Plan of Talks

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

1. Overview of week. Nick's notes
   
2. (Ian Shipman) Tame geometry. Subanalytic geometry. Defining functions. Stratifications and triangulations. Thom isotopy lemmas. Example of real line. Refs: [BM88], [VM96] Nick's notes
   
3. (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
   
4. (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
   
5. (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

1. (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
   
2. (Thomas Bitoun) Characteristic cycles. From constructible sheaves to conical Lagrangian cycles. Functoriality under maps. Refs: [KS84], [SV96] Nick's notes
   
3. (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
   
4. (Sarah Kitchen) Riemann-Hilbert correspondence. Differential operators as quantization of functions on cotangent bundle. Algebraic model of constructible sheaves: regular holo- nomic D-modules. Refs: [Be], [Kap] Nick's notes
   
5. Discussion Nick's notes
   

Day 3: Wednesday 6/15

1. Overview Nick's notes
   
2. (Agnes Gadbled) Exact Floer-Fukaya theory. Fukaya category of compact exact Lagrangians in exact symplectic target. Brane structures. Moduli spaces of disks. Organization into A∞ -category. Refs: [S] Nick's notes
   
3. (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
   
4. (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
   
5. (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

1. (James Pascaleff) Mirror symmetry for toric varieties. Fukaya category of cotangent bundle of torus. Refs: [FLTZ] and related papers. Nick's notes
   
2. (Nick Rozenblyum) Springer theory. Fukaya category of cotangent bundle of Lie algebra. Fourier transform from Floer perspective. Refs: [BoM81], [Nspr] Nick's notes
   
3. (Travis Schedler) Microlocalization and Hamiltonian reduction. Formalism of microlocaliza- tion and Hamiltonian reduction. Introduction to crepant resolutions and their quanti- zations. Nick's notes
   
4. (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

1. (David Nadler) Gauge theory setting. Hitchin integrable system. Relation to talks of previous day. Challenge of quantization of fibers. Refs: [BD], [KW], [Kap]
2. (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,” preprint. link
• [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 e d’intersection pour les vari´t´s nilpotentes,” C. R. Acad. Sci. Paris S´r. I Math. 292 (1981), no. ee e 15, 707–710.
• [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, 77–129. link
• [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 u
• [KhS] Mikhail Khovanov, Paul Seidel “Quivers, Floer cohomology, and braid group actions”, arXiv:math/0006056. link
• [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, 2001. link
• [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– 619. link
• [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”, arXiv:math/0405089. link
• [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 a
• [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