Symplectic Field Theory VIII: Symplectic Homology

该奖项为将于2016年8月8日至12日在柏林洪堡大学举行的辛同调研讨会的美国与会者提供部分旅行支持。辛几何是经典力学和动力学的基础几何。宏观物理系统的一组可能的状态由辛流形描述，而这个辛流形的几何对于理解物理系统如何随时间演化是必不可少的。辛场理论(SFT)是由Eliashberg-Givental-Hofer于2000年创立的，它是通过对辛流形中的伪全纯曲线(一种面积最小的曲面)进行计数，并通过将辛流形分割成更简单的片段来执行这些计数来理解辛几何的主要程序。自2005年以来，在德国不同地点举办了一系列关于SFT特定方面的讲习班，这些讲习班在这一主题的发展过程中发挥了关键作用。每个这样的研讨会包括一周的迷你课程、研究讲座和讨论，在此之前还有一个周末的入门课程，回顾必要的背景材料。2016年8月，第八届SFT研讨会将在柏林举行，这笔赠款将提供旅行支持，使美国的初级参与者能够参加研讨会。研讨会的具体主题是辛同调。在过去的十年里，辛同调已经成为辛几何和接触几何和拓扑学的中心工具，并有许多引人注目的应用。辛同调的用途包括区分奇异的Stein结构和接触结构，检测向量场的Reeb轨道，以及阻挡拉格朗日协边带。此外，一个密切相关的不变量，包装Fukaya范畴，在镜像对称性最近的发展中扮演着重要的角色。这次研讨会将以深入和协调的方式汇集辛同调的结构和应用方面的主要最新进展，以便将这一重要主题形成一个连贯的整体，并使更广泛的受众能够接触到它。研讨会的中心课程将包括Tobias Ekholm关于Legendrian外科公式及其应用的迷你课程，Alexandru Oancea关于Eilenberg-Steenrod辛公理的课程，以及Mark McLean关于包裹Floer上同调和仿射变种的课程。此外，将有五个研究讲座和一个周末的预科课程，其中有六个背景讲座。派遣美国人参加这个研讨会将使美国数学家能够跟上辛几何的最新发展，并为该学科的未来做出贡献。有关研讨会的更多详细信息，请访问https://www.mathematik.hu-berlin.de/~wendl/SFT8/.。

This award provides partial travel support for US based participants of the workshop on Symplectic Homology to be held on August 8 - 12, 2016, at the Humboldt University, Berlin. Symplectic geometry is the fundamental geometry underlying classical mechanics and dynamics. The set of possible states of a macroscopic physical system is described by a symplectic manifold, and the geometry of this symplectic manifold is essential for understanding how the physical system evolves over time. Symplectic field theory (SFT), initiated in 2000 by Eliashberg-Givental-Hofer, is a major program for understanding symplectic geometry by counting pseudoholomorphic curves (a kind of area-minimizing surfaces) in symplectic manifolds, and performing these counts by cutting symplectic manifolds into simpler pieces. Since 2005, a series of workshops on particular aspects of SFT has been held at various locations in Germany, and these workshops have played a key role in the development of the subject. Each such workshop consists of a week of mini-courses, research lectures, and discussions, preceded by a weekend of introductory courses reviewing necessary background material. In August 2016, the eighth SFT workshop will be held in Berlin, and this grant will provide travel support to enable junior US-based participants to participate in the workshop.The specific topic of the workshop is symplectic homology. In the past decade, symplectic homology has emerged as a central tool in symplectic and contact geometry and topology, with a number of spectacular applications. The uses of symplectic homology include distinguishing exotic Stein structures and contact structures, detecting Reeb orbits of vector fields, and obstructing Lagrangian cobordisms. In addition, a closely related invariant, the wrapped Fukaya category, plays an essential role in recent developments in mirror symmetry. This workshop will bring together the major recent advances in the structure and applications of symplectic homology, in an in-depth and coordinated manner, in order to form this vital subject into a coherent whole and make it accessible to a wider audience. The centerpiece of the workshop will consist of mini-courses by Tobias Ekholm on the Legendrian surgery formula and its applications, Alexandru Oancea on the symplectic Eilenberg-Steenrod axioms, and Mark McLean on wrapped Floer cohomology and affine varieties. In addition there will be five research lectures and a weekend precourse with six background talks. Sending US participants to this workshop will enable US mathematicians to keep up with the latest developments in symplectic geometry and contribute to the future of the subject. More details about the workshop can be found at https://www.mathematik.hu-berlin.de/~wendl/SFT8/.