CAREER: Knot Theory and Dynamics in Contact Geometry

提案DMS-0239600PI：John Etnyre，宾夕法尼亚大学题目：职业：接触几何中的纽结理论和动力学摘要本提案的重点是研究所有(奇)维拓扑与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勒让德结(这些是与接触结构相切的纽结)调节的，因此拟议的研究的第一个主要主题是勒让德结。作为这项建议的一部分，我们将研究传奇结的一般结构。预期的结果将是对某些Legendrian结和接触结构的不同分类结果；此外，还将更好地理解Legendrian手术(接触结构的一种重要外科构造)。更高维度的传奇结也将用接触同调来研究。关于三维以上的接触结构或传奇结，人们知之甚少。通过对这些维度上的传奇结的研究，应该能够阐明接触结构的本质，就像相应的研究揭示了许多关于三维接触结构的情况一样。拟议研究的最后部分集中在接触结构和流体动力学之间的联系，这是首席研究员和R.Ghrist几年前发现的。在这里，将继续与Ghrist合作，目的是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。这自然地引出了流体流动的能量最小化以及接触和黎曼几何之间的关系的研究。接触结构是非常自然的物体，诞生于两个多世纪前的几何光学和偏微分方程式的研究中。几个世纪以来，接触结构已经触及了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，当人们在溜冰、平行停车、使用冰箱或只是在玻璃中观看光的美丽运动时，人们都会遇到接触几何。许多伟大的数学家在这个问题上投入了大量的精力，但直到最近一二十年，它才进入数学的前台。这种复兴是由于最近接触拓扑学的显著突破，产生了具有许多应用的丰富而美丽的理论。最近所有这些工作中最显著的特征是三维接触结构和拓扑之间的密切联系。因此，通过研究这种联系结构的抽象概念，人们可以了解到关于我们生活的宇宙的许多微妙的事情。例如，接触几何的研究最近在我们对理想流体流动的理解上取得了一些意想不到的进展。主要研究人员将探索接触结构和所有(奇)维拓扑之间的联系，通过接触几何继续他对理想化流体流动(流体动力学)的研究，并分析关于弦理论和接触几何的有趣的新猜想。首席研究员还将从事一些教育工作，包括对研究生的支持和鼓励，以及创建介绍和调查材料，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.