This seminar is aimed at graduate students interested in symplectic geometry and related areas, and will try to give a comprehensive introduction to pseudoholomorphic curves. Most of the talks will be given by students. (For a more advanced seminar on a similar subject, try the symplectic dinner seminar organized by Chris Wendl).

The main reference for the first part of the seminar is: D. McDuff, D. Salamon, J-holomorphic curves and symplectic topology, American Math. Society, 2004.

One of the two weekly meetings (Tuesday) will be more structured, with postdocs and faculty welcome to attend, while the other weekly meeting (Thursday) will be completely informal.

Contacts: Y. Lekili, D. Auroux, K. Wehrheim

Program of the seminar

• Tuesday September 12: Denis Auroux, Katrin Wehrheim: Introductory talk; organization of the seminar.
• Tuesday September 19: Ana Rita Pires: symplectic manifolds (Darboux theorem, ...), symplectomorphisms and Hamiltonian vector fields, Lagrangian submanifolds, almost complex structures (tame/compatible, contractible spaces), first Chern class. [Cannas da Silva]
• Tuesday September 26: Yanki Lekili: J-holomorphic maps (global and in local coordinates), energy and energy identity, comparison with holomorphic functions: unique continuation, isolated critical points, somewhere injective/simple/multiply covered, positivity of intersections (statements only). [McDuff-Salamon, ch. 2]
• Tuesday October 3: Maksim Maydanskiy: moduli spaces (with marked points, modulo reparametrization, as zero set of a section), implicit function theorem for Fredholm section of Banach bundle, linearized Cauchy-Riemann operator, statement of Fredholm and transversality properties. [McDuff-Salamon, 3.1 and Thm.A.3.3.]
• Tuesday October 17: Fangyun Yang: real Cauchy-Riemann operators (elliptic estimates, Fredholm property, Riemann-Roch); including with boundary conditions. [McDuff-Salamon, C.1-3]
• Thursday October 26: Maksim Lipyanskiy: nonlinear elliptic regularity and compactness (for bounded gradient), energy. [Ivashkovich-Shevchishin]
• Tuesday October 31: Maksim Lipyanskiy: bubbling effect (rescaling), removable singularity (sketch of proof?), compactness modulo bubbling. [Ivashkovich-Shevchishin]
• Tuesday November 7: Maksim Maydanskiy: compactified moduli space, evaluation maps, Gromov-Witten invariants, stable curves.
• Tuesday November 14: Yanki Lekili: non-squeezing theorem.
• Tuesday November 21: Ronen Mukamel: Morse homology.
• Tuesday November 28: Matt Hedden: Lagrangian Floer homology - action functional and holomorphic strips.
• Tuesday December 5: Aleksandar Subotic: Lagrangian Floer homology - bubbling and compactness.
• Tuesday December 12: Katrin Wehrheim: Lagrangian Floer homology - gluing.