Collaborative Research: RUI: Knotting Phenomena in Contact and Symplectic Topology

奖项：DMS-0909273， dms -0909021首席研究员：Joshua M. Sabloff， Lisa traynor2009美国复苏与再投资法案（公法111-5）资助。主要研究人员建议通过探索拉格朗日和勒让子流形的结现象来研究柔性和刚性问题，这些问题是辛和接触拓扑特征的核心。通过拓扑透镜来接近辛几何和接触几何已经产生了一个年轻而蓬勃发展的学科，它探索了灵活性（当辛世界表现为拓扑时）和刚性（当辛世界表现为几何时）之间的边界。特别地，计划研究关于Legendrian子流形的Arnold猜想，拉格朗日盘的压缩现象，以及拉格朗日子流形的协同理论。我们将特别关注接触3流形中的Legendrian结和辛4流形中的lagrange曲面。为解决上述问题所使用的方法本身就值得关注。一方面，pi计划发展和使用由Gromov和Floer提出的拟全纯技术，并将其扩展到eliashberg、Givental和Hofer提出的辛场理论（SFT）框架。另一方面，pi计划开发和使用生成家庭技术。研究全纯不变量的结构和基于族的不变量的生成将使我们对几何柔性和刚性有更深入的了解；在这些不同类型的变量之间建立联系将增加对每个变量的理解。辛拓扑和接触拓扑作为经典力学和几何光学的语言在物理学中有其根源。在与这些根保持联系的同时，辛拓扑学和接触拓扑学已经发展成为一个中心数学领域，它结合了几何学（测量科学）和拓扑学（空间形状的研究）的特点。这个领域有各种各样的应用，包括流体力学，微分方程，以及对我们所处的三维空间和四维时空的可能形状的研究。pi项目的目标是更好地理解辛拓扑和接触拓扑是如何介于几何和拓扑之间的，从而加强上述应用程序的基础。pi的研究活动将为本科生和研究生带来发现新数学的过程，pi与本科生的研究将作为一个教学实验室，将数学研究整合到pi的课程中。

AbstractAward: DMS-0909273, DMS-0909021Principal Investigator: Joshua M. Sabloff, Lisa TraynorThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The principal investigators propose to investigate flexibilityand rigidity questions that are central to the character ofsymplectic and contact topology by exploring knotting phenomenaof Lagrangian and Legendrian submanifolds. Approachingsymplectic and contact geometry through a topological lens hasgiven rise to a young and thriving discipline with interestingquestions that explore the boundary between flexibility (when thesymplectic world behaves topologically) and rigidity (when thesymplectic world behaves geometrically). In particular, the PIsplan to study the Arnold Conjecture for Legendrian submanifolds,squeezing phenomena for Lagrangian disks, and the cobordismtheory of Lagrangian submanifolds. Special attention will bepaid to Legendrian knots in contact 3-manifolds and Lagrangiansurfaces in symplectic 4-manifolds. The methods used inaddressing the aforementioned problems are of interest in theirown right. On one hand, the PIs plan to develop and usepseudo-holomorphic techniques initiated by Gromov and Floer, andexpanded to the Symplectic Field Theory (SFT) framework byEliashberg, Givental, and Hofer. On the other, the PIs plan todevelop and use generating family techniques. Investigationsinto the structure of the holomorphic and the generating familybased invariants will yield insight into geometric flexibilityand rigidity; making connections between these different types ofinvariants will increase understanding of each.Symplectic and contact topology have their roots in physics asthe language of classical mechanics and geometric optics. Whileremaining in touch with those roots, symplectic and contacttopology have blossomed into a central mathematical field thatcombines features of geometry (the science of measurement) andtopology (the study of the shape of space). This field has avariety of applications including fluid mechanics, differentialequations, and the study of the possible shape of the3-dimensional space and the 4-dimensional space-time in which welive. The goal of the PIs' project is to achieve a betterunderstanding of how symplectic and contact topology sit betweengeometry and topology, and thereby strengthening the foundationfor the aforementioned applications. The PIs'research activitieswill bring a diverse set of students at both the undergraduateand graduate levels into the process of discovering newmathematics, with the PIs' research with undergraduates servingas a pedagogical laboratory for integrating mathematical researchinto the PIs' curricula.