**Kylerec 2022 happened in person June 19-23, 2022. All notes can be found by visiting this page (thanks Laura Wakelin for the notes)!**

**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 𝜔^{𝑛} 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 kylerec2022@gmail.com.

**Past Workshops: **For the webpages from the previous Kylerec workshops, see the following pages.

2019 Kylerec on sheafy symplectic topology

2018 Kylerec on the nearby Lagrangian conjecture

2017 Kylerec on symplectic fillings

2016 Workshop on Lefschetz fibrations: rigidity and flexibility