Math 290 - Seminar on Mirror Symmetry - Fall 2011
D. Auroux -
Thu., 5-6:30pm, Room 891 Evans
Outline
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.
What's happening
- 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.
Further optional topics (not this semester!):
- The other half of HMS for CP^1 and CP^2: Clifford tori and
matrix factorizations.
- Algebraic structures on Fukaya categories of Lefschetz fibrations.
or in a different direction:
- Lagrangian correspondences, quilts, and Fukaya categories
- Correspondences, symmetric products, and Heegaard-Floer homology.
References
The main reference is:
- P. Seidel, Fukaya categories and Picard-Lefschetz theory,
Zurich Lectures in Advanced Mathematics, European Math. Soc., Zurich,
2008.
Other references:
- 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)
For the optional later topics:
- 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