Course Syllabus

The course will meet in Northwest Building room B108, Mon & Wed 9:00-10:15am.

Course goals:

This graduate topics course will survey various aspects of Lagrangian Floer theory, starting with its introduction to address classical conjectures in symplectic geometry, and the rich algebraic structures captured by the Fukaya category. These will also be discussed through the lens of homological mirror symmetry. Further topics include partially wrapped Fukaya categories, family Floer cohomology, and local-to-global principles for Fukaya categories.

Prerequisites: graduate-level differential geometry and algebraic topology (Math 230a and 231a or equivalent).  Some prior familiarity with symplectic geometry strongly recommended. (Suggested reading: Ana Cannas, Lectures on Symplectic Geometry, through section 18). Prior familiarity with algebraic geometry and homological algebra will be helpful but is not strictly required.  Basic complex analysis.

Students who need a grade will be expected to attend lectures and to complete assignments, write a final paper, and for undergraduates (or non-math grad students), a brief oral examination related to one of the assignments.

References:

• P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Math. Soc., 2008.
• D. Auroux, A beginner's introduction to Fukaya categories
• more to be added

Lecture Notes

I will try to provide handwritten lecture notes for most lectures.

• Lecture 1 (Wed Sept 2): overview of the course: from Lagrangian Floer cohomology to Fukaya categories and homological mirror symmetry
• Lecture 2 (Wed Sept 9): Lagrangian Floer cohomology: setup, Floer's equation, transversality, compactness, definition of the differential.
• Lecture 3 (Mon Sept 14): Lagrangian Floer cohomology continued: coefficients, orientations, index.
• Lecture 4 (Wed Sept 16): Index and grading; examples of Lagrangian Floer cohomology; monotone Lagrangians