CAREER: Knot Theory and Dynamics in Contact Geometry

提案 DMS-0239600PI：John Etnyre，宾夕法尼亚大学 标题：职业：接触几何中的结理论和动力学 摘要 该提案的重点是研究所有（奇数）维度中的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动力学问题。拓扑和接触几何之间的许多连接都是由勒让德结（这些结与接触结构相切）介导的，因此本研究的第一个主题是勒让德结。作为该提案的一部分，将研究勒让德结的一般结构。预期结果将是某些勒让德结和接触结构的各种分类结果；此外，更好地理解传奇手术（一种重要的接触结构手术结构）。更高维度的传奇结也将使用接触同源性进行研究。对于三维以上的接触结构或传奇结，人们知之甚少。通过研究这些维度中的勒让德结，应该阐明接触结构的本质，正如相应的研究揭示了许多关于三维接触结构的信息一样。拟议研究的最后一部分集中于首席研究员和 R. Ghrist 几年前发现的接触结构和流体动力学之间的联系。在这里，与 Ghrist 的合作将继续进行，目的是了解流体流动中何时以及何种类型出现闭合流线。我们还将从接触拓扑的角度研究流体动力学不稳定性。这自然导致了对流体流动的能量最小化以及接触与黎曼几何之间关系的研究。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物体。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，当滑冰、平行停车、使用冰箱或只是观看玻璃中的光的美丽变化时，人们会遇到接触几何。水。许多伟大的数学家为这个学科投入了大量的工作，但直到最近一两年，它才进入数学的前台。这种复兴归功于最近接触拓扑的显着突破，产生了丰富而美丽的理论并具有许多应用。所有这些最近的工作最显着的特点是接触结构和第三维拓扑之间的密切联系。因此，通过研究接触结构这一抽象概念，人们可以了解有关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在我们对理想化流体流动的理解方面取得了一些意想不到的进展。首席研究员将探索所有（奇数）维度的接触结构和拓扑之间的联系，继续通过接触几何研究理想化流体流动（流体动​​力学），并分析有关弦理论和接触几何的有趣新猜想。首席研究员还将参与多项教育工作，包括研究生的支持和鼓励以及介绍性和调查材料的创建，以便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.