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

摘要奖：DMS-0909273，DMS-0909021主要调查者：约书亚·M·萨布洛夫，丽莎·特雷诺该奖项由2009年美国复苏和再投资法案(公法第111-5条)资助。主要研究者建议通过探索Lagrangian和Legendrian子流形的纽结现象来研究对辛拓扑和接触拓扑学特征至关重要的柔韧性和刚性问题。通过拓扑透镜探讨辛几何和接触几何，产生了一门年轻而蓬勃发展的学科，其有趣的问题是探索灵活性(当辛世界表现为拓扑时)和刚性(当辛界表现为几何行为时)之间的界限。特别是，我们计划研究Legendrian子流形的Arnold猜想，拉格朗日圆盘的挤压现象，以及拉格朗日子流形的余边理论。我们将特别注意接触三维流形中的Legendrian纽结和辛四维流形中的拉格朗日曲面。解决上述问题所使用的方法与它们各自的权利有关。一方面，PI计划发展和使用由Gromov和Floer开创的伪全纯技术，并由Eliashberg，Givental和Hofer扩展到辛场论(SFT)框架。另一方面，PI计划开发和使用生成族技术。对全纯不变量和基于生成族的不变量的结构的研究将使我们深入了解几何的灵活性和刚性；在这些不同类型的变量之间建立联系将增加对每一种变量的理解。辛拓扑和接触拓扑学作为经典力学和几何光学的语言，在物理学中有其根源。在与这些根源保持联系的同时，辛拓扑和接触拓扑学已经发展成为一个结合了几何(测量科学)和拓扑学(研究空间形状)特征的中心数学领域。这一领域有着广泛的应用，包括流体力学、微分方程式，以及研究三维空间和四维时空的可能形状。PI项目的目标是更好地理解辛和接触拓扑如何位于几何和拓扑之间，从而加强上述应用的基础。PIS的研究活动将把本科生和研究生阶段的不同学生带入发现新数学的过程，PIS对本科生的研究作为将数学研究纳入PIS课程的教学实验室。

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.