Workshop on Low-Dimensional Contact Geometry

摘要：“低维接触几何”项目由美国数学学会（AIM）和斯坦福大学数学系联合组织。展览由八月中至十二月中举行。该计划将以接触几何研讨会结束。该计划的目标是利用接触几何的最新进展。现在看来，在该领域取得新突破的时机已经成熟，在接触几何和辛几何及相关领域的成熟和年轻研究人员的集中，以及几名研究生的参与，可能有助于产生新的显著成果。组织者认为，三维接触几何的连贯图像是触手可及的。特别是，参与者将尝试实现大类别3-流形的接触结构的完整分类，进一步推进接触3-流形的Legendrian和横向结理论，并进一步建立作为辛4-流形边界的接触3-流形的性质。接触几何诞生于两个多世纪前惠更斯，汉密尔顿，雅可比的工作中，作为光学的几何语言。人们很快意识到它在许多其他领域也有应用，包括非完整力学和热力学。在日常生活中，停车、滑冰、使用冰箱、观看光在一杯水中的美丽游戏，都会遇到接触几何。索菲斯·李、埃利·卡尔坦、达布克斯和其他许多伟大的数学家在这个问题上投入了大量的工作。然而，直到最近，大多数结果都是地方性的。随着80年代辛和接触拓扑学的诞生，这门学科获得了新生，在过去的十年里见证了许多突破性的发现。发现了与哈密顿力学、辛和亚黎曼几何、叶理理论、复杂几何和分析、拓扑流体力学、三维拓扑学和结理论之间新的重要相互作用。该计划的详细信息可以在http:/math.stanford.edu/contact.html上找到

Proposal: DMS-0075477PI: Yakov eliashberg et alAbstract:The program ``Low-dimensional contact geometry" is organized jointlyby the American Institute of Mathematics (AIM) and the Departmentof Mathematics of Stanford University. It will run frommid-August till mid-December 2000. The program will end with aa Contact Geometry Workshop. The goal of the program is to capitalize on recent progress in contact geometry. It seems that the time is ripe nowfor new breakthroughs in the field, and that the concentration of bothestablished and young researchers in contact and symplectic geometry and related areas, as well as participation of several graduate students, may help to produce new remarkable results. The organizers feel that a coherent picture of contact geometry in 3 dimensions is within reach.In particular, the participants will try to achieve a complete classification of contact structures for large classes of 3-manifolds, to make further advances in the theory of Legendrian and transversalknots in contact 3-manifolds, and to establish further properties of contact3-manifolds which serve as boundaries of symplectic 4-manifolds.Contact geometry was born more than two centuries ago in the work of Huygens, Hamilton, Jacobi as a geometric language for optics. It was soon realized that it has applications in many other areas, includingnon-holonomic mechanics and thermodynamics. One encounters contact geometry in everyday life when parking a car, skating, using a refrigerator, or watching the beautiful play of light in a glass of water. Sophus Lie, Elie Cartan, Darboux and many other great mathematicians devoted a lot of their work to this subject. However, till very recently most of the results were of a local nature. With the birth of symplectic and contact topology in the Eighties, the subject was reborn, and the last decade witnessed anumber of breakthrough discoveries. There were found new important interactions with Hamiltonian mechanics, symplectic and sub-Riemannian geometry, foliation theory, complex geometry and analysis, topological hydrodynamics, 3-dimensional topology, and knot theory.Details of the program can be found at http:/math.stanford.edu/contact.html