Math 290 - Seminar on Mirror Symmetry

(and Heegaard-Floer homology)

Spring 2012

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

Outline

This is a continuation of the seminar held last semester (on Lagrangian Floer homology, Fukaya categories, and Lefschetz fibrations).

The plan for this semester is to focus on applications of Floer homology and Fukaya categories in low-dimensional topology, specifically "bordered Heegaard-Floer homology". We will also have a few talks more directly on mirror symmetry.

Prerequisite: prior knowledge of Lagrangian Floer homology (eg from having attended the seminar last semester) will be very helpful; but if there are interested newcomers we can have a review talk on the topic.

What's happening

• 1/19: Overview: Heegaard-Floer homology, Lagrangian correspondences, and Fukaya categories (D. Auroux)
• 1/26: no meeting
• 2/2: Heegaard-Floer homology (Z. Sylvan) [Notes]
• 2/9: Heegaard-Floer homology continued (Z. Sylvan, V. Gripp)
• 2/16: Cylindrical reformulation (V. Gripp) [Notes]; relative Hilbert schemes (H. Lee) [Notes]
• 2/23: Helge Ruddat (Univ. Mainz): Perverse curves and mirror symmetry [Notes]
• 3/1: Lagrangian correspondences between symmetric products (H. Lee) [Notes]
• 3/8: Heegaard-Floer homology from correspondences (D. Auroux); Intro to bordered Floer homology (K. Choi) [Notes]
• 3/15: Bordered Heegaard-Floer homology: strands algebra and CFD (K. Choi) [Notes]
• 3/22: Bordered Heegaard-Floer homology continued (K. Choi)
• 4/5: Siu-Cheong Lau (IPMU): SYZ mirror symmetry for toric Calabi-Yaus
• 4/12: The Fukaya category of the symmetric product of an open Riemann surface (D. Auroux) [Notes]
• 4/19: Bordered-sutured Floer homology (R. Zarev)
• 4/26: Bordered-sutured Floer homology continued (R. Zarev)
• 5/3: Mohammed Abouzaid (Simons Center): Fukaya categories of Stein manifolds

Our core topic:

• Heegaard-Floer homology of closed 3-manifolds
• Lagrangian correspondences in symmetric products
• An alternative description of Heegaard-Floer homology
• Bordered Heegaard-Floer homology
• (Wrapped) Fukaya categories of symmetric products

Other potential talk topics:

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

Guest lectures:

• February 23: Helge Ruddat (Univ. Mainz)
• April 5: Siu-Cheong Lau (IPMU Tokyo)

References

• P. Ozsváth, Z. Szabó, An introduction to Heegaard Floer homology, Clay Math Proc. 5, AMS, Providence, 2006, pp. 3-27; available here.
• P. Ozsváth, Z. Szabó, Heegaard diagrams and holomorphic disks, Different faces of geometry, Int. Math. Ser. 3, Kluwer, New York, 2004, pp. 301-348; arXiv:math.GT/0403029.
• T. Perutz, Lagrangian matching invariants for fibred four-manifolds I, Geom. Topol. 11 (2007), 759-828; arXiv:math.SG/0606061 (section 3).
• K. Wehrheim, C. Woodward, Functoriality for Lagrangian correspondences in Floer theory, arXiv:0708.2851
• K. Wehrheim, C. Woodward, Pseudoholomorphic Quilts, arXiv:0905.1369
• R. Lipshitz, P. Ozsváth, D. Thurston, A tour of bordered Floer theory, Proc. Nat. Acad. Sci. 108 (2011) 8085-8092; arXiv:1107.5621.
• D. Auroux, Fukaya categories and bordered Heegaard-Floer homology, Proc. ICM 2010, vol. II, Hindustan Book Agency, 2010, pp. 917-941; arXiv:1003.2962.

• 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