Heegaard Splitting and Topology of 3-Manifolds

三维流形是在我们生活的三维空间中建模的物体。 甜甜圈和空间宇宙都是三维流形的例子。这些对象在物理和其他自然科学的许多背景下自然出现，并可用于模拟许多有趣的现象。本项目的主要目标是研究三维流形的数学性质。PI计划调查数学分支中的一些中心问题，称为低维拓扑。这些问题与三维流形在某些映射下如何变化以及称为外科手术的操作有关。PI使用的主要工具是一种称为Heegaard分裂的拓扑结构，它是一种将复杂的三维流形分解成沿着二维表面的简单部分。本研究的目标是低维拓扑学和纽结理论中的一些基本问题。它还对其他科学研究领域产生潜在影响，例如DNA的拓扑结构。在这个项目中，PI将研究三维流形的拓扑结构。该项目有三个主要部分。第一部分是探索证明Berge猜想的一种新方法。Berge猜想可以分为两部分：第一部分是证明隧道数为1的纽结的Berge猜想，第二部分是证明如果三个球面中的一个非平凡纽结允许透镜空间的Dehn手术，那么这个纽结必须有隧道数为1。PI及其合作者对上半年进行了深入研究。同样的方法可能导致Berge猜想的后半部分的证明。第二部分研究了一个关于Heegaard亏格和一次映射的猜想。PI将使用特殊类型的手术调查该推测。本文的最后一部分研究了三流形拓扑中与Heegaard分裂和曲线复形有关的几个基本问题。PI计划开发新的工具，并使用他以前工作中的技术来实现这些目标。该奖项反映了NSF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。

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.