Metastable Pseudoisotopy, G-Manifolds, and Functor Calculus

亚稳态赝同位素、G 流形和函子微积分

• 批准号: 1608259
• 负责人: Thomas Goodwillie
• 金额: $ 34.5万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608259&HistoricalAwards=false
• 关键词: Metastable; Pseudoisotopy; Manifolds; Functor; Calculus

AbstractAward: DMS 1608259, Principal Investigator: Thomas G. GoodwillieThis award provides support for principal investigator's research in algebraic topology, which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to certain equivalence. Manifolds are topological spaces that locally resemble Euclidean spaces. The first part of the project holds the promise of new invariants for families of manifolds and new connections between higher-dimensional geometric topology and string topology. The second part should lead to other new invariants for families of manifolds with a group action. The third part will continue the study of "functor calculus", a theory developed by the PI previously, from a geometric point of view. Graduate students will be involved and trained in the project.The planned research has three components. The first involves systematically detecting that about manifolds which is invisible to surgery and algebraic K-theory, beginning with the homotopy fiber of the Hatcher suspension map for pseudoisotopy spaces. Though the appropriate analogue of K-theory is elusive, the corresponding analogue of topological cyclic homology and the cyclotomic trace seems to be within reach. The second is concerned with actions of finite groups on manifolds; the goal is to develop G versions of various constructions related to spaces of manifolds, and especially of spaces of h-cobordisms. The third is about the "geometric" view of homotopy functor calculus, and especially about some fundamental questions about jets and differential operators in that context.

摘要奖：DMS 1608259，主要研究者：托马斯G. GoodwillieThis奖提供支持首席研究员的研究代数拓扑，它使用代数工具来研究拓扑空间。其基本目标是找到代数不变量，将拓扑空间分类到一定的等价性。流形是局部类似于欧几里得空间的拓扑空间。该项目的第一部分持有的承诺，新的不变量的家庭流形和新的连接之间的高维几何拓扑和字符串拓扑。第二部分应导致其他新的不变量的家庭流形的一组行动。第三部分将继续从几何的角度研究“函子演算”，这是PI以前开发的理论。研究生将参与该项目并接受培训。计划的研究有三个组成部分。第一个涉及系统地检测，这是不可见的外科手术和代数K-理论的流形，从同伦纤维的Hatcher悬挂映射的伪合痕空间。虽然K-理论的适当类比是难以捉摸的，拓扑循环同调和分圆迹的相应类比似乎是触手可及的。第二个是关于有限群在流形上的作用;目标是发展与流形空间有关的各种构造的G版本，特别是h-协边空间。第三个是关于同伦函子演算的“几何”观点，特别是关于在这种背景下的喷流和微分算子的一些基本问题。