Submanifolds and Cobordisms in Contact and Symplectic Topology

接触拓扑和辛拓扑中的子流形和配边

• 批准号: 1906414
• 负责人: John Etnyre
• 金额: $ 46.82万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906414&HistoricalAwards=false
• 关键词: Submanifolds; Cobordisms; Contact; Symplectic; Topology

Contact structures are natural objects, born over two centuries ago, in the work of Lie concerning solving differential equations, Gibbs concerning thermodynamics, Huygens concerning geometric optics, and Hamilton concerning classical mechanics. They have been studied by many mathematicians and seem to touch on diverse areas of mathematics and physics, but only in the last few decades have they moved into the foreground of mathematics. This is due to the remarkable breakthroughs in field, resulting in a rich and beautiful theory with many applications both inside mathematics and to science and engineering. In this project the PI will consider a variety of questions about various spaces with contact structures, focusing on objects inside of them, relations between them, and other structures on them. This will not only further our understanding of the field, but also its impacts on other areas of study. The PI will also devote significant time to helping graduate students and postdoctoral scholars become productive researchers in the field.The PI will investigate contact and symplectic structures through a variety of techniques, but focusing on their submanifolds and connections to Riemannian metrics. Recall that in dimension 3 understanding Legendrian and transverse knots in a contact manifold has gone hand in hand with advances in our understanding of contact structures and their subtle links with topology. For example the first proof of existence of contact structures came from surgery on transverse knots and the famed tight versus overtwisted dichotomy comes down to the types of Legendrian or transverse knots a contact structure supports. The PI will continue his investigations of such knots in 3 manifolds, focusing on qualitative features of them. Also recall, that many important concepts in contact geometry are expressed in terms of submanifolds of the contact structure (for example, Giroux torsion, open book decompositions, etc). Trying to understand how these various submanifolds interact and how various surgery constructions affect them will be another focus of the PI. The PI will also consider higher dimensional contact manifolds where much less is known. Here, basic questions about the existence and isotopy classification of contact submanifolds (a generalization of transverse knots) and isotropic submanifolds will be considered - as will surgery constructions and how they affect various properties of contact manifolds. Riemannian metrics have long been known to have deep connections with the smooth topology of manifolds and more recently it has been shown that contact structures do as well. The PI will continue to explore relations between these two geometric structures with the goal of seeing key properties of a contact structure (such as tightness) reflected in Riemannian metrics that are adapted to them. This will hopefully lead to a more complete understanding of the general picture of contact structures on 3 manifolds and create new tools for studying higher dimensional contact manifolds. The PI will also explore recent conjectures of Eliashberg about the existence of symplectic structures by explicitly verifying them in some nontrivial cases and exploring inductive approaches to proving them in some general settings.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.

接触结构是两个多世纪以前诞生的自然对象，在李关于解微分方程、吉布斯关于热力学、惠更斯关于几何光学和汉密尔顿关于经典力学的工作中。它们被许多数学家研究过，似乎触及了数学和物理的不同领域，但只是在过去的几十年里，它们才进入数学的前台。这是由于领域的重大突破，产生了丰富而美丽的理论，在数学和科学工程中有许多应用。在这个项目中，PI将考虑各种关于接触结构的各种空间的问题，重点是它们内部的物体，它们之间的关系以及它们上的其他结构。这不仅将进一步加深我们对该领域的理解，而且还将对其他研究领域产生影响。PI还将投入大量的时间来帮助研究生和博士后学者成为该领域的富有成效的研究人员。PI将通过各种技术研究接触和辛结构，但专注于它们的子流形和与黎曼度量的联系。回想一下，在3维中，对接触流形中勒让德结和横结的理解，与我们对接触结构及其与拓扑的微妙联系的理解的进步是齐头并进的。例如，接触结构存在的第一个证据来自横结的手术，著名的紧与过扭曲二分法归结为接触结构支持的勒让德或横结的类型。PI将继续对3个流形中的此类结进行调查，重点关注它们的定性特征。还记得，接触几何中的许多重要概念都是用接触结构的子流形来表达的（例如，吉鲁挠，开卷分解等）。试图了解这些不同的子流形如何相互作用，以及各种手术构造如何影响它们将是PI的另一个重点。PI还将考虑更高维的接触流形，其中知之甚少。在这里，基本问题的存在性和同伦分类的接触子流形（一个推广的横向结）和各向同性子流形将被认为是将手术建设和他们如何影响各种性质的接触流形。黎曼度量长期以来被认为与流形的光滑拓扑有很深的联系，最近又被证明接触结构也有很深的联系。PI将继续探索这两种几何结构之间的关系，目标是看到接触结构的关键属性（如紧密性）反映在适应它们的黎曼度量中。这将有助于我们更全面地理解三维流形上的切触结构，并为研究高维切触流形创造新的工具。PI还将探索Eliashberg最近关于辛结构存在性的理论，通过在一些非平凡的情况下明确验证它们，并探索在一些一般情况下证明它们的归纳方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Symplectic fillings and cobordisms of lens spaces

• DOI: 10.1090/tran/8474
• 发表时间: 2020-06
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: John B. Etnyre;Agniva Roy

Legendrian contact homology in $\mathbb{R}^3$

$mathbb{R}^3$ 中的传奇接触同源性

• DOI: 10.4310/sdg.2020.v25.n1.a4
• 发表时间: 2020
• 期刊: Surveys in Differential Geometry
• 作者: Etnyre, John B.;Ng, Lenhard L.
• 通讯作者: Ng, Lenhard L.

On 3-manifolds that are boundaries of exotic 4-manifolds

在作为奇异 4 流形边界的 3 流形上

• DOI: 10.1090/tran/8586
• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Etnyre, John;Min, Hyunki;Mukherjee, Anubhav
• 通讯作者: Mukherjee, Anubhav

Knot Colorings: Coloring and Goeritz Matrices

结着色：着色和 Goeritz 矩阵

• DOI: 10.1080/00029890.2023.2174352
• 发表时间: 2023
• 期刊: The American Mathematical Monthly
• 作者: Kolay, Sudipta

Contact surgery and symplectic caps

接触手术和辛帽

• DOI: 10.1112/blms.12332
• 发表时间: 2020
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Conway, James;Etnyre, John B.
• 通讯作者: Etnyre, John B.