Submanifolds and Metrics in Contact Geometry

接触几何中的子流形和度量

• 批准号: 1608684
• 负责人: John Etnyre
• 金额: $ 31.76万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608684&HistoricalAwards=false
• 关键词: Submanifolds; Metrics; Contact; Geometry

Contact structures on manifolds are natural objects, born over two centuries ago, in the work of Huygens, Hamilton, and Jacobi, on geometric optics. 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 contact topology, resulting in a rich and beautiful theory with many applications. Studying subsets and their interactions with such structures was instrumental in the understanding of three-dimensional spaces, and it led to profound progress. The Principal Investigator will now extend this to higher dimensions, where this exploration is likely to prove equally illuminating. The Principal Investigator will also continue to study properties of contact structures on low-dimensional spaces and their interaction with topology and Riemannian geometry, and he will train the next generation of researchers by working with a large group of graduate students and organizing conferences and seminars.The research supported by this award will focus on problems centered around three broad topics: the interactions of contact geometry and topology in low dimensions, properties and constructions of contact manifolds in higher dimensions, and connections between contact geometry and the more familiar Riemannian geometry. In low dimensions the main motivating question is to determine which three-manifolds admit a tight contact structure. Currently quite a bit is known about this question, but very little is known about it for hyperbolic homology spheres. This problem will be studied using a variety of techniques, from convex surfaces, to holomorphic curves and Riemannian geometry. In addition, understanding interactions between various properties a contact structure can have will be studied. While much is known about contact geometry in low dimensions, there is very little known in higher dimensions. The principal investigator will study constructions and properties of high-dimensional contact manifolds. The starting point for this will be the study of isotropic and contact submanifolds of contact manifolds. Such considerations have led to a wealth of information in low-dimensions and it is expected to be similarly fruitful in higher dimensions as well. In the past few years there have been some interesting and subtle connections between contact geometry and Riemannian geometry. The Principal Investigator will explore this further hoping to find contact geometric analogs of classical results relating topology to Riemannian geometry.

歧管上的接触结构是自然物体，出生于两个世纪前，在Huygens，Hamilton和Jacobi在几何光学方面的作品中。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。这是由于接触拓扑中的显着突破，从而带来了许多应用的丰富而美丽的理论。研究子集及其与此类结构的相互作用对三维空间的理解起着重要作用，这导致了深刻的进步。主要研究人员现在将其扩展到更高的维度，在此探索可能同样启发时。首席研究者还将继续研究接触结构在低维空间上的特性及其与拓扑结构和riemannian几何形状的相互作用，他将通过与大量的研究生合作以及组织会议和研讨会来培训下一代研究人员。该奖项支持的研究将集中于围绕三个广泛主题的问题：在较高维度的低维度，属性和构造中的接触几何学和拓扑之间的相互作用，以及接触几何形状与更熟悉的Riemannian几何形状之间的联系。在低维度中，主要的激励问题是确定哪种三序列的接触结构紧密。目前，对这个问题有很多了解，但是对于双曲线同源性领域知之甚少。从凸表面到全态曲线和riemannian几何形状，将使用多种技术研究此问题。此外，将研究了解触点结构之间的各种属性之间的相互作用。虽然对低维度的接触几何形状知之甚少，但在较高的维度中鲜为人知。主要研究者将研究高维接触歧管的结构和特性。此的起点将是对触点歧管的各向同性和接触式子延伸的研究。这些考虑因素导致了低维度的大量信息，预计它在更高的维度也将同样富有成果。在过去的几年中，接触几何形状与黎曼几何形状之间存在一些有趣而微妙的联系。首席研究者将探索这一进一步的希望，以找到与拓扑结构与Riemannian几何形状有关的经典结果的接触几何类似物。

项目成果

Symplectic hats

• DOI: 10.1112/topo.12258
• 发表时间: 2020-01
• 期刊: Journal of Topology
• 影响因子: 1.1
• 作者: John B. Etnyre;Marco Golla

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