Math 290 - Seminar on Mirror Symmetry - Fall 2015

D. Auroux - Tuesdays, 3:30-5pm, Room 891 Evans

Outline

This seminar will revisit the themes of past seminars (see here, here and here) on geometric aspects of mirror symmetry.

The seminar will continue into Spring 2016 -- still planning for Tuesdays 3:30-5 in 891 Evans.

The details are to be determined based on the participants' interests. A possible plan for this semester is as follows:

• Lagrangian Floer homology
• Fukaya categories (ordinary and wrapped)
• Deformation theory, generation criteria, etc.
• Homological mirror symmetry
• Computations: pair of pants, genus 2 curve, etc.
• (time permitting): family Floer homology and family Floer functor

Prerequisite: some knowledge of symplectic geometry, ideally including prior exposure to J-holomorphic curves.

Course control number: 54554.

What's happening

• Tuesday 9/1: Overview (D. Auroux)
• Tuesday 9/8: Lagrangian Floer cohomology (A. Hanlon)
• Tuesday 9/15: Lagrangian Floer cohomology continued (A. Hanlon)
• Tuesday 9/22: PSS isomorphisms in Lagrangian Floer homology (B. Filippenko)
• Tuesday 9/29: Products in Lagrangian Floer theory, Fukaya category (M. Weiler)
• Tuesday 10/6: Twisted complexes, mapping cones, and generators (D. Auroux)
• Tuesday 10/13: Lagrangian surgery and mapping cones (K.-Y. Fang)
• Tuesday 10/20: Hochschild cohomology and classification of A_infty-structures (J. Hicks)
• Tuesday 10/27: The Fukaya category of the genus 2 curve (C. Cannizzo) (Notes)
• Tuesday 11/3: no meeting
• Tuesday 11/10: The mirror of the genus 2 curve (C. Cannizzo) (Notes)
• Tuesday 11/17: The wrapped Fukaya category (B. Gammage)
• Tuesday 11/24: The wrapped category of the pair of pants and HMS (D. Auroux) (Notes)
• Tuesday 12/1: Open-closed maps and Abouzaid's generation criterion (J. Hicks)
• Tuesday 12/8: no meeting

Possible References

Notes from past seminars and courses on the topic:

• Fall 2009 topics course on mirror symmetry (Oct 15-Dec 1 lectures)
• Fall 2011 seminar (see also the Spring 2011 seminar)

• P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Math. Soc., Zurich, 2008.
• D. Auroux, A beginner's introduction to Fukaya categories, Contact and Symplectic Topology, Bolyai Soc. Math. Stud. 26, Springer, 2014, pp. 85-136. (arXiv:1301.7056)

• M. Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. IHES 112 (2010), 191-240 (arXiv:1001.4593)
• M. Abouzaid, A cotangent fiber generates the Fukaya category, Adv. Math. 228 (2011), 894-939 (arXiv:1003.4449)
• M. Abouzaid, D. Auroux, A. Efimov, L. Katzarkov, D. Orlov, Homological mirror symmetry for punctured spheres., J. Amer. Math. Soc. 26 (2013), 1051-1083 (arXiv:1103.4322)
• P. Seidel, Fukaya categories and deformations, Proceedings of 2002 ICM (arXiv:0206155)
• P. Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), 727-769 (arXiv:0812.1171)
• R. Thomas, Derived categories for the working mathematician, Proc. Winter School on Mirror Symmetry (Harvard, 1999), International Press, 2001 (arXiv:0001045)
• M. Abouzaid, Family Floer cohomology and mirror symmetry, Proceedings of 2014 ICM (arXiv:1404.2659)