Kylerec is happening in person June 19-23, 2022, pending COVID restrictions. Applications are open and the deadline is April 10, 2022.
Topic: Quantitative symplectic geometry
Description: We will investigate Quantitative Symplectic Geometry, focusing on the problem of when a given symplectic manifold embeds inside another, and how the answer to this question may depend on quantitative parameters of the spaces in question. A symplectic manifold 𝑀 is a smooth manifold with a closed, non-degenerate two-form 𝜔. A symplectomorphism between two symplectic manifolds is a diffeomorphism preserving the symplectic forms. When the dimension of 𝑀 is 2𝑛, the 𝑛-th exterior power 𝜔𝑛 is a volume form on 𝑀. Consequently, a symplectomorphism is volume-preserving. In other words, a numerical obstruction to the existence of a symplectic embedding from a symplectic manifold 𝑀 to 𝑀′ is the comparison between their volumes. The first example of an interesting obstruction other than volume was found by Gromov in 1985, which states that a symplectic ball of radius 𝑅 cannot be embedded into a symplectic cylinder of radius 𝑟 if 𝑅 > 𝑟. Gromov’s celebrated “non-squeezing theorem” led Ekeland and Hofer to define the general definition of a symplectic capacity, a numerical invariant that can be used to obstruct symplectic embeddings.
After covering the basics of symplectic geometry, we will survey this circle of ideas on various symplectic capacities: the Hofer-Zehnder capacity, Viterbo’s spectral capacities, capacities from rational symplectic field theory, Guth’s construction of embeddings of higher dimensional ellipsoids, Hutchings-Taubes’ embedded contact homology (ECH) capacities, McDuff’s four-dimensional ellipsoid embedding problem, Cristofaro-Gardiner’s sharp embedding results using toric geometry, and recent elementary alternatives to capacities by McDuff-Siegel and Hutchings.
Format: Kylerec is a student-led and student-run workshop. We will live in a communal setting, sharing cooking and cleaning responsibilities. Talks will be given by a majority of the participants, with guidance from our mentors. Our vision is to foster a healthy, relaxed and creative atmosphere where we can learn mathematics together and make human connections in the process. There are no spectators, only participants!
Mentors: Dan Cristofaro-Gardiner (University of Maryland), Kyler Siegel (University of Southern California), Morgan Weiler (Cornell)
Organizing committee: Daren Chen (Stanford), Clair Xinle Dai (Harvard), Jae Hee Lee (MIT), Luya Wang (UC Berkeley), Jiajun Yan (University of Virginia).
Funding: Local expenses (including lodging and food) and partial travel expenses will be covered for participants. We are grateful to the NSF for their support under Grant DMS-2002676.
Contact: You are welcome to ask any questions by sending an email to firstname.lastname@example.org.
Past Workshops: For the webpages from the previous Kylerec workshops, see the following pages.