Contact geometry, Heegaard Floer homology and open book decompositions

主要研究者努力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，接触结构在解决了几个长期存在的猜想后，已经成为数学研究的前沿。这个项目的第一个目标是探索连接接触结构和heegard flower不变量的连接。自从大约十年前Heegaard flower理论被引入以来，它已经彻底改变了对结、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性质如何在heeggaard flower不变量的代数形式中留下印记。该项目的第二个目标是研究接触结构的几何特征和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1的接触结构的障碍，并找到结合数的下界。最后，该项目旨在拓宽我们对Legendrian和横向结理论的理解。为了实现这一点，主要研究者的目标是开发新的勒让德和横向不变量，并应用这些不变量和其他已知的不变量来分类勒让德和横向代表在一个广泛的结类型类别中。主要研究者试图通过研究被称为接触结构的几何对象来扩大我们对三维和四维空间的理解。接触结构最早出现在物理学中是通过汉密尔顿、惠更斯和雅可比对几何光学的研究。它们为研究光学、经典力学和热力学提供了一种自然语言，并在物理学和数学的许多子领域中得到了应用。它们是一种工具，可以用来探测三维和四维空间，以更好地了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的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.