Open string topology and holomorphic curves

主要研究接触流形中勒让子流形和光滑流形中任意子流形的不变量自然产生的代数结构。更具体地说，勒让子流形的不变量来自某些辛流形中的全纯曲线，而光滑子流形的不变量是基于流形的路空间的交集理论的开弦拓扑。这两种理论都有拓扑场理论指导的代数结构。本项目的一部分是改进微分分次算子代数的语言，以便用类似的代数语言来表示这两个理论。越来越多的证据表明流形的弦拓扑与它在辛余切丛或接触单位余切丛中的升力的全纯不变量之间存在联系。主要的研究者计划定义和计算一些不变量，当光滑子流形是欧氏空间中的一个纽结时，希望将这两个理论联系起来。主要研究人员还提出了几个与三维空间中的纽结和四空间中的光滑曲面有关的计算方案，也是用全纯曲线定义的。接触几何在物理学中有许多出现，从光学到热力学，再到经典力学。例如，遵循力学中最小作用量原理的粒子后来变成接触几何(或其密切相关的场，辛几何)中的物体，称为全纯曲线。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大的，有时甚至是令人惊讶的发现。纽结理论在理解宇宙的大小方面以及限制在小空间中的长DNA链方面都有应用。纽结理论中的一个中心问题是确定纽结的复杂性，这反过来需要开发可计算的非平凡纽结不变量。最近，全纯曲线提供了大量这样的有用的纽结不变量。首席研究员计划在弦理论的推动下开发其他纽结不变量，这些纽结不变量应该基于全纯曲线与这些不变量相联系。

The Principal Investigator plans to study algebraic structures that arisenaturally from invariants of Legendrian submanifolds in contact manifoldsas well as arbitrary submanifolds in smooth manifolds.More specifically, the invariants for a Legendrian submanifold come fromholomorphic curves in certain symplectic manifolds, while the invariantsfor a smooth submanifold arise from open string topology whichis based on the intersection theory of the manifold's path space.Both theories have algebraic structures guided by topological field theory.Part of this project is to refine thelanguage of differential graded operad algebras to express the twotheories in the similar algebraic language.There is a growing body of evidence that suggests a connection between thestring topology of a manifold and the holomorphic invariants for its lift inthe symplectic cotangent bundle or contact unit cotangent bundle.The Principal Investigator plans to define and compute some of theseinvariants, hopefully leading to a connection between the two theorieswhen the smooth submanifold is a knot in Euclidean 3-space.The Principal Investigator also proposes several more computationalprojects related to knots in 3-space and smooth surfaces in 4-space,also defined using holomorphic curves.Contact geometry makes many appearances in physics, from opticsto thermodynamics to classical mechanics.For example, particles obeying the Least Action Principal from mechanicstranslate into objects in contact geometry (or its closely related field,symplectic geometry) known as holomorphic curves.Studying these holomorphic curves have led to some powerfuland sometimes surprising discoveries about contact rigidity and contactdynamics. Knot theory has applications in understanding large and small aspects of the universe, as well as long DNA strands confined to small space.A central question in knot theory is determiningthe complexity of knots which in turn requires developing computablenon-trivial knot invariants.Again holomorphic curves has recently provided a plethora of such usefulknot invariants.The Principal Investigator plans to develop other knot invariants,motivated by string theory, that should be connected to these invariantsbased on holomorphic curves.