May 25 – 31, 2018, Leavenworth Lodge in Washington State, USA
Symplectic geometry of cotangent bundles: Nearby Lagrangian Conjecture
Kylerec is a student-led and student-run workshop. We will live in a communal setting, sharing cooking and cleaning duties. Talks will be given by participants, with guidance from our mentors. Our vision is to curate a healthy, relaxed and creative atmosphere where we can learn mathematics together and make human connections in the process.
The topic this year is on the nearby Lagrangian conjecture, that every closed exact Lagrangian L in a cotangent bundle T*Q is Hamiltonian isotopic to the zero-section. We will look at this via work of Thomas Kragh that the projection of the Lagrangian to the base of the cotangent bundle is a homotopy equivalence, in https://arxiv.org/abs/1505.07359. This work is accessible for beginner students, and worthwhile for more advanced students. We will spend some time on the wrapped Fukaya category, one of the most important tools in the entire subject and reasonably tractable here. Proving the homotopy equivalence from the homology equivalence is then a great opportunity to learn about local systems in Floer theory, and how they can be used to encode homotopical information. Finally, we plan to look at proofs of the conjecture in low dimensional cases, in particular the cotangent bundles of exotic spheres. These provide beautiful examples of how a delicate study of moduli spaces of pseudo-holomorphic curves can produce stunning results in symplectic geometry. The workshop will be mentored by Roberta Guadagni (U Penn), Thomas Kragh (Uppsala), Kyler Siegel (MIT), and Jingyu Zhao (Brandeis). For more mathematical detail see our website https://kylerec.wordpress.com/.