Holomorphic Methods in Symplectic Geometry

The aim of this 2-hour course is a systematic introduction of the method of (pseudo-)holomorphic curves in symplectic geometry. This method was introduced by Gromov in 1985 with a tremendous impact not only in symplectic geometry but also through its applications in Hamiltonian Dynamical Systems and mathematical physics. This method not only leads directly to the important class of Gromov-Witten invariants, in their more algebraic form also known as quantum cohomology, but the same analysis plays a central role in Floer theory and its more general version known as Symplectic Field Theory. This course will not treat any of these deep theories - GW-invariants, quantum cohomology, Floer homology, etc. in total. The aim is to deal in detail with the analytical set-up of holomorphic curves, the emphasis lies on the notion of Fredholm theory in this Cauchy-Riemann context.
Cornerstones of this course will be:

• local elliptic estimates for Cauchy-Riemann operators
• the suitably adapted boundary value problems
• selected topics from the nonlinear theory, e.g. bubbling phenomenon

Literatur:

• D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, Coll. Publ. 52, AMS, 2004.
• H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994.

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