Heegaard Splitting and Topology of 3-Manifolds

三维流形是以我们生活的三维空间为模型的物体。甜甜圈和空间宇宙都是三种流形的例子。这些物体在物理和其他自然科学中的许多环境中自然出现，并可以用来模拟许多有趣的现象。本项目的主要目的是研究三维流形的数学性质。PI计划研究数学的一个分支--低维拓扑学中的一些核心问题。这些问题涉及三维流形在特定地图下的变化，以及被称为外科手术的手术。PI使用的主要工具是一种名为Heegaard Split的拓扑结构，它是将复杂的三维流形分解为沿二维曲面的较简单的部分。本研究针对低维拓扑和纽结理论中的一些基本问题进行研究。它对科学研究的其他领域也有潜在的影响，例如DNA的拓扑结构。在这个项目中，PI将研究三个流形的拓扑。该项目由三个主要部分组成。第一部分是探索一种证明Berge猜想的新方法。Berge猜想可以分为两个部分：前半部分证明了Berge猜想关于隧道数为1的纽结，后半部分证明了如果三维球面上的一个非平凡纽结允许进行透镜空间Dehn手术，则该纽结一定有隧道数1。PI和他的合作者对上半场进行了深入的研究。同样的方法可能导致对Berge猜想的后半部分的证明。研究的第二部分是关于Heegaard亏格和一次映射的一个长期存在的猜想。私家侦探将使用特殊类型的手术来调查这一猜测。研究的最后一部分是研究三维流形拓扑中关于Heegaard分裂和曲线复形的几个基本问题。PI计划开发新的工具和使用他以前工作中的技术来实现这些目标。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Three-manifolds are objects modeled on the three-dimensional space that we live in. A donut and the spatial universe are both examples of three-manifolds. These objects arise naturally in many contexts in physical and other natural sciences, and can be used to model many interesting phenomena. The main goal of this project is to study the mathematical properties of three-manifolds. The PI plans to investigate some central questions in a branch of mathematics, known as low dimensional topology. These questions are concerned with how three-manifolds change under certain maps as well as operations called surgeries. The major tool that the PI uses is a topological structure called Heegaard splitting, which is a decomposition of a complicated three-manifold into simpler pieces along a two-dimensional surface. This research targets some of the fundamental questions in low-dimensional topology and knot theory. It also has a potential impact on other areas of scientific investigations, such as the topological structures of DNA. In this project, the PI will study the topology of three-manifolds. The project has three major parts. The first part is to explore a new approach to proving the Berge Conjecture. The Berge Conjecture can be divided into two halves: the first half is to prove the Berge Conjecture for knots with tunnel number one, and the second half is to show that if a nontrivial knot in the three-sphere admits a lens-space Dehn surgery, then the knot must have tunnel number one. The PI and his collaborators have carried out an in-depth study on the first half. The same approach may lead to a proof of the second half of the Berge Conjecture. The second part of the research is to study a long-standing conjecture concerning Heegaard genus and degree-one map. The PI will investigate this conjecture using special type of surgeries. The objective of the last part of the research is to study several fundamental questions in three-manifold topology concerning Heegaard splittings and curve complex. The PI plans to develop new tools and use techniques from his previous work to achieve these goals.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.