Surfaces in 4-manifolds and modified surgery theory

拓扑学是研究空间直到连续变形的学科。一维、二维和三维的空间以及五维和更大维度的空间是很好理解的。然而，在第四维度，仍有许多未知之处。PI计划使用在高维拓扑学中取得成功的技术来提高对四维空间的理解。除了该项目的研究方面，PI还将组织会议，为他的课程撰写讲稿，维护一个会议清单网站，撰写研究水平调查并指导初级研究人员。该项目专注于拓扑4维流形，位于低维拓扑和高维外科理论的交叉点上。根据所涉及的类别，维度4中的拓扑有两种截然不同的风格：平滑或拓扑。虽然光滑的4-流形仍然非常神秘，但拓扑范畴是服从广义分类的。PI的目标是分类具有固定边界和良好基本群的拓扑4-流形，以及分类具有固定结点群的4-流形中的局部平坦曲面。特别是，PI计划证明不可定向的解开猜想：如果4-球面中的不可定向局部平坦表面有二阶纽结群，那么它是拓扑无结的。这一奖项反映了NSF的法定使命，并通过使用基金会的智力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.