Collaborative Research: Factorization Homology and the Cobordism Hypothesis

拓扑学研究流形，即空间或时空的抽象概念。流形的拓扑学自19世纪诞生以来，一直与理论物理交织在一起。这种联系随着爱因斯坦的广义相对论而发展起来，广义相对论要求时空因质量的存在而弯曲，因此时空的几何比欧几里得的经典几何更复杂。这种联系随着量子场论的出现而进一步发展，量子场论产生了流形上最有辨识力的不变量。在这里，不变量意味着一次分析所有流形的统一技术。协边假设是20世纪90年代中期对流形的这些不变量--称为拓扑量子场论--提出的一种分类，它可能产生于理论物理。我们的项目证明了协带主义假说。主要研究人员通过在代数中发展一种新的方法-因式分解同调来从局部不变量中理论组装全局不变量来做到这一点。当代拓扑量子场的中心结构原则是由Lurie，Hopkins和Baez-Dolan发展的协边假设。这表明，在对称一元n-范畴中取值的拓扑量子场论与该n-范畴的完全可对偶化对象是双射的。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带主义假说。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。这个改进的理论允许系数系统是对称的么半n-范畴，推广了以前系数是n-圆盘代数的因式分解同调。它们增强的因式分解同调为基于分层模的场论中的局部性提供了一个新的基础，而不是自20世纪80年代以来形成局部性基础的Morse理论和外科手术表示。这项工作的技术基础是家族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.