A holomorphic building in symplectic field theory (SFT), a particular flavor of Floer theory!

This is a graduate course on Floer homology in symplectic geometry. This course is intended to prepare PhD students for research in symplectic and low-dimensional topology.

The first part of the course will focus on the detailed construction of Hamiltonian-Floer homology. Specific topics include

• basic Fredholm theory. Banach spaces/manifolds, Fredholm maps, Fredholm index, Smale's implicit function theorem.
• Floer equation, linearization and index calculations.
• generic transversality
• broken Floer trajectories and gluing

The second part of the course will be a general survey of important Floer theories, including Lagrangian-Floer homology, Fukaya categories, Heegaard-Floer homology, symplectic homology and symplectic field theory (SFT).

• Instructor: Julian Chaidez (julian.chaidez@usc.edu)
• Lectures: WF 1:30-3 in KAP 427
• Office Hours: MF 4-5 in KAP 245
• Grade Scheme: 50% HW + 30% final paper + 20% presentation.
• Final Paper (Due 5/2): A final paper (10-15 pages) is required of all students. The paper should be a survey on a Floer theory and an application of the students choice.
• Final Presentation (Due 5/2): A final presentation (15 minutes) is required of all students.
• For more information, see the course syllabus.

• HW1 (W 1/29). [S] 1.2, 1.7, 1.8, 1.9, 1.10, 1.11
• HW2 (W 2/5). [S] 1.22, 1.23, 2.1, [AD] 6.28, 6.30
• HW3 (W 2/12). [AD] 7.38, 7.39, 7.40, 7.41, 7.43
• HW4 (W 2/19). [S] 2.8, 2.9, 2.10 [AD] 10.46
• HW5 (W 3/5). [AD] 6.25, 11.47 & W 3/5
• HW6 (W 4/9). [OS] 4.7, 4.9, 5.4, 5.9, 5.11
• HW7 (W 4/16). [OS] 7.7, 8.4, 10.8

• W 1/15. intro, Morse homology, Arnold conjecture. [S] 1.3 [AD] 1.1-3.5.
• F 1/17. Floer overview, action, Floer solutions. [S] 1.4-1.6 [AD] 6.1-6.4.
• W 1/22. NO CLASS. Hamiltonians, complex structures. [AD] 5.1-5.6.
• F 1/24. Banach manifolds, Fredholm theory [S] 2.1, [AD] C.1-C.5.
• W 1/29. Properties of Floer solutions: energy, compactness, asymptotics. [S] 1.4-1.6, 2.7 [AD] 6.5-6.6, 8.2.
• F 1/31. linearization, Fredholmness, decay. [S] 2.2-2.3 [AD] 8.4,8.7,8.9.
• W 2/5. rotation number, CZ index, index of orbits. [S] 2.4 [AD] 7.1-7.2.
• F 2/7. Fredholm index formula. [S] 2.5 [AD] 8.7.
• W 2/12. perturbation space, somewhere injectivity. [AD] 8.3, 8.6.
• F 2/14. generic tranversality of Floer map. [AD] 8.5.
• W 2/19. space of trajectories, pre-gluing map. [S] 3.1 [AD] 9.1-9.3.
• F 2/21. construction of gluing, immersion property. [S] 3.3 [AD] 9.4-9.5.
• W 2/26. gluing map is injective. [S] 3.3. [AD] 9.6 .
• F 2/28. linearized Morse operator. [S] 3.2, 3.5 [AD] 10.1-10.2.
• W 3/5. generic regularity, trajectory bijection. [S] 3.5 [AD] 10.3-10.4.
• F 3/7. continuation map, isotopy homotopy. [S] 3.4 [AD] 11.1-11.6.
• W 3/12. Lagrangians, Lagrangian-Floer homology. [Au] § 1.
• F 3/14. A-infinity algebras/categories/functors. [Au] § 2.
• W 3/26. Disk trees, Fukaya category A-infinity-operations & [Au] § 2-4
• F 3/28. Heegaard splittings/diagrams/moves, spin-c structures. [OS] § 1-6
• W 4/2. Heegaard-Floer groups, applications. [OS] § 7-10
• F 4/4. filtered complexes/chain maps/homotopies, examples.
• W 4/9. spectral invariants in Morse/Floer homology, applications.
• F 4/11. Liouville domains, max ppl, SH and properties. [We] § 2-3.
• W 4/16. S1-equivariant SH, applications (dynamics, capacities). [We] § 4
• F 4/18. Hamiltonian manifolds, symplectic cobordisms. [BE+] § 1-3
• W 4/23. Deligne-Mumford space, energy, buildings. [BE+] § 4-8
• F 4/25. SFT index, SFT compactness, applications. [BE+] § 10-12.
• W 4/30. flavors of SFT, applications.
• F 5/2. Final Presentations.