Surfaces in 4-manifolds and modified surgery theory

拓扑学是研究空间的连续变形。一维、二维和三维的空间是很好理解的，五维和更大维的空间也是如此。然而，在第四维度中，仍有许多未知之处。PI计划使用在高维拓扑中已经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将组织会议，为他的班级写讲义，维护会议列表网站，编写研究级调查和指导初级研究人员。该项目的重点是拓扑四维流形和低维拓扑和高维手术理论之间的交叉点。根据所涉及的类别，第四维中的拓扑有两种不同的风格：光滑的或拓扑的。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本群的拓扑4-流形进行分类，以及对具有固定纽结群的4-流形中的局部平坦曲面进行分类。特别是，PI计划证明不可定向的unknotting猜想：如果一个不可定向的局部平面在4-sphere有二阶纽结群，那么它是拓扑unknotted。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.