Factorization homology and the topology of manifolds

因式分解同调（英语：Factorization homology）是拓扑n-流形的同调理论，是Beilinson和Drinfeld代数几何意义上的因式分解代数的同调的拓扑类似物。这种理论的系数系统是由n-圆盘代数的选择提供的，n-圆盘代数是一种由n重环空间、李代数和某些量子场论等来源产生的结构。该项目旨在进一步发展这一相对较新的理论，既带来新的技术来承担以前研究的不变量，并找到新的流形不变量。一个特别的焦点是3-流形拓扑，其中一个目标是表达量子结和3-流形不变量，如Reshetikhin-Turaev不变量，Vassiliev结不变量和Chern-Simons不变量，在一个统一的方式在因式分解同源性方面，第二个目标是通过因式分解同源性构建新的结同源性理论。另一个重点是关系的因式分解同源手术分类的高维拓扑流形。这个项目在于拓扑，研究抽象概念的空间，由数学物理的动机-一个交集，导致了大量的工作和交叉授粉。这些问题涉及到在物理空间或时空的理论模型中可能存在什么样的全局几何，以及如何检测这种全局几何。试图通过局部观测来描述这样一个候选时空模型的全局几何是一个有趣的问题。该项目是解决这一问题的一种方法。也就是说，如果允许人们在整个时空的不同点上进行某种局部观测，然后将它们进行比较，那么人们如何从这些相容的观测集合中重建全局几何呢？因子分解同源性，这个项目的重点，提供了解决这个问题的途径之一。

Factorization homology is a homology theory for topological n-manifolds, constructed as a topological analogue of the homology of a factorization algebra in the algebro-geometric sense of Beilinson and Drinfeld. The coefficient system for such a theory is provided by a choice of n-disk algebra, a structure arising from n-fold loop spaces, Lie algebras, and certain quantum field theories, among other sources. This project aims to further develop this relatively new theory, to both bring new techniques to bear on previously studied invariants and to find new manifold invariants. A particular focus is 3-manifold topology, where one goal is to express quantum knot and 3-manifold invariants, such as Reshetikhin-Turaev invariants, Vassiliev knot invariants, and Chern-Simons invariants, in a uniform way in terms of factorization homology, and a second goal is to construct new knot homology theories via factorization homology. Another focus is the relation of factorization homology to the surgery classification of higher-dimensional topological manifolds.This project lies in topology, which studies abstract notions of space, motivated by mathematical physics - an intersection which had led to a great deal of work and cross-pollination. The problems are concerned with what possible global geometry may exist in a theoretical model for physical space or space-time and how this global geometry can be detected. It is interesting problem to try to describe the global geometry of such a candidate model for space-time by local observations. This project is one approach to this problem. Namely, if one were allowed to make local observations of some kind at different points throughout space-time, then compare them, how could one then reconstruct the global geometry from these compatible collections of observations? Factorization homology, the focus of this project, provides one avenue toward addressing this question.