``Sections'' refers to the relevant sections in the book ``J-holomorphic Curves and Symplectic Topology'' by D. McDuff and D. A. Salamon.

day | sections | material | homework | solutions | |
---|---|---|---|---|---|

February 6 | 1.1-12.7 | overview over lecture course, questions in symplectic topology | |||

February 13 | some highlights of symplectic topology: symplectic nonsqueezing, group of symplectomorphisms C^0-closed, existence of exotic symplectic form on R^{2n} | ||||

February 20 | existence of leafwise fixed points, regular presymplectic manifolds, symplectic quotients, minimal action of a coisotropic submanifold, Hofer norm | ||||

February 27 | Extreme cases of leafwise fixed point result: fixed points, Lagrangian intersections. Examples. Comparison with results from algebraic topology/ differential topology. Review of Hamiltonian Lie group actions. | ||||

March 6 | more about Hamiltonian Lie group actions, proof of the leafwise fixed point result using the Lagrangian intersection result, proof of skinny nonsqueezing | ||||

March 13 | proof of C^0_loc-closedness of the group of symplectomorphisms, symplectic capacities | ||||

March 20 | obstruction to presymplectic embeddings, exact Lagrangian embeddings, proof of existence of an exotic symplectic form on R^{2n} | ||||

March 27 | two more applications of Lagrangian intersections: exact Lagrangians in the cotangent bundle, nondegeneracy of the Hofer norm | ||||

April 10 | almost complex structures, compatibility, convexity at infinity, idea of proof of the Lagrangian intersection result | ||||

April 17 | 9.2 | existence of a solution of Floer's equation, minimal holomorphic action, quantization of energy, proof of the Lagrangian intersection result | |||

April 24 | 2.1, 2.2, 4.3, 9.2 | proof of quantization of energy for holomorphic maps, holomorphic maps stay inside a compact set, Dirichlet energy, energy identity, mean value inequality for holomorphic maps | |||

May 1 | 9.2, 4.3 | proof of the fact that holomorphic maps stay inside a compact set | |||

May 8 | 4.3, 4.6 | proof of the mean value inequality for holomorphic maps, preparation for the proof of existence of a solution of Floer's equation: Hölder and Sobolev spaces, Sobolev's and Morrey's inequalities | |||

May 22 | compactness of the space of solutions of the parameter-dependent Floer equation, manifold structure on this space, proof of the existence of a solution of Floer's equation | ||||

May 29 | review of proof of the existence of a solution of Floer's equation, regularity of a holomorphic map, compactness for holomorphic maps, proof of compactness of the space of solutions of the parameter-dependent Floer equation | ||||

June 12 | proof of the uniform Sobolev bound for solutions of Floer's equation, begin of the proof that there exists a good manifold structure on the space of solutions of the parameter-dependent Floer equation | ||||

June 19 | Fredholm property for the vertical derivative of the Floer section, real linear Cauchy-Riemann operators, Maslov index, Riemann-Roch Theorem | ||||

June 25, 9:00 - 11:00, BBG 079 | transversality for the vertical derivative of the Floer section, end of the proof that there exists a good manifold structure on the space of solutions of the parameter-dependent Floer equation | ||||

tentamen: June 26, 8:30 - 13:00. The exam will be oral, take place at HFG606, and last 45 minutes per person. |