Symplectic Field Theory VIII: Symplectic Homology

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

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/.