MATH 270Z: Symplectic geometry

The precise topics discussed in the course may decide on the interests of the class (and the whims of the instructor), but to give a flavor of the material to be covered, I have listed here some references which suggest the central concerns of the course:

• Kashiwara-Schapira, Sheaves on Manifolds
• Guillermou, Sheaves and symplectic geometry of cotangent bundles, https://arxiv.org/pdf/1905.07341.pdf
• Tamarkin, Microlocal criterion for non-displaceability, https://arxiv.org/pdf/0809.1584.pdf
• Zhang, Quantitative Tamarkin category, https://arxiv.org/pdf/1807.09878.pdf
• Kuwagaki, Introduction to sheaf quantization https://arxiv.org/pdf/2205.02661.pdf
• Nadler-Shende, Sheaf quantization in Weinstein symplectic manifolds, https://arxiv.org/pdf/2007.10154.pdf

Possible additional topics to be discussed: The role of the J-homomorphism in microlocal sheaf theory; irregular Riemann-Hilbert correspondence; perverse sheaves and perverse schobers.

Undergraduate students interested in taking this course should seek permission from me first. Undergraduates and pre-quals graduate students, who require grades for the course, will be required to prepare a final project or presentation, and possibly also to complete exercises I assign during the course.

Further references:

Hyperfunctions:

• Komatsu, An introduction to the theory of hyperfunctions https://link-springer-com.ezp-prod1.hul.harvard.edu/content/pdf/10.1007/BFb0068144
• Martineau, Les hyperfonctions de M. Sato http://www.numdam.org/item/SB_1960-1961__6__127_0.pdf
• Bony, Hyperfonctions et équations aux dérivées partielles http://archive.numdam.org/article/SB_1976-1977__19__73_0.pdf

Background on sheaves:

• Dimca, Sheaves in Topology
• Virk, some operations on sheaves: http://rvirk.com/notes/topology2012/operations.pdf

Background on derived categories:

• Mazel-Gee, An invitation to higher algebra https://etale.site/teaching/w21/math-128-lecture-notes.pdf
• Mazel-Gee, The Zen of infinity-categories, https://etale.site/writing/zen-of-infty-cats.pdf

Helpful notes:

• Li, Sheaf theory in symplectic geometry
• Notes from Nadler seminar on microlocal sheaves: https://math.berkeley.edu/~phaine/#GRT-2021-2022

Legendrian knots and front projections:

• Ng, Gallery of Legendrian knots
• Entyre, Legendrian and transversal knots
• Casals, Mastering the Art of Front Cooking
• Shende-Treumann-Zaslow, Legendrian knots and constructible sheaves

Persistent homology:

• Jun Zhang, Quantitative Tamarkin Category (linked at top)
• Žiga Virk, Introduction to persistent homology
• Gary Koplik Persistent Homology: A Non-Mathy Introduction with Examples
• Gunnar Carlsson, Topology and data