Knot Theory and Dynamics in Contact Geometry

本提案的重点是更好地理解所有（奇数）维度的接触结构的性质，特别注意三维，并将接触拓扑技术应用于流体动力学问题。本研究的第一个主题是Legendrian knot。Legendrian结是与接触结构相切的结，似乎包含了大量关于接触结构的信息。例如，著名的三维紧扭和超扭二分法可以用Legendrian结来理解。这些结也给出了接触结构的一个重要不变量。作为本建议的一部分，我们将研究Legendrian结的一般结构。预期的结果将是对某些Legendrian结点和接触结构的各种分类结果；此外，更好地理解了Legendrian surgery（一种重要的接触结构的外科构造）。高维的legendrin结也将使用接触同调来研究。在三维以上的维度中，接触结构或勒让德结所知甚少。通过研究这些维度上的Legendrian结，接触结构的本质应该得到阐明，就像相应的研究揭示了三维接触结构一样。提议研究的最后一部分集中在几年前由首席研究员和R. Ghrist发现的接触结构和流体力学之间的联系。在这里，我们将继续与克里斯特合作，目的是了解流体流动中何时以及何种类型的封闭流线发生。我们还将从接触拓扑学的角度研究水动力不稳定性。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。接触结构是非常自然的物体，诞生于两个多世纪前惠更斯、汉密尔顿和雅可比对几何光学的研究中。几个世纪以来，接触结构涉及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，滑冰、平行泊车、驾驶潜艇、使用冰箱，或者只是观察一杯水中美丽的光线，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工作，但直到最近一二十年，它才进入数学的前景。这种复兴是由于最近接触拓扑的显著突破，导致了一个丰富而美丽的理论与许多应用。最近所有这些工作中最显著的特点是接触结构和三维拓扑之间的密切联系。此外，在哈密顿力学、辛和亚黎曼几何、叶理理论、复杂几何和分析、拓扑流体力学和结理论之间也有重要的新发现的相互作用。首席研究员将在所有（奇数）维度上探索接触结构和拓扑结构之间的新联系，并通过接触几何继续他的理想流体流动（流体动力学）研究。

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.