Math 274: Topics in algebra | Constructible sheaves and symplectic geometry

Instructor: Vivek Shende

Lectures: Tu-Th 12:30 - 2:00 PM, 5 Evans
Office Hours: by appointment

Prerequisites: Basic understanding of manifolds and algebra.

Syllabus: This course will be an introduction to the microlocal study of constructible sheaves with an emphasis on their recent interactions with questions of symplectic geometry. The course will begin from the beginning (you needn't know what a sheaf is) but will be mostly focused on examples, applications, and open questions.

These come in various flavors. On the one hand, constructible sheaf categories serve as invariants which are strong enough to address classical symplecto-geometric questions such as nonsqueezing, nondisplaceability, capacities, the distinguishing of Legendrian knots, etc. One main goal of the course is to explain the ``sheaf quantization'' approach of Tamarkin and Guillermou-Kashiwara-Schapira to these questions.

On the other hand, many algebraic structures and their relations arise naturally from these symplectic manipulations. The sense in which the constructible sheaf theory (and this class) is algebraic in flavor is that these structures and invariants are eminently calculable. Examples include: categorical representations of braid groups (and consequently certain knot homologies), and cluster algebras.

A final, or underlying, topic is the role of constructible sheaf theory in homological mirror symmetry; or in other words, the relationship between the two ``hands'' above.

Grading: Undergraduates taking this class for a grade will be required to give final presentations to the class on a subject to be agreed on with the instructor.

Course texts:

Kashiwara and Schapira, Sheaves on Manifolds.

Tamarkin, Microlocal condition for non-displaceablility

Viterbo, An introduction to Symplectic topology through sheaf theory.

Guillemou, Kashiwara, Schapira, Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems, Duke Math J. 161 (2012) 201–245.

Guillermou, Schapira, Microlocal theory of sheaves and Tamarkin’s non displaceability theorem

Guillermou, Quantization of conic Lagrangian submanifolds of cotangent bundles

Shende, Generating families and constructible sheaves

Shende, The conormal torus is a complete knot invariant

Other useful texts:

Various lecture notes of Schapira, particularly this review of microlocal sheaf theory

Nadler, Zaslow, Constructible sheaves and the Fukaya category

Nadler, Microlocal branes are constructible sheaves

Shende, Treumann, Zaslow, Legendrian knots and constructible sheaves

Qiaochu Yuan's notes from David Nadler's 2013 class on microlocal geometry.

Weinstein's Lectures on the geometry of quantization
(Question: how is the use of generating families for producing solutions to the Schroedinger equation related to their use in producing sheaves?)

Lecture notes

Lecture 1: One dimensional examples.

Lecture 2: Sheafification, Stratifications, and Cech cohomology.

Lecture 3: Sheaves on the real plane constructible with respect to the stratification by the axes.

Lecture 4: more two dimensional examples. The pictures were somewhat elaborate, so no notes may appear for some time, but the examples are drawn from this paper.

Lecture 5: categorical generalities and some homological algebra.

Lecture 6: derived functors, derived pushforward, a quiver example of a derived equivalence.

Lecture 7: lower shriek and upper shriek.

Lecture 7.5: introduction to integral kernels: the Fourier transform in one variable.

Lecture 8: Tools of the trade: base change, devissage, Verdier duality.

Lecture 8.5: The geodesic flow kernel

Lecture 9: Microsupport and the push-forward estimate (stuff from chapter 5 of Kashiwara-Schapira)

Lecture 10: Cotangent bundles, jet bundles, generating families, Hamiltonian isotopy, . . .

Lecture 11: Deducing the nonexistence of compact exact embedded Lagrangians in R^2n from the existence of sheaves with given microsupport.

Lecture 12: Last time, we saw that Hom(F, F) is a good invariant of a sheaf to be used for the purposes of doing symplectic geometry. In this class I explained the more classical interpretation of this invariant in case the sheaf comes from a generating family (from sections 2 and 3 of this paper).

Exercise: read more about generating families, e.g. in Viterbo's Symplectic topology as the geometry of generating functions. By replacing "generating function" with "sheaf" and the Laudenbach-Sikorav theorems on the existence of generating functions with the Guillermou-Kashiwara-Schapira and Guillermou theorems on sheaf quantization, remove any hypotheses that the symplectomorphisms which appear need be Hamiltonian.

Lecture 13: Statement of the sheaf quantization theorems of Guillermou-Kashiwara-Schapira and Guillermou and outline of the strategy of the proof; some discussion of the Maslov class

Lecture 14: More microsupport estimates (from chap. 5 of Kashiwara-Schapira).

Lecture 15: Another application of sheaf quantization: the conormal torus is a complete knot invariant.

Lectures 16 and 17: Microlocalization. (about the mu-hom functor; various things in chapters 3, 4, 6 of Kashiwara-Schapira)

Lecture 18: Quantization of Hamiltonian isotopies, after Guillermou, Kashiwara, Schapira.

Lectures 19, 20: Cluster varieties from Legendrian knots (we saw how various things, e.g. positroid varieties, wild character varieties could arise as moduli of sheaves with prescribed microlocal constraints; and also how structures on them arise from isotopy. The reference is my paper with the same name.)

Lecture 21: Quantization of Lagrangians, after Guillermou (I)