Contact geometry, Heegaard Floer homology and open book decompositions

接触几何、Heegaard Floer 同调和开卷分解

• 批准号: 1205933
• 负责人: David Vela-Vick
• 金额: $ 13.16万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-15 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205933&HistoricalAwards=false
• 关键词: Contact; geometry; Heegaard; Floer; homology

The principle investigator endeavors to deepen our understanding of geometric objects on 3-manifolds called contact structures. In recent years, contact structures have moved to the forefront of mathematical interest after featuring prominently in the resolution of several long-standing conjectures. The first goal of this project is to probe connections linking contact structures and Heegaard Floer invariants. Since it's introduction roughly a decade ago, Heegaard Floer theory has revolutionized the study of knots, 3-manifolds and smooth 4-manifolds. This project seeks to better understand how geometric properties of contact structures imprint themselves in the algebraic formalism of Heegaard Floer invariants. The project's second goal is to study connections between geometric characteristics of contact structures and topological properties of the open book decompositions that support them. Specifically, the principle investigator aims to develop obstructions to contact structures having support genus one and to find lower bounds for the binding number. Finally, the project seeks to broaden our understanding of Legendrian and transverse knot theory. To accomplish this, the principle investigator aims to develop new Legendrian and transverse invariants and to apply these and other known invariants to classify Legendrian and transverse representatives in a broad class of knot types.The principle investigator seeks to broaden our understanding of 3 and 4-dimensional spaces by studying geometric objects called contact structures. Contact structures first appeared in physics through the work of Hamilton, Huygens and Jacobi on geometric optics. They provide a natural language for studying optics, classical mechanics and thermodynamics, and have applications in many subfields of physics and mathematics. They are a tool one can use to probe 3 and 4-dimensional spaces to better understand their shape and geometric structure. The development of techniques for studying these spaces ultimately helps to informs us about the topological and geometric characteristics of our own universe.

主要研究者致力于加深我们对称为接触结构的 3 流形上的几何对象的理解。 近年来，接触结构在解决几个长期存在的猜想中发挥了重要作用后，已成为数学兴趣的前沿。 该项目的第一个目标是探测连接接触结构和 Heegaard Floer 不变量的连接。 自大约十年前推出以来，Heegaard Floer 理论彻底改变了纽结、3 流形和光滑 4 流形的研究。 该项目旨在更好地理解接触结构的几何特性如何在 Heegaard Floer 不变量的代数形式主义中留下印记。 该项目的第二个目标是研究接触结构的几何特性与支持它们的开卷分解的拓扑特性之间的联系。 具体来说，主要研究者的目标是开发具有支持属一的接触结构的障碍，并找到结合数的下限。 最后，该项目旨在扩大我们对勒让德理论和横结理论的理解。 为了实现这一目标，首席研究员的目标是开发新的勒让德和横向不变量，并应用这些和其他已知的不变量来对广泛的结类型中的勒让德和横向代表进行分类。首席研究员试图扩大我们对 3 和 4- 的理解通过研究称为接触结构的几何对象来研究维度空间。 接触结构首次出现在物理学中是通过汉密尔顿、惠更斯和雅可比在几何光学方面的工作。 它们为研究光学、经典力学和热力学提供了一种自然语言，并在物理和数学的许多子领域都有应用。 它们是一种可以用来探测 3 维和 4 维空间以更好地理解它们的形状和几何结构的工具。 研究这些空间的技术的发展最终有助于我们了解宇宙的拓扑和几何特征。