CAREER: Knot Theory and Dynamics in Contact Geometry

提案DMS-0239600 PI：John Etnyre，宾夕法尼亚大学标题：职业生涯：接触几何中的纽结理论和动力学摘要该提案的重点是研究所有（奇数）维中拓扑和接触几何之间的关系，并将接触几何技术应用于流体力学问题。拓扑和接触几何之间的许多联系是由Legendrian结（这些结与接触结构相切）介导的，因此所提出的研究的第一个主题是Legendrian结。作为这个建议的一部分，勒让德结的一般结构将被研究.预期的结果将是对某些Legendrian结和接触结构的各种分类结果;此外，更好地理解Legendrian手术（接触结构的重要手术构造）。我们也将用接触同调来研究更高维度的勒让德结。对于三维以上的接触结构或勒让德结，我们知之甚少。通过对这些维的勒让德结的研究，可以揭示接触结构的本质，而相应的研究也揭示了三维接触结构的许多问题。建议的研究的最后一部分集中在接触结构和流体力学之间的联系上，这是几年前由首席研究员和R。格里斯特在这里，我们将继续与Ghrist合作，以了解流体流动中何时以及何种类型的闭合流线。我们还将从接触拓扑的角度研究流体动力学不稳定性。这自然会导致研究流体流动的能量最小化以及接触与黎曼几何之间的关系。接触结构是两个多世纪前在几何光学和偏微分方程研究中诞生的非常自然的对象。几个世纪以来，接触结构已经触及数学和物理的许多不同领域，包括经典力学和热力学。在日常生活中，人们在滑冰时遇到接触几何，平行停车，使用冰箱，或者只是在一杯水中观看美丽的光线。许多伟大的数学家对这一课题做了大量的研究，但只是在最近一二十年里，它才成为数学的前沿.这种复兴是由于最近接触拓扑学的重大突破，产生了丰富而美丽的理论，具有许多应用。最近所有这些工作的最显著的特点是三维接触结构和拓扑之间的密切联系。因此，通过研究接触结构的抽象概念，人们可以了解我们所生活的宇宙的许多微妙的事情。例如，接触几何学的研究最近使我们对理想流体流动的理解有了一些意想不到的进展。首席研究员将探索接触结构和拓扑结构之间的连接在所有（奇数）维，通过接触几何继续他的理想化流体流动（流体力学）的研究，并分析有趣的新的acuptures concerningstring理论和接触几何。首席研究员还将从事几项教育工作，包括支持和鼓励研究生，以及创建介绍和调查材料，以将快速发展的接触几何领域带给更广泛的受众。

Proposal DMS-0239600PI: John Etnyre, University of PennsylvaniaTitle: CAREER: Knot Theory and Dynamics in Contact GeometryABSTRACTThe focus of this proposal is to study the relation between topologyand contact geometry in all (odd) dimensions and to apply contactgeometric techniques to questions in hydrodynamics. Many of theconnections between topology and contact geometry are mediated byLegendrian knots (these are knots that are tangent to a contactstructure), thus the first main theme of the proposed research isLegendrian knots. As part of this proposal the general structureof Legendrian knots will be studied. The expected outcome will bevarious classification results for certainLegendrian knots and contactstructures; and, moreover, a betterunderstanding of Legendrian surgery(an important surgery construction of contact structures). Legendrian knots in higher dimensions will also be studied using contact homology.There is very little known about contactstructures, or Legendrian knots,in dimensions above three. By investigating Legendrian knots in these dimensions the nature of contactstructures should be illuminated, justas the corresponding study revealed much about three dimensional contactstructures. The final part of the proposed research centers on the connection between contact structures and hydrodynamics discovered a fewyears ago by the Principal Investigator and R. Ghrist. Here work withGhrist will continue with the aim of understanding when, and what type of, closed flow lines occur in fluid flows. We shall also study hydrodynamicinstability from the contact topological perspective. This naturally leadsinto the study of energy minimization for fluid flows and relations betweencontact and Riemannian geometry.Contact structures are very natural objects, born over two centuries ago,in the study of geometric optics and partial differential equations.Through the centuries contactstructures have touched on many diverse areasof mathematics and physics,including classical mechanics and thermodynamics.In everyday life oneencounters contact geometry when ice skating, parallel parking a car,using a refrigerator, or simply watching the beautiful play of light ina glass of water. Many great mathematicians have devoted a lot of their work to this subject but only in the last decade or two has it movedinto the foreground of mathematics. This renaissance is due to the recentremarkable breakthroughs in contact topology, resulting in a rich andbeautiful theory with many applications. The most remarkable feature ofall this recent work is the intimate connections between contactstructures and topology in dimension three. Thus by studying this abstractnotion of a contact structure one can learn many subtle things about theuniverse in which we live. For example, the study of contact geometryhas recently lead to some unexpected advances in our understanding ofthe flow of idealized fluids. The Principal Investigator will exploreconnections between contact structures and topology in all (odd)dimensions, continue his study of idealized fluid flows (hydrodynamics)via contact geometry and analyze intriguing new conjectures concerningstring theory and contact geometry. The Principal Investigator willalso engage in several educational endeavors, including the support andencouragement of graduates students and the creation of introductoryand survey materials to bring the rapidly developing field of contactgeometry to a wider audience.