Collaborative Research: Factorization Homology and the Cobordism Hypothesis

拓扑学领域研究流形，这是空间或时空的抽象概念。自 19 世纪诞生以来，流形拓扑就与理论物理学紧密相连。这种联系随着爱因斯坦的广义相对论而发展，该理论要求时空因质量的存在而弯曲，因此时空的几何形状比欧几里得的经典几何形状更复杂。随着量子场论的出现，这种联系进一步发展，量子场论产生了最敏锐的流形不变量。这里，不变量是指同时分析所有流形的统一技术。共边假说是 20 世纪 90 年代中期提出的对流形不变量（称为拓扑量子场论）的分类，它可能源自理论物理学。我们的项目证明了协同假设。主要研究人员通过开发一种新的代数方法（因式分解同调）来实现这一目标，用于从局部不变量理论组装全局不变量。当代拓扑量子场的核心结构原则是由 Lurie、Hopkins 和 Baez-Dolan 提出的协边假说。这断言对称幺半群 n 范畴中的拓扑量子场论与该 n 范畴的完全对偶对象处于双射状态。特别是，它断言场论是由它在一点上的值决定的。我们的项目证明了协同假设。主要研究人员通过进一步发展并应用因式分解同调理论来做到这一点。这种增强的理论允许系数系统为对称幺半群 n 类别，概括了先前系数为 n 盘代数的因式分解同调。它们增强的分解同调性为基于分层模的场论中的局部性提供了新的基础，这与自 20 世纪 80 年代 Atiyah 公理以来形成局部性基础的莫尔斯理论和手术演示相反。这项工作的技术基础是家庭分层的这种差异拓扑。

The field of topology studies manifolds, an abstract notion of space or space-time. Since its inception in the 19th century, the topology of manifolds has been intertwined with theoretical physics. This connection grew with Einstein's theory of general relativity, which requires that space-time is curved by the presence of mass, and therefore that the geometry of space-time is more complex than the classical geometry of Euclid. This connection grew further with the advent of quantum field theories, which have given rise to the most discerning invariants of manifolds. Here, an invariant means a uniform technique for analyzing all manifolds at once. The cobordism hypothesis is a proposed classification from the mid 1990s of these invariants of manifolds - called topological quantum field theories - which could arise from theoretical physics. Our project proves the cobordism hypothesis. The principal investigators do this by developing a new method in algebra, factorization homology, for the theoretical assembly of global invariants from local invariants.The central structural tenet of contemporary topological quantum field is the cobordism hypothesis, developed by Lurie, Hopkins, and Baez-Dolan. This asserts that topological quantum field theories valued in a symmetric monoidal n-category are in bijection with fully dualizable objects of that n-category. In particular, it asserts that a field theory is determined by its value on a point. Our project proves the cobordism hypothesis. The principal investigators do this by further developing, and then applying, the theory of factorization homology. This enhanced theory allows for coefficient systems which are symmetric monoidal n-categories, generalizing the previous factorization homology whose coefficients are n-disk algebras. Their enhanced factorization homology offers a new basis for locality in field theory based on moduli of stratifications, as opposed to the Morse theory and surgery presentations which have formed the basis for locality since Atiyah's axioms from the 1980s. The technical basis for this work is this differential topology of stratifications in families.