Fall 2015
Mathematics 277
Fukaya Categories, Sheaves, and Cosheaves

Harvard College/GSAS: 159626
Location: Science Center 411
Meeting Time: M., W., F. 1:00 pm- 2:00 pm
Exam Group: FAS08_A
Office Hours for Hiro: Tuesdays 13:30 - 14:30, Wednesdays 14:00 - 15:00.

After setting up the foundations for defining Fukaya categories, we will explore conjectures showing that various Fukaya categories "glue". Little analytic background will be assumed, but we will attempt to cover the foundations.

The syllabus

Click here for the very tentative syllabus.

Talk survey

Fill out this survey to let me know which talk topics you'd like to work on. You may fill out this form even if you don't want to give a talk (e.g., if you're not a student taking this class for a grade) but would still like to work with a group to learn about certain topics.

Piazza

To discuss with other students, go to Piazza. There you will also find possible talk topics for your talks later in this class.

Important dates

  • September 2, 2015 (Wed): Class begins.
  • October 12 - 16 (Mon - Fri): Oct 12 is Columbus Day, so there is no class. And I will not be around the whole week. There will either be student talks, or substitute talks this week.
  • November 25-27 (Wed - Fri): Thanksgiving break
  • December 2 (Wed): Final class

Notes

I will attempt to post notes from all the lectures. Every lecture, two students will also be asked to take notes.
  1. Wed Sept 2. Toward Fukaya Categories. Symplectic manifolds, compatible almost complex structures, Lagrangians, Morse theory introduction to Floer theory. Notes by Boyu Zhang. Notes by Geoff Smith. Comments by Hiro.
  2. Fri Sept 4. Morse Theory. Notes by Koji Shimizu. Notes by Lynnelle Ye. Hiro comments.
  3. Mon Sept 7. Labor day, no class. Do the survey.
  4. Wed Sept 9. Toward the Fukaya category, II. What you get when you compactify one-dimensional components of Floer trajectories. Notes by Svetlana Makarova. Notes by Krishanu Sankar.
  5. Fri Sept 11. dg categories. Notes by Kevin Sackel. Notes by Svetlana Makarova.
  6. Mon Sept 14. Finally seeing the Aoo structures of the Fukaya category. Notes by Svetlana Makarova.
  7. Wed Sept 16. Statement of homological mirror symmetry (baby version). Discussion of the talk topics. Notes by Svetlana Makarova.
  8. Fri Sept 18. Transversality. Notes by Eduard Duryev.
  9. Mon Sept 21. Grading of Lagrangians. Notes by Eduard Duryev. Notes by Jun Hou Fung.
  10. Wed Sept 23. Dimensions of moduli spaces. Notes by Eduard Duryev.
  11. Fri Sept 25. The case of a point, Yoneda embedding. Notes by Eduard Duryev. Notes by Boyu Zhang.
  12. Mon Sept 28. The case of a point, Karoubi completion and Perf Notes by Meng Guo. Notes by Geoffrey Smith (possibly with errors, according to Geoffrey).
  13. Wed Sept 30. Examples in C. Notes by Jun Hou Fung.
  14. Fri Oct 2. Introduction to Arnold conjecture. PSS isomorphism. Notes by Yu-Wei Fan. Notes by Lynnelle Ye.
  15. Mon Oct 5. Partially wrapped Fukaya categories, Abouzaid's theorem, cotangent bundle of the circle. Notes by Jun Hou Fung.
  16. Wed Oct 7. Split-generating categories. Notes by Lynnelle Ye. Notes by Kevin Sackel.
  17. Fri Oct 9. Abouzaid's theorem on cotangent bundles.
  18. Mon Oct 12. No class (holiday).
  19. Wed Oct 14. Guest lecture by Fabian Haiden. Notes by Kevin Sackel.
  20. Fri Oct 16. Guest lecture by Siu-Cheong Lau. Introduction to mirror symmetry and the mirror construction. Notes by Kevin Sackel.
  21. Mon Oct 19. Introduction to TFTs and Hochschild homology, I. Notes by Nati Blaier.
  22. Wed Oct 21. Introduction to TFTs and Hochschild homology, II. Notes by Jun Hou Fung.
  23. Fri Oct 23. TFTs in dimension two. Calabi-Yau and Yau-Calabi. Notes by David Yang. Notes by Kevin Sackel.
  24. Mon Oct 26. Student Talk: Baris (Abouzaid's generation theorem).
  25. Wed Oct 28. Student Talk: Baris, II.
  26. Fri Oct 30. Open-closed CFTs. Notes by Yu-Wei. Notes by Kevin Sackel.
  27. Mon Nov 2. Calabi-Yau algebra examples. Notes by Nati Blaier.
  28. Wed Nov 4. Introduction to oo-categories. Notes by Nati Blaier.
  29. Fri Nov 6. Spaces as oo-categories, local systems. Notes by Baris Kartal.
  30. Mon Nov 9. Local systems. Notes by Nati Blaier.
  31. Wed Nov 11. Costello's theorem about Open-Closed TCFTs and CY categories.
    Fri Nov 13. No class.
    Mon Nov 16. No class.
  32. Wed Nov 18. Proof of Costello's theorem. Notes by Jun Hou Fung.
  33. Fri Nov 20. Fukaya-Seidel category. Notes by Kevin Sackel.
  34. Mon Nov 23. Mirror symmetry for CP^1, Seidel's theorem. Notes by Jun Hou Fung.
    Wed Nov 25. No class
    Fri Nov 27. No class
  35. Mon Nov 30. Student lecture: David Yang, Nadler-Zaslow theorem. Notes by Jun Hou Fung.
  36. Wed Dec 2. Ideas from Kapranov-Kontsevich-Soibelman. Notes by Jun Hou Fung.
  37. Fri Dec 4. Student lecture: Justin Campbell, Nadler-Zaslow theorem.

References

A great reference is Audin and Damian's book on Morse theory and Floer homology. It is free with a Harvard log-in, and if you are analytically inclined, you can find detailed proofs there of the kinds of compactification and gluing results you need in Floer theory, and in Morse theory. They do not deal in-depth with the study of holomorphic curves with boundary on Lagrangians. For the basic idea of how one compactifies moduli spaces and deduces the $A_\infty$ relations for these, using perturbation techniques, you can see Seidel's book (see below). It will help to have a fairly good understanding of Audin's material before trying to understand Seidel, however.

Another good reference, in the case where the analysis is simplest, is Paul Seidel's book on Fukaya categories and Picard-Lefschetz theory. You can find an excellent review of it by Ivan Smith here.

A reference for Morse Theory is Bott's Morse Theory indomitable.

A reference for the ideas of deformation quantization, and the role of Hochschild cohomology, is Kontsevich's Poisson manifold paper.