Symplectic Field Theory VIII: Symplectic Homology

该奖项为2016年8月8日至12日在柏林洪堡大学举行的辛同源学研讨会的美国参与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。宏观物理系统的可能状态集由辛流形描述，辛流形的几何对于理解物理系统如何随时间演变至关重要。辛场论（SFT）于2000年由Eliashberg-Givental-Hofer发起，是一个通过计算辛流形中的伪全纯曲线（一种面积最小化曲面）并通过将辛流形切割成更简单的块来执行这些计数来理解辛几何的主要程序。自2005年以来，在德国的不同地点举办了一系列关于SFT特定方面的讲习班，这些讲习班在该主题的发展中发挥了关键作用。每个这样的研讨会包括一个星期的迷你课程，研究讲座和讨论，在周末的入门课程复习必要的背景材料之前。2016年8月，第八届SFT研讨会将在柏林举行，这笔资金将为美国的年轻参与者提供旅行支持。研讨会的具体主题是辛同调。在过去的十年中，辛同调已经成为辛几何和接触几何和拓扑学的核心工具，并有了许多引人注目的应用。辛同调的应用包括区分奇异斯坦结构和接触结构、检测矢量场的Reeb轨道和阻挡拉格朗日协律。此外，一个密切相关的不变量，包裹的深谷范畴，在镜像对称的最新发展中起着至关重要的作用。本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。研讨会的核心将包括Tobias Ekholm关于Legendrian手术公式及其应用的迷你课程，Alexandru Oancea关于辛Eilenberg-Steenrod公理的迷你课程，以及Mark McLean关于包裹花上同调和仿射变体的迷你课程。此外，还将有五次研究讲座和一个周末的预备课程，包括六次背景讲座。派美国人参加本次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。有关研讨会的更多细节，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/.