Plan of Talks
This is an outline of the schedule of talks at the workshop.
Update 5/25: An expanded
syllabus is now online. It contains detailed descriptions of what
each talk ought to cover, as well as a summary of goals for the workshop. Please have a look!
Day 1: Monday 6/13
Singularities and constructible sheaves
- Overview of week. Nick's notes
- (Ian Shipman) Tame geometry. Subanalytic geometry. Deﬁning functions. Stratiﬁcations and
triangulations. Thom isotopy lemmas. Example of real line.
Refs: [BM88], [VM96] Nick's notes
- (Hiro Tanaka) Homotopical categories. Differential graded and A∞ -categories. Functors and
modules. Linear structure: shifts and cones. Localization with respect to collection of
morphisms. Homological perturbation theory.
Refs: [Ke06], [S], [L] Nick's notes
- (Justin Curry) Constructible sheaves. Differential graded category of sheaves. Functoriality
under maps. Standard triangles and bases. Relation to constructible functions.
Refs: [KS84], [GM83] Nick's notes
- (Daniel Rowe) Examples. Constructible sheaves on S^1 stratiﬁed with a single marked point.
Constructible sheaves on A^1 stratiﬁed with a single marked point. Constructible sheaves
on Schubert stratiﬁcations. Nick's notes
Day 2: Tuesday 6/14
Microlocal geometry of sheaves
- (Aliakbar Daemi) Cotangent bundles. Exact symplectic structure. Geodesic ﬂow. Examples of
Lagrangians: conormals, graphs and generalizations. Conormals to stratiﬁcation. Con-
ical Lagrangian cycles.
Refs: [A], [KS84] Nick's notes
- (Thomas Bitoun) Characteristic cycles. From constructible sheaves to conical Lagrangian cycles.
Functoriality under maps.
Refs: [KS84], [SV96] Nick's notes
- (Faisal Al-Faisal) Intersection of Lagrangian cycles. Perturbations near inﬁnity. Gradings. In-
tersections of characteristic cycles: index theorems, compatibility with multiplication
of constructible functions.
Refs: [GrM97], [NZ09] Nick's notes
- (Sarah Kitchen) Riemann-Hilbert correspondence. Diﬀerential operators as quantization of
functions on cotangent bundle. Algebraic model of constructible sheaves: regular holo-
Refs: [Be], [Kap] Nick's notes
- Discussion Nick's notes
Day 3: Wednesday 6/15
Exact Lagrangians in cotangent bundles
- Overview Nick's notes
- (Agnes Gadbled) Exact Floer-Fukaya theory. Fukaya category of compact exact Lagrangians in
exact symplectic target. Brane structures. Moduli spaces of disks. Organization into
Refs: [S] Nick's notes
- (Thomas Kragh) Morse category of submanifolds. Gradient tree A∞ -category of submanifolds
with local systems. Equivalence with constructible sheaves.
Refs: [KS01], [NZ09] Nick's notes
- (Toly Preygel) Inﬁnitesimal Fukaya category of cotangent bundle. Noncompact branes:
perturbations, taming, bounds on disks. Comparisons with directed and wrapped
Fukaya categories. Equivalence of subcategory of standard branes with Morse cate-
gory of submanifolds.
Refs: [S], [Sik94], [FO97], [NZ09], [Nspr] Nick's notes
- (Dario Beraldo) Equivalence of sheaves and branes. Formalism of Yoneda lemma and bimod-
ules. Beilinson’s argument. Decomposition of diagonal. Noncharacteristic motions.
Refs: [B78], [N09], [Nspr] Nick's notes
Day 4: Thursday 6/16
Some examples and applications
- (James Pascaleff) Mirror symmetry for toric varieties. Fukaya category of cotangent bundle of
Refs: [FLTZ] and related papers. Nick's notes
- (Nick Rozenblyum) Springer theory. Fukaya category of cotangent bundle of Lie algebra. Fourier
transform from Floer perspective.
Refs: [BoM81], [Nspr] Nick's notes
- (Travis Schedler) Microlocalization and Hamiltonian reduction. Formalism of microlocaliza-
tion and Hamiltonian reduction. Introduction to crepant resolutions and their quanti-
zations. Nick's notes
- (Chris Dodd and Sheel Ganatra) W-algebras from topological viewpoint. Fukaya category beyond compact
branes in Slodowy slices.
Refs: [KhS], [SS], [Ma], [Lo] among many related papers. Nick's notes on Chris' talk, Nick's notes on Sheel's talk
Day 5: Friday 6/17
- (David Nadler) Gauge theory setting. Hitchin integrable system. Relation to talks of previous
day. Challenge of quantization of ﬁbers.
Refs: [BD], [KW], [Kap]
- (David Nadler) Where to go from here. Nick's notes (incomplete)
- [A] Arnold, V. I. “Mathematical methods of classical mechanics.” Translated from the Russian by
K. Vogtmann and A. Weinstein. Graduate Texts in Mathematics, 60. Springer-Verlag, New York-
Heidelberg, 1978. x+462 pp. ISBN: 0-387-90314-3
- [B78] A. A. Beilinson, “Coherent sheaves on P n and problems in linear algebra,” (Russian) Funktsional.
Anal. i Prilozhen. 12 (1978), no. 3, 68–69; English translation: Functional Anal. Appl. 12 (1978),
no. 3, 214–216 (1979). link
- [BD] A. Beilinson and V. Drinfeld, “Quantization of Hitchin Hamiltonians and Hecke Eigensheaves,”
- [Be] J. Bernstein, “Algebraic theory of D-modules.” link
- [BM88] E. Bierstone and P. Milman, “Semianalytic and subanalytic sets,” Inst. Hautes Etudes Sci.
Publ. Math. 67 (1988), 5–42. link
- [FLTZ] Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, Eric Zaslow. “T-Duality and Homolog-
ical Mirror Symmetry of Toric Varieties”, arXiv:0811.1228 link
- [BoM81] W. Borho and R. MacPherson, “Repr´sentations des groupes de Weyl et homologie
d’intersection pour les vari´t´s nilpotentes,” C. R. Acad. Sci. Paris S´r. I Math. 292 (1981), no.
- [FO97] K. Fukaya and Y.-G. Oh, “Zero-loop open strings in the cotangent bundle and Morse homo-
topy,” Asian. J. Math. 1 (1997) 96–180. link
- [FSS] Kenji Fukaya, Paul Seidel, Ivan Smith, “The symplectic geometry of cotangent bundles from a
categorical viewpoint”. arXiv:0705.3450. link
- [GM83] M. Goresky and R.MacPherson. “Intersection homology. II.” Invent. Math. 72 (1983), no. 1,
- [GrM97] M. Grinberg and R. MacPherson. “Euler characteristics and Lagrangian intersections.” Sym-
plectic geometry and topology (Park City, UT, 1997), 265–293, IAS/Park City Math. Ser., 7,
Amer. Math. Soc., Providence, RI, 1999.
- [Kap] A. Kapustin, “A-branes and noncommutative geometry,” arXiv:hep-th/0502212. link
- [KW] A. Kapustin and E. Witten, “Electric-Magnetic Duality And The Geometric Langlands Pro-
gram,” arXiv:hep-th/0604151. link
- [KS84] M. Kashiwara and P. Schapira, Sheaves on manifolds. Grundlehren der Mathematischen Wis-
senschaften 292, Springer-Verlag (1994).
- [Ke06] B. Keller, On differential graded categories. arXiv:math.AG/0601185. International Congress
of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Z¨rich, 2006. link
- [KhS] Mikhail Khovanov, Paul Seidel “Quivers, Floer cohomology, and braid group actions”,
- [KS01] M. Kontsevich and Y. Soibelman, “Homological Mirror Symmetry and Torus Fibrations,” Sym-
plectic geometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publ., River Edge, NJ,
- [Lo] I. Losev, “Finite W-algebras”, arXiv:1003.5811. link
- [L] J. Lurie, “Stable Inﬁnity Categories,” arXiv:math/0608228. link
- [Ltft] J. Lurie, “On the Classiﬁcation of Topological Field Theories,” arXiv:0905.0465 link
- [Ma] Ciprian Manolescu “Link homology theories from symplectic geometry”, arXiv:math/0601629. link
- [N09] D. Nadler, “Microlocal Branes are Constructible Sheaves,” Selecta Math. 15 (2009), no. 4, 563–
- [Nspr] D. Nadler, “Springer theory via the Hitchin ﬁbration”, arXiv:0806.4566. link
- [NZ09] D. Nadler, E. Zaslow, “Constructible sheaves and the Fukaya category.” J. Amer. Math. Soc.
22 (2009), no. 1, 233–286. link
- [SV96] W. Schmid and K. Vilonen, “Characteristic cycles of constructible sheaves,” Invent. Math. 124
(1996), 451–502. link
- [S] P. Seidel, Fukaya Categories and Picard-Lefschetz Theory.
- [SS] Paul Seidel, Ivan Smith “A link invariant from the symplectic geometry of nilpotent slices”,
- [STZ] N. Sibilla, D. Treumann, E. Zaslow, “Ribbon Graphs and Mirror Symmetry I”, arXiv:1103.2462. link
- [Sik94] J.-C. Sikorav, “Some properties of holomorphic curves in almost complex manifolds,” in Holo-
morphic Curves in Symplectic Geometry, Birkh¨user (1994), 165–189. link
- [Sl80] P. Slodowy, Simple singularities and simple algebraic groups. Lecture Notes in Mathematics,
815. Springer, Berlin, 1980.
- [VM96] L. van den Dries and C. Miller, “Geometric categories and o-minimal structures,” Duke Math.
J. 84, no. 2 (1996), 497–539. link