了分类，包括两种例外情况：具有完整G_2的7-流形和具有完整自旋的8-流形(7)。布赖恩特在1985年首次证明了这一点，乔伊斯在1995年证明了具有特殊完整的闭合流形的存在。从那时起，人们对例外完整流形的理解有了越来越多的进展。在具有例外完整的闭流形的背景下，G_2方面近年来在理解方面取得了更多的进展，例如，微分同态类型，以及例子的校准几何和规范理论。该项目将研究自旋(7)-流形的类似问题，特别是从Joyce的加权射影空间结构。在最初的12个月中，重点将集中在拓扑问题上：例如，计算不变量，研究可应用的分类结果以及与有理同伦理论的关系。这将建立对该地区和特定建筑的熟悉，同时仍有可能在短期内导致一些可发布的结果。这项工作的主要来源是Joyce的“具有特殊全息的紧致流形”，而拓扑学的背景材料将包括Hatcher的“代数拓扑学”，以及Milnor和Stasheff的“特征类”。更多关于校准几何(例如，通过校准的子流形搜索纤维)和规范理论的分析问题将在较长期内被考虑。为了解决这些问题，将通过TCC课程和独立阅读来开发必要的背景材料，例如对Sobolev空间和其他泛函分析工具的理解，以及包括K3曲面和Cayle子流形在内的对象。

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.