**Kylerec 2023 will happen in person June 19-23, 2023, near Yosemite National Park. Please apply here by April 10, 2023. We are committed to promoting diversity in mathematics. We especially encourage people from groups that are underrepresented in math to apply.**

**Topic: **Homological Mirror Symmetry and Symplectomorphisms

**Description: **The topic of Kylerec 2023 will be Homological Mirror Symmetry, with eyes on applications towards understanding the symplectomorphism groups and autoequivalences of Fukaya categories. For more detailed description, please see below.

**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!

**Pre-workshop:** Leading up to the main workshop, there will be a 1-day pre-workshop over Zoom to be announced.

**Mentors: **Catherine Cannizzo (University of California, Riverside), Heather Lee, Abigail Ward (MIT)

**Organizing committee: **Daren Chen (Stanford), Clair Xinle Dai (Harvard), Jae Hee Lee (MIT), Thomas Massoni (Princeton), Luya Wang (UC Berkeley).

**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 kylerec2023@gmail.com.

**More Detailed Description of This Year’s Program:**

Mirror symmetry originates from dualities in string theory, which relate the two topological twists of a superconformal field theory. While physically a sound proposal, the mathematical incarnations of the two descriptions of the dual theories are very different; hence the physics produces remarkable mathematical conjectures relating two seemingly completely different areas of mathematics. The early triumphs of the theory (which one may dub classical, or Hodge-theoretic mirror symmetry) involved the enumerative symplectic geometry of a Calabi–Yau threefold , computed by the Hodge theory of its mirror Calabi–Yau .

In 1994, Kontsevich proposed a completely new perspective on mirror symmetry, claiming that the relationship involves an equivalence of two triangulated categories (i.e. non-commutative spaces) , given by the derived Fukaya category of Calabi–Yau and the derived category of coherent sheaves on . This formulation is known as homological mirror symmetry, and has been generalized and verified in many settings. Notably, the generalization includes similar statements for non-compact and non-Calabi–Yau manifolds, and each side of the HMS proposal must be replaced by suitable triangulated categories (of the similar flavor).

While the program of verifying the HMS conjecture for a wide variety of examples is a developed industry on its own, one can also use homological mirror symmetry as a source of insights towards understanding symplectic topology (or algebraic geometry, for that matter). In particular, there are new information about the symplectomorphism groups revealed through the lens of homological mirror symmetry, by studying their induced actions on the Fukaya category. One can leverage the information on the B-side (the autoequivalences of the derived category), typically better understood, to understand the A-side (the autoequivalences of the Fukaya category). Via this approach, for example, we now have new examples of symplectomorphisms on Weinstein 4-manifolds Hacking–Keating, examples of infinitely generated symplectic Torelli groups Sheridan–Smith, and examples of symplectic manifolds with symplectic mapping class groups of unbounded ranks by Auroux–Smith.

In the 5-day workshop, we aim to discuss various aspects of the subject, starting from surveying homological mirror symmetry results and building on them towards applications in the study of symplectomorphisms. Hopefully by the end of the program we will touch on some of the frontiers of current research in the area.

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

2022 Kylerec on Quantitative symplectic geometry

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