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.

接触结构是自然物体，出生于两个世纪前，在有关解决微分方程的谎言中，有关热力学的吉布斯，有关几何学光学的Huygens以及有关经典力学的huygens。许多数学家都对他们进行了研究，并且似乎涉及数学和物理学的各个领域，但是只有在过去的几十年中，他们才进入数学的前景。这是由于领域的显着突破，导致了一种丰富而美丽的理论，在数学和科学和工程学中都有许多应用。在这个项目中，PI将考虑有关具有接触结构的各种空间的各种问题，重点关注它们内部的对象，它们之间的关系以及它们上的其他结构。这不仅将进一步了解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还将花费大量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将通过各种技术调查接触和合成结构，但专注于他们的子曼群和与Riemannian指标的联系。回想一下，在维度3中，了解触点歧管中的传奇和横向结与我们对接触结构的理解及其与拓扑的微妙联系的进步息息相关。例如，接触结构存在的第一个证据来自横向结的手术，而著名的紧密二分法与二分法的二分法归结为legendrian或横向结的类型。 PI将继续对3个流形的这种结进行调查，重点关注其定性特征。还回想起，接触几何学上的许多重要概念都以接触结构的子序列表示（例如，giroux扭转，开放式书籍分解等）。试图了解这些各种子术如何相互作用以及各种手术结构如何影响它们将是PI的另一个重点。 PI还将考虑较少已知的较高维度接触歧管。在这里，将考虑有关接触亚策略（横向结的概括）和各向同性亚策略的存在和同位素分类的基本问题 - 手术结构以及它们如何影响接触歧管的各种特性。长期以来，人们已经众所周知，里曼尼亚的指标与流形的平滑拓扑具有很深的联系，最近已经表明，接触结构也可以。 PI将继续探索这两种几何结构之间的关系，目的是查看在适应它们的Riemannian指标中反映的接触结构的关键特性（例如紧密度）。希望这将使人们对3个歧管上的接触结构的一般情况有更完整的了解，并创建新的工具来研究更高的维度接触歧管。 PI还将通过在某些非琐事的情况下明确验证它们，并在某些一般环境中明确验证埃里亚斯贝格的最新猜想，以明确验证它们的存在，并探索归纳方法来证明它们。此奖项反映了NSF的法规任务，并被认为通过评估值得评估。利用基金会的知识分子和更广泛的影响审查标准。

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

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.

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