Quantitative Symplectic Geometry: May 8-12, 2017

Organized by: Dan Cristofaro-Gardiner, Richard Hind, Michael Hutchings.

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Despite the fundamental importance of symplectic geometry, many basic questions about it are not well understood. Quantitative symplectic geometry is concerned with closely related questions of size and time in symplectic geometry. Specifically, when can one symplectic manifold with boundary (such as a domain in ) be symplectically embedded into another? And what can one say about the number and properties of the closed orbits of the Reeb vector field on a contact manifold (such as the boundary of a star-shaped domain in ), or more generally of the Hamiltonian vector field on a level set of the Hamiltonian?

There has recently been a renaissance in the area of quantitative symplectic geometry. A number of new techniques have been introduced, especially using various kinds of contact homology and related theories such as S1-equivariant symplectic homology, and some longstanding conjectures may be coming within reach. For example, one important conjecture is Viterbo’s conjecture, asserting that all normalized symplectic capacities agree on convex domains in . Another conjecture of interest asserts that the boundary of every star-shaped domain in has at least n Reeb orbits.

The goal of this workshop is to bring together researchers in quantitative symplectic geometry to compare approaches and set the agenda for the field. Some specific questions to be discussed include: identifying gaps between known constructions of and obstructions to symplectic embeddings, especially in dimensions greater than four; studying symplectic embedding obstructions, and lower bounds on the number of Reeb orbits, coming from various versions of contact homology; cataloging and relating various symplectic capacities, and developing tools for computing them in order to approach Viterbo’s conjecture; in situations where symplectic embeddings exist, understanding the topology and Hofer geometry of the space of all symplectic embeddings; and understanding when various kinds of Lagrangian embeddings are possible.