Microlocal theory of sheaves in symplectic geometry

Microlocal theory of sheaves in symplectic geometry
(Fall 2022)

Organizers: Johan Asplund and Yash Deshmukh.

Description: The goal of this seminar is to study some basics of the microlocal theory of sheaves and its relationship with symplectic geometry. The first half of the semester will be devoted to basic definitions, some applications and also computations. The second half of the semester will consist of understanding the proof of a theorem by Ganatra–Pardon–Shende which relates the partially wrapped Fukaya category of a Weinstein manifold to a category of microlocal sheaves on its skeleton.

Time and location: Tuesdays 12:00–1:00 pm ET, Math 407.

Schedule

Note: Non-standard seminar dates or locations are marked with ※.

Date	Speaker	Topic	Ref
Sep 27^※	Yash Deshmukh and Sebastian Haney	Basics of microlocal sheaves I Show/hide abstract In the next 1.5 talks, we will begin discussing the basics of microlocal sheaf theory. After recalling some facts about Whitney stratifications and the shriek functors, we will define the microsupport of a sheaf. We will also prove a result characterizing the microsupport of a constructible sheaf.	[GPS18, KS90] Notes
Oct 4	Sebastian Haney	Basics of microlocal sheaves II Show/hide abstract We continue with discussing of the basics of microlocal sheaf theory. After recalling some facts about Whitney stratifications and the shriek functors, we will define the microsupport of a sheaf. We will also prove a result characterizing the microsupport of a constructible sheaf.	[GPS18, KS90] Notes
Oct 11	Alex Xu	Basics of microlocal sheaves III Show/hide abstract We continue developing the terminology for microlocal sheaf theory. We begin by constructing the microstalk 'functor' and discuss various geometric aspects of the construction. Afterwards, we will discuss key properties, such as co-representability and their role within the category Sh_X(M). If time permits, we will attempt to place this into the broader picture of [GPS18].	[GPS18, KS90] Notes
Oct 13^※	Juan Muñoz-Echániz	Nadler–Zaslow correspondence Show/hide abstract It is a basic principle that the symplectic topology of a cotangent bundle reduces to differential topology of the base, by rescaling along the fibres. A particular instance of this is the Nadler–Zaslow correspondence, which asserts an equivalence between the derived category of constructible sheaves on a manifold and an appropriate version of the derived Fukaya category of its cotangent bundle. I will explain how to build the correspondence, sketch some proofs and discuss some applications.	[NZ09] Notes
Oct 25	Johan Asplund	Legendrian knots and constructible sheaves Show/hide abstract Based on the paper by Shende–Treumann–Zaslow we study sheaves with singular support in the positive cone over a null-homologous Legendrian knot in the unit cotangent bundle of R^2. This leads to a combinatorial description of the sheaf category given a front diagram of the Legendrian. It might be regarded as a special case of the Nadler–Zaslow correspondence from the point of view of the Legendrian.	[STZ17] Notes
Nov 1	Yin Li	Wrapped microlocal sheaves on pairs of pants Show/hide abstract Given a stopped Liouville domain, one way to define its (derived) partially wrapped Fukaya category is taking the subcategory of compact objects in the derived category of all A_\infty-modules over the endomorphism algebra of Lagrangian cocores and linking discs. Applying the same idea to sheaves, we get the definition of the dg category of wrapped microlocal sheaves. We will describe its basic properties and explain its calculation for pair-of-pants.	[Nad16] Notes
Nov 8	Sebastian Haney	Arboreal singularities Show/hide abstract Arboreal singularities are a class of Legendrian singularities associated to trees whose microlocal sheaf categories are equivalent to modules over a quiver given by the directed tree. I will explain Nadler's construction of these singularities and the computation of their microlocal sheaf categories. If time permits, we will also see how to use arboreal singularities to compute microlocal sheaf categories of Lagrangian skeleta.	[Nad17]
Nov 15	No seminar
Nov 22	Yash Deshmukh	Ganatra–Pardon–Shende I Show/hide abstract We will begin with the proof of the main theorem (Thm 1.1) in [GPS18]. I will recall some discussion from earlier talks showing that the sheaf categories form a microlocal Morse theater. I will then start discussing the ingredients, such as the stop removal formula, that go into proving that (partially) wrapped Fukaya categories form a microlocal Morse theater.	[GPS18] Notes
Nov 29	Yash Deshmukh	Ganatra–Pardon–Shende II Show/hide abstract We will continue the discussion from the last time and prove that (partially) wrapped Fukaya categories form a microlocal Morse theater.	[GPS18] Notes
Dec 6	Andrew Hanlon	Ganatra–Pardon–Shende III Show/hide abstract We will explore how the correspondence between sheaf categories and wrapped Fukaya categories on cotangent bundles can be extended to certain stopped Liouville manifolds. This follows Section 7 of [GPS18].	[GPS18] Notes

References

[GPS18] S. Ganatra, J. Pardon, and V. Shende. Microlocal Morse theory of wrapped Fukaya categories, (2018). [arxiv:1809.08807]
[KS90] M. Kashiwara and P. Schapira. Sheaves on manifolds, Springer-Verlag, Berlin (1990).
[Nad16] D. Nadler. Wrapped microlocal sheaves on pairs of pants, (2016). [arxiv:1604.00114]
[Nad17] D. Nadler. Arboreal singularities, Geom. Topol., 21 (2017). [arxiv:1309.4122]
[NZ09] D. Nadler and E. Zaslow. Constructible sheaves and the Fukaya category, J. Amer. Math. Soc., 22 (2009). [arxiv:0604379]
[STZ17] V. Shende, D. Treumann, and E. Zaslow. Legendrian knots and constructible sheaves, Invent. Math., 207 (2017). [arxiv:1402.0490]