Knot Theory and Dynamics in Contact Geometry

DMS-0203941 John Etnyre本提案的重点是更好地理解所有（奇数）维中接触结构的性质，特别注意三维，并将接触拓扑技术应用于流体力学问题。拟议研究的第一个主题是Legendrian结。勒让德结是与接触结构相切的结，似乎编码了关于接触结构的大量信息。例如，三维空间中著名的紧与过扭曲二分法可以用勒让德结来理解。这些纽结也给出了接触结构的一个重要的不变量。作为这项建议的一部分，勒让德结的一般结构将进行研究。预期结果将是某些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）。更高维度的勒让德结也将使用接触同调进行研究。对于三维以上的接触结构或勒让德结，我们知之甚少。通过研究这些维度上的勒让德结，接触结构的本质应该得到阐明，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。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学的角度研究流体动力学不稳定性。接触结构是两个多世纪以前惠更斯、汉密尔顿和雅可比在几何光学方面的工作中诞生的非常自然的对象。几个世纪以来，接触结构已经涉及到数学和物理学的许多不同领域，包括经典力学和热力学。在日常生活中，人们在滑冰、平行停车、驾驶潜艇、使用冰箱或只是观看一杯水中的光的美丽游戏时都会遇到接触几何。许多伟大的数学家都致力于他们的大量工作，这一主题，但只有在过去的十年或二十年已进入前台的数学。这种复兴是由于最近在接触拓扑学上的重大突破，产生了丰富而美丽的理论，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三维拓扑之间的密切联系。此外，还有重要的新发现的相互作用与哈密顿力学，辛和亚黎曼几何，叶理理论，复杂的几何和分析，拓扑流体力学和纽结理论。主要研究者将探索接触结构和拓扑结构之间的新联系，并通过接触几何形状继续研究理想化流体流动（流体动力学）。

DMS-0203941John EtnyreThe focus of this proposal is to better understand the nature of contact structures in all (odd) dimensions, with special attention given to dimension three, and to apply contact topological techniques to questions in hydrodynamics. The first main theme of the proposed research is Legendrian knots. Legendrian knots are knots that are tangent to a contact structure and seem to encode a great deal of information about the contact structure. For example, the famed tight vs. overtwisted dichotomy in dimension three can be understood in terms of Legendrian knots. These knots also give an important invariant of a contact structure. As part of this proposal the general structure of Legendrian knots will be studied. The expected outcome will be various classification result for certain Legendrian knots and contact structures; and, moreover, a better understanding 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 contact structures, or Legendrian knots, in dimensions above three. By investigating Legendrian knots in these dimensions the nature of contact structures should be illuminated, just as the corresponding study revealed much about three dimensional contact structures. The final part of the proposed research centers on the connection between contact structures and hydrodynamics discovered a few years ago by the Principal Investigator and R. Ghrist. Here work with Ghrist will continue with the aim of understanding when, and what type of, closed flow lines occur in fluid flows. We shall also study hydrodynamic instability from the contact topological perspective. This naturally leads into the study of energy minimization for fluid flows and relations between contact and Riemannian geometry.Contact structures are very natural objects, born over two centuries ago, in the work of Huygens, Hamilton and Jacobi on geometric optics. Through the centuries contact structures have touched on many diverse areas of mathematics and physics, including classical mechanics and thermodynamics. In everyday life one encounters contact geometry when ice skating, parallel parking a car, navigating a submarine, using a refrigerator, or simply watching the beautiful play of light in a 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 moved into the foreground of mathematics. This renaissance is due to the recent remarkable breakthroughs in contact topology, resulting in a rich and beautiful theory with many applications. The most remarkable feature of all this recent work is the intimate connections between contact structures and topology in dimension three. Moreover, there were important newfound interactions with Hamiltonian mechanics, symplectic and sub-Riemannian geometry, foliation theory, complex geometry and analysis, topological hydrodynamics, and knot theory. The Principal Investigator will explore new connections between contact structures and topology in all (odd) dimensions and continue his study of idealized fluid flows (hydrodynamics) via contact geometry.