Submanifolds and Cobordisms in Contact and Symplectic Topology

接触结构是自然的物体，诞生于两个多世纪以前，在李关于解微分方程的工作中，在吉布斯关于热力学的工作中，在惠更斯关于几何光学的工作中，在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，似乎涉及数学和物理的不同领域，但直到最近几十年它们才进入数学的前景。这是由于该领域的重大突破，产生了一个丰富而美丽的理论，在数学和科学与工程领域都有许多应用。在这个项目中，PI将考虑关于各种具有接触结构的空间的各种问题，重点关注其中的物体，它们之间的关系以及它们上面的其他结构。这不仅会加深我们对该领域的理解，而且会对其他研究领域产生影响。PI还将投入大量时间帮助研究生和博士后学者成为该领域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，但重点是它们的子流形和与黎曼度量的联系。回想一下，在三维空间中，对接触流形中的勒让德结和横结的理解与我们对接触结构及其与拓扑学的微妙联系的理解携手并进。例如，接触结构存在的第一个证据来自于对横节的手术，而著名的紧与过扭二分法可以归结为接触结构支持的Legendrian节或横节的类型。PI将继续研究3个流形中的这种结，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示的（例如，吉鲁扭转，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不同的手术结构如何影响它们将是PI的另一个重点。PI也将考虑更高维的接触流形，在那里我们知道的要少得多。在这里，关于接触子流形（横向结的推广）和各向同性子流形的存在和同位素分类的基本问题将被考虑，以及手术结构和它们如何影响接触流形的各种性质。黎曼度量长期以来一直被认为与流形的光滑拓扑有很深的联系，最近也证明了接触结构也是如此。PI将继续探索这两种几何结构之间的关系，目的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。这将有希望使我们对三维流形上的接触结构有一个更全面的了解，并为研究高维接触流形创造新的工具。PI还将探索Eliashberg最近关于辛结构存在的猜想，通过在一些非平凡的情况下明确地验证它们，并探索在一些一般情况下证明它们的归纳方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.