Surfaces in 4-manifolds and modified surgery theory

拓扑学研究的是连续变形的空间。一、二、三维空间以及五维或更大的空间都很容易理解。然而，在四维空间，仍有许多未知之处。PI计划使用已经在高维拓扑中取得成功的技术来提高对四维空间的理解。除了项目的研究方面，PI还将组织会议，为他的班级撰写讲义，维护会议列表网站，撰写研究级调查并指导初级研究人员。该项目主要研究拓扑四维流形，处于低维拓扑和高维外科理论的交叉点。根据所涉及的类别，维度4中的拓扑有两种不同的风格：平滑的或拓扑的。虽然光滑4流形仍然非常神秘，但拓扑范畴可以进行广泛的分类。PI的目标是对具有固定边界和良好基群的拓扑4-流形进行分类，并对具有固定结群的4-流形中的局部平面进行分类。特别地，PI计划证明非定向解结猜想：如果一个非定向局部平面在4球上有二阶结群，那么它是拓扑解结的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is the study of spaces up to continuous deformation. Spaces of dimensions one, two and three are well understood, as are spaces in dimension five and greater. In dimension four however, much remains unknown. The PI plans to use techniques that have been successful in high dimensional topology to improve the understanding of four-dimensional spaces. In addition to the research aspects of the project, the PI will organize conferences, write lecture notes for his classes, maintain a conference listing website, write research-level surveys and mentor junior researchers. The project focuses on topological 4-dimensional manifolds and lies at the intersection between low dimensional topology and high dimensional surgery theory. Topology in dimension 4 has two distinct flavors according to the category involved: smooth or topological. While smooth 4-manifolds remain very mysterious, the topological category is amenable to broad classifications. The PI’s goals are to classify topological 4-manifolds with a fixed boundary and good fundamental group as well as to classify locally flat surfaces in 4-manifolds with a fixed knot group. In particular, the PI plans to prove the non-orientable unknotting conjecture: if a non-orientable locally flat surface in the 4-sphere has knot group of order two, then it is topologically unknotted.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.