Contact homology and String topology

首席调查者计划计算并应用Legendrian接触同调来研究单射空间中的Legendrian子流形。这种同调是基于辛流形中的全纯曲线。在Legendrian子流形是光滑子流形的法线升力的特殊情况下，主要研究者计划将接触同调与光滑子流形的开弦拓扑上的在建A-无穷结构联系起来。该项目的下一步将是将更一般的结构从开弦拓扑转换到接触几何，从而形成辛场理论的相对版本，作为Legendrian接触同调的推广。通过这种余正交构造，研究者计划利用同调来研究欧氏空间或射影空间中的光滑子流形，如纽结。主要研究者还将应用Legendrian接触同调来研究Legendrian子流形的高同伦群。全纯曲线和梯度流树之间的关系应该有助于该项目的许多计算。从光学到热力学再到经典力学，接触几何在物理学中出现了许多形式。例如，遵循力学中的最小作用量原理的粒子转化为接触几何(或其密切相关的场，辛几何)中的对象，称为全纯曲线。其中一些与物理学的联系早在几个世纪前就已为人所知；然而，直到最近，接触几何才从数学界的重大进步中受益。具体地说，研究这些全纯曲线导致了关于接触刚性和接触动力学的一些强有力的、有时令人惊讶的发现。在过去的几年里，一个活跃的研究领域围绕着应用这些全纯曲线来研究三维和四维拓扑中尚未解决的问题。如上所述，低维拓扑中的这些问题似乎与全纯曲线无关。然而，来自最近结果的技术已经牢固地建立了一种联系。首席研究员计划进一步研究全纯曲线的有效性，重点是将其与三维拓扑纽结理论联系起来。

The Principal Investigator plans to compute and apply Legendrian contact homology to study Legendrian submanifolds in one-jet spaces. This homology is based on holomorphic curves in symplectic manifolds. In the special case when the Legendrian submanifold is the conormal lift of a smooth submanifold, the Principal Investigator plans to relate the contact homology with an under-construction A-infinity-structure on the open string topology of the smooth submanifold. The next step in the project will be to translate more general structures from open string topology to contact geometry, thereby formulating a relative version of symplectic field theory as a generalization of Legendrian contact homology. Via this conormal construction, the investigator plans to use the homology to study smooth submanifolds, such as knots, in Euclidean or projective space. The Principal Investigator will also apply Legendrian contact homology to study higher homotopy groups of Legendrian submanifolds. The relationship between holomorphic curves and gradient flow trees should facilitate many of the project's computations.Contact geometry makes many appearances in physics, from optics to thermodynamics to classical mechanics. For example, particles obeying the Least Action Principal from mechanics translate into objects in contact geometry (or its closely related field, symplectic geometry) known as holomorphic curves. Some of these connections to physics have been known for centuries; however, only recently has contact geometry benefited from significant advances within the mathematical community. Specifically, studying these holomorphic curves have led to some powerful and sometimes surprising discoveries about contact rigidity and contact dynamics. In the last couple of years, an active area of research has evolved around applying these holomorphic curves towards studying unresolved problems in three and four-dimensional topology. As stated, these problems in low-dimensional topology seem to have nothing to do with holomorphic curves. Yet, the techniques from recent results have firmly established a connection. The Principal Investigator plans to further study the effectiveness of holomorphic curves, with an emphasis on connecting it to the theory of topological knots in three dimensions.