Collaborative Research: Factorization homology and the cobordism hypothesis

拓扑学领域研究流形，空间或时空的抽象概念。自世纪开始，流形的拓扑学就与理论物理学交织在一起。这种联系随着爱因斯坦的广义相对论而增长，该理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增长，量子场论产生了最有辨识力的流形不变量。在这里，不变量意味着一次分析所有流形的统一技术。配边假设是1990年代中期提出的流形不变量的分类，称为拓扑量子场论，可能来自理论物理学。我们的项目证明了配边假说。主要研究者通过发展一种新的代数方法，因式分解同调，从局部不变量理论上组装全局不变量。当代拓扑量子场的中心结构原则是协边假说，由Lurie，霍普金斯和Baez-Dolan发展。这断言，在对称monoidal n-范畴中取值的拓扑量子场论与该n-范畴的完全可对偶对象是双射的。特别地，它断言场论由它在一点上的值决定。我们的项目证明了配边假说。主要研究者通过进一步发展，然后应用因子分解同调理论来做到这一点。这个增强的理论允许系数系统是对称monoidal n-范畴，推广了以前的因子分解同调，其系数是n-圆盘代数。他们的增强因式分解同源性提供了一个新的基础上局部性领域理论的基础上模量的分层，而不是莫尔斯理论和外科手术介绍已形成的基础上局部性自阿蒂亚的公理从20世纪80年代。这项工作的技术基础是这种微分拓扑分层的家庭。

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.