Contact homology and String topology

接触同调和弦拓扑

• 批准号: 0707091
• 负责人: Michael Sullivan
• 金额: $ 9.71万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707091&HistoricalAwards=false
• 关键词: Contact; homology; String; topology

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.

首席研究员计划计算并应用勒让接触同调来研究单射流空间中的勒让子流形。这种同源性基于辛流形中的全纯曲线。在特殊情况下，当勒让德子流形是光滑子流形的共法升力时，首席研究员计划将接触同源性与光滑子流形的开弦拓扑上正在构建的 A 无穷大结构联系起来。该项目的下一步是将更一般的结构从开弦拓扑转化为接触几何，从而制定辛场论的相对版本作为勒让德接触同调的推广。通过这种共正规结构，研究人员计划利用同源性来研究欧几里德空间或射影空间中的光滑子流形，例如结。首席研究员还将应用勒让接触同调来研究勒让子流形的更高同伦群。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。接触几何在物理学中多次出现，从光学到热力学再到经典力学。例如，遵守力学中最小作用原理的粒子转换为接触几何（或其密切相关的领域，辛几何）中的对象，称为全纯曲线。其中一些与物理学的联系已经为人所知几个世纪了。然而，直到最近，接触几何才从数学界的重大进步中受益。具体来说，研究这些全纯曲线已经在接触刚度和接触动力学方面取得了一些强有力的、有时令人惊讶的发现。在过去的几年中，围绕应用这些全纯曲线来研究三维和四维拓扑中未解决的问题，一个活跃的研究领域已经发展起来。如前所述，低维拓扑中的这些问题似乎与全纯曲线无关。然而，从最近的结果来看，技术已经牢固地建立了联系。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结理论联系起来。