**Topic: **Quantitative symplectic geometry

**Date: **July 19 – 25, 2020

**Location: **Big Bear Lake, CA

**Applications: **Fill out this form to apply. The deadline is April 5, 2020. We are committed to promoting diversity in mathematics and especially encourage people from underrepresented groups to apply.

**Description:** This year, 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 đť‘€ is a smooth manifold with a closed, non-degenerate two-form đťś”. A symplectomorphism between two symplectic manifolds is a diffeomorphism preserving the symplectic forms. When the dimension of đť‘€ is 2đť‘›, the đť‘›-th exterior power đťś”^{đť‘›} is a volume form on đť‘€. Consequently, a symplectomorphism is volume-preserving. In other words, a numerical obstruction to the existence of a symplectic embedding from a symplectic manifold đť‘€ to đť‘€â€˛ 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 đť‘… cannot be embedded into a symplectic cylinder of radius đť‘ź if đť‘… > đť‘ź. Gromovâ€™s celebrated â€śnon-squeezing theoremâ€ť 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, Viterboâ€™s spectral capacities, capacities from rational symplectic field theory, Guthâ€™s construction of embeddings of higher dimensional ellipsoids, Hutchings-Taubesâ€™ embedded contact homology (ECH) capacities, McDuffâ€™s four-dimensional ellipsoid embedding problem, and Cristofaro-Gardinerâ€™s sharp embedding results using toric geometry.

**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:Â **Dan Cristofaro-Gardiner (UC Santa Cruz), Sara Venkatesh (Stanford).

**Organizing committee: **Orsola Capovilla-Searle (Duke), Dahye Cho (Stony Brook), FranĂ§ois-Simon Fauteux-Chapleau (Stanford), Tim Large (MIT), Sarah McConnell (Stanford).

**Funding:Â **Local expenses (including lodging and food) and partial travel expenses will be covered for participants.

**Contact: **You are welcome to ask any questions by sending an email to kylerec2020@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