D. Auroux - Thu., 5-6:30pm, Room 891 Evans

This is a continuation of the seminar held last semester (on geometric aspects of mirror symmetry, the SYZ conjecture, and homological mirror symmetry). The plan for this semester is to focus on geometric and algebraic aspects of Fukaya categories by going over Seidel's book on the subject. If time permits we will then see some applications to homological mirror symmetry.

Prerequisite: some knowledge of symplectic geometry. Having attended last spring's seminar will help in the later part of the semester; but we will aim to remain self-contained as long as possible.

**8/25:**Introduction: Lefschetz fibrations, Fukaya categories, and homological mirror symmetry (D. Auroux) (Notes)**9/1:**A∞ categories (definition, bar complex), modules, functors; Yoneda embedding (D. Auroux) (Notes)**9/8:**Yoneda embedding, exact triangles, mapping cones; twisted complexes (R. Vianna) (Notes)**9/15:**Twisted complexes (R. Vianna) (Notes: see 9/8); Lagrangian Floer homology (C. Gerig) (Notes: see 9/22).**9/22:**Lagrangian Floer homology continued; products and higher products (C. Gerig) (Notes)**9/29:**The Fukaya category: graded Lagrangians, perturbation data (Z. Sylvan) (Notes)**10/6:**Lefschetz fibrations, vanishing cycles (D. Auroux) (Notes)**10/13:**Monodromy, Dehn twists (D. Auroux) (Notes)**10/20:**Exact triangles from Dehn twists (H. Lee) (Notes #1, Notes #2)**10/27:**Exact triangles from Dehn twists continued (H. Lee) (Notes); Application: the vanishing cycles of a pencil generate the Fukaya category of the fiber (D. Auroux) (Notes)**11/3:**The Fukaya category of a Lefschetz fibration (D. Auroux) (Notes)- 11/10: no seminar
**11/16:**M. Abouzaid: HMS for toric varieties (in topology seminar) (Notes)**11/17:**M. Abouzaid: Khovanov homology from Fukaya categories of Hilbert schemes (Notes)**12/1:**S. Ma'u: Lagrangian correspondences and pseudoholomorphic quilts (Notes)**12/8:**S. Ma'u: Lagrangian correspondences and pseudoholomorphic quilts II (Notes)

**Topics we didn't cover:**

- Application: homological mirror symmetry for CP^1 and CP^2: the Beilinson resolution of the diagonal, generators of the derived category; the mirror, its vanishing cycles, the Fukaya category.
- Application: exotic symplectic manifolds (after Maydanskiy-Seidel).

**Optional topics for even later** (depending on interests and time):

- HMS for the genus 2 curve.
- The wrapped Fukaya category (definition and examples).
- Abouzaid's generation criterion.

- The other half of HMS for CP^1 and CP^2: Clifford tori and matrix factorizations.
- Algebraic structures on Fukaya categories of Lefschetz fibrations.

- Lagrangian correspondences, quilts, and Fukaya categories
- Correspondences, symmetric products, and Heegaard-Floer homology.

The main reference is:

- P. Seidel,
*Fukaya categories and Picard-Lefschetz theory*, Zurich Lectures in Advanced Mathematics, European Math. Soc., Zurich, 2008.

- P. Seidel,
*Vanishing cycles and mutation*, European Congress of Mathematics Vol II (Barcelona 2000), Progr. Math. 202, Birkhäuser, 2001, pp. 65-85; arXiv:math.SG/0007115 - P. Seidel,
*A long exact sequence for symplectic Floer cohomology*, Topology, 42 (2003) 1003-1063; arXiv:math.SG/0105186 - D. Auroux, L. Katzarkov, D. Orlov,
*Mirror symmetry for weighted projective planes and their noncommutative deformations*, Ann. Math., 167 (2008) 867-943; arXiv:math.AG/0404281 - M. Maydanskiy,
*Exotic symplectic manifolds from Lefschetz fibrations,*arXiv:0906.2224 (also: M. Maydanskiy and P. Seidel, arXiv:0906.2230)

- P. Seidel,
*Homological mirror symmetry for the genus two curve*, arXiv:0812.1171 - M. Abouzaid,
*A geometric criterion for generating the Fukaya category*, arXiv:1001.4593 - P. Seidel,
*Fukaya A_∞ structures associated to Lefschetz fibrations I*, arXiv:0912.3932