Math 235:  Fukaya Categories
Winter 2021

Syllabus

Fukaya categories occupy a central role in modern mathematics, at the junction of algebraic geometry, symplectic geometry, low-dimensional topology, and mathematical physics.  The goal of this course is to give an introduction to Lagrangian Floer homology, Fukaya categories, and homological mirror symmetry.

It is highly recommended that you concurrently take Sucharit Sarkar's Math 236 on Heegaard Floer homology.  Heegaard Floer homology is an example of a Lagrangian Floer homology theory with an outsize influence on low-dimensional topology. 

Instructor: Ko Honda
Office Hours: Mondays 2:30-3:30pm or by appointment
E-mail: honda at math dot ucla dot edu
URL: http://www.math.ucla.edu/~honda

Class Meetings:  I plan to record the lectures.  Lectures are MWF 12 noon - 12:50pm on Zoom.

Topics
  1. Some symplectic geometry
  2. Lagrangian Floer (co-)homology
  3. A-infinity algebras and A-infinity categories
  4. Construction of the Fukaya categories and some variants including the wrapped Fukaya category
  5. Exact triangles, twists, split generation
  6. Homological mirror symmetry e.g. of the torus
Prerequisites: Math 225 sequence or equivalent (a good knowledge of differentiable manifolds and homology).  Some knowledge of symplectic geometry is helpful, but not necessary.

Grading: TBA

References

Basics of symplectic geometry:
  1. D. McDuff and D. Salamon, Introduction to symplectic topology, 2nd edition, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998.
  2. A. Cannas da Silva, Lectures on symplectic geometry.
A-infinity algebras:
  1. B. Keller, Introduction to A-infinity algebras and modules.

Fukaya categories:

  1. D. Auroux, A beginner's introduction to Fukaya categories.
  2. P. Seidel, Fukaya categories and Picard-Lefschetz theory.

WARNING:  The course syllabus provides a general plan for the course; deviations may become necessary.
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Last modified: January 3, 2021.