Contact geometry, Heegaard Floer homology and open book decompositions

主要研究者致力于加深我们对三维流形上的几何对象（称为接触结构）的理解。 近年来，接触结构在几个长期存在的问题的解决中表现出突出的特点后，已经成为数学兴趣的前沿。 这个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。 自从大约十年前引入以来，Heegaard Floer理论已经彻底改变了对纽结，3-流形和光滑4-流形的研究。 该项目旨在更好地了解接触结构的几何特性如何在Heegaard Floer不变量的代数形式中体现出来。 该项目的第二个目标是研究接触结构的几何特征和支持它们的开卷分解的拓扑性质之间的联系。 具体而言，主要研究者的目的是开发障碍接触结构，支持属一，并找到下限的结合数。 最后，这个项目旨在扩大我们对勒让德和横结理论的理解。 为了实现这一目标，主要调查的目的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分类勒让德和横向代表在一个广泛的类结types.The主要调查旨在扩大我们的理解3和4维空间通过研究几何对象称为接触结构。 接触结构最早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。 它们是一种工具，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。 研究这些空间的技术的发展最终有助于我们了解我们自己宇宙的拓扑和几何特征。

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.