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.

主要研究者努力加深我们对3个曼尼属的几何对象的理解，称为接触结构。 近年来，在解决了几种长期存在的猜想中，接触结构已成为数学兴趣的最前沿。 该项目的第一个目标是探测连接连接，将联系人结构和Heegaard浮动不变性链接起来。 由于大约是十年前的引言，因此Heegaard Floer理论彻底改变了对结，3个manifolds和Smooth 4 manifolds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浮动不变的代数形式上印象自己。 该项目的第二个目标是研究接触结构的几何特征与支持它们的开放书籍分解的拓扑特性之间的联系。 具体而言，原则研究者旨在发展障碍物，以接触一个支持属的结构，并找到结合数的下限。 最后，该项目旨在扩大我们对传奇和横向结理论的理解。 为此，主要研究者旨在开发新的传奇和横向不变式，并应用这些和其他已知的不变式，以在广泛的打结类型中对传奇和横向代表进行分类。原则研究者旨在通过研究称为3和4维的空间来扩大我们对3和4维空间的理解，以研究称为接触接触结构。 接触结构首先是通过汉密尔顿，霍根斯和雅各比在几何光学元件上的工作中出现的。 它们提供了一种自然语言，用于研究光学，经典力学和热力学，并在许多物理和数学子领域都有应用。 它们是可以用来探测3和4维空间的工具，以更好地了解其形状和几何结构。 研究这些空间的技术的发展最终有助于告知我们自己宇宙的拓扑和几何特征。