Open string topology and holomorphic curves

主要研究者计划研究代数结构，这些代数结构自然来自于切触流形中的勒让德子流形以及光滑流形中的任意子流形的不变量。更具体地说，勒让德子流形的不变量来自于某些辛流形中的全纯曲线，而光滑子流形的不变量则来自于开弦拓扑，而开弦拓扑是以流形的路空间的交理论为基础的。理论具有由拓扑场论指导的代数结构。本项目的一部分是改进微分分次运算代数的语言，以便用类似的代数语言来表达这两种理论。越来越多的证据表明，流形的弦拓扑与其在辛余切丛或接触单位余切丛中的提升的全纯不变量之间存在联系。主要研究者计划定义和计算其中的一些不变量，希望在光滑子流形是欧几里得三维空间中的一个纽结时，将这两个理论联系起来。首席研究员还提出了几个与三维空间中的纽结和四维空间中的光滑曲面相关的计算项目，也是使用全纯曲线定义的。接触几何在物理学中有很多出现，从光学到热力学再到经典力学。例如，从力学的最小作用原理出发，将粒子转化为接触几何（或与之密切相关的领域，辛几何）中的物体，称为全纯曲线，研究这些全纯曲线已经导致了一些关于接触刚性和接触动力学的强有力的，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。纽结理论的一个中心问题是确定纽结的复杂性，这反过来又需要开发可计算的非平凡纽结不变量。最近，全纯曲线再次提供了大量这样有用的纽结不变量。首席研究员计划开发其他纽结不变量，受到弦理论的启发，that should be connected连接to these invariants不变量based基础on holomorphic全纯curves曲线.

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.