# Quantitative Symplectic Geometry: May 8-12, 2017

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

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 ${R}^{2n}$) 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 ${R}^{2n}$), 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 ${R}^{2n}$. Another conjecture of interest asserts that the boundary of every star-shaped domain in ${R}^{2n}$ 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.

#### Talk Schedule

 Time Title Presenter Location 9:30am The Length and Width of Lagrangian Cobordisms Lisa Traynor SCGP 102 11:00am TBA Egor Shelukhin SCGP 102 12:00pm Lunch N/A SCGP Cafe 2:15pm Computing EHZ Capacities Of 4d Polytopes Julian Chaidez SCGP 102 3:15pm Tea N/A SCGP Lobby 4:00pm Infinite Staircases in Symplectic Embeddings Ana Rita Pires SCGP 102

 Time Title Presenter Location 9:00am Unobstructed symplectic packing of tori by ellipsoids Michael Entov SCGP 102 10:00am Coffee Break N/A SCGP Cafe 10:30am Symplectic packing stability beyond four dimensions Olguta Buse SCGP 102 11:30am Lunch N/A SCGP Cafe 1:00pm SCGP Weekly Talk: Measuring space and time in symplectic geometry Michael Hutchings SCGP 102 2:30pm The symplectic size of a random convex body Yaron Ostrover SCGP 102 3:30pm Tea N/A SCGP Lobby 4:00pm Discussion Session Discussion Session SCGP 102

 Time Title Presenter Location 9:30am The stabilized symplectic embedding problem Dusa McDuff SCGP 102 10:30am Coffee Break N/A SCGP Cafe 11:00am PDEs and symplectic embedding obstructions Felix Schlenk SCGP 102 12:00pm Lunch N/A SCGP Cafe 3:15pm Tea N/A SCGP Lobby

 Time Title Presenter Location 9:30am Knotted symplectic embeddings between domains in R^4 Michael Usher SCGP 102 10:30am Coffee Break N/A SCGP Cafe 11:00am A quantitative perspective on the classification of Lagrangian tori Georgios Dimitroglou Rizell SCGP 102 12:00pm Lunch N/A SCGP Cafe 2:15pm Symplectic embeddings of Lagrangian products Vinicius Gripp Ramos SCGP 102 3:15pm Tea N/A SCGP Lobby 4:00pm Discussion Session Discussion Session SCGP 102

 Time Title Presenter Location 9:30am Lusternik-Schnirelmann Theory, the Shift Operator and Closed Reeb Orbits Viktor Ginzburg SCGP 102 10:30am Coffee Break N/A SCGP Cafe 11:00am Poincare duality for free loop spaces Kai Cieliebak SCGP 102 12:00pm Lunch N/A SCGP Cafe 2:15pm The size of a neighborhood of a Lagrangian in C^n. Mark Mclean SCGP 102 3:15pm Tea N/A SCGP Lobby 4:00pm Equivariant symplectic capacities Jean Gutt SCGP 102