• Jun Zhang
Part of the CRM Short Courses book series (CRMSC)


This chapter gives a brief overview of the primary materials in this book. It starts from the background of symplectic geometry with two famous results: Gromov’s non-squeezing theorem and Arnold’s conjecture (Lagrangian version). Then a discussion on the key concept of singular support follows, with an emphasis on its geometric interpretation. With the concept of singular support, Tamarkin categories will be described, and the Guillermou-Kashiwara-Schapira sheaf quantization will be formulated. These form the underlying platform where various symplectic objects can be expressed in terms of sheaves. Moreover, there is a section devoted to the background material on persistence k-modules, which can be viewed as elements in a special Tamarkin category; there is another section introducing Hofer’s geometry, which is an iconic quantitative apparatus in symplectic geometry. Finally, a brief argument showing that the sheaf counterpart of the standard symplectic homology can be constructed from a certain projector in a Tamarkin category will be provided. This yield an alternative approach to study domains of Euclidean spaces.


  1. 1.
    Asano, T., Ike, Y.: Persistence-like distance on Tamarkin’s category and symplectic displacement energy (2017). Preprint. arXiv: 1712.06847Google Scholar
  2. 7.
    Chaperon, M.: Phases génératrices en géométrie symplectique. In: Les rencontres physiciens-mathématiciens de Strasbourg-RCP25, vol. 41, pp. 191–197 (1990)Google Scholar
  3. 10.
    Chiu, S.F.: Nonsqueezing property of contact balls. Duke Math. J. 166(4), 605–655 (2017)MathSciNetCrossRefGoogle Scholar
  4. 14.
    D’Agnolo, A., Kashiwara, M.: Riemann-Hilbert correspondence for holonomic \({\mathcal D}\)-modules. Publications mathématiques de l’IHÉS 123(1), 69–197 (2016)Google Scholar
  5. 15.
    Eliashberg, Y., Kim, S.S., Polterovich, L.: Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol. 10(3), 1635–1747 (2006)MathSciNetCrossRefGoogle Scholar
  6. 18.
    Fraser, M.: Contact non-squeezing at large scale in \(\mathbb {R}^{2n} \times S^1\). Int. J. Math. 27(13), 1650107 (2016)Google Scholar
  7. 22.
    Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)MathSciNetCrossRefGoogle Scholar
  8. 25.
    Guillermou, S.: Sheaves and symplectic geometry of cotangent bundles (2019). Preprint. arXiv: 1905.07341Google Scholar
  9. 26.
    Guillermou, S., Kashiwara, M., Schapira, P.: Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems. Duke Math. J. 161(2), 201–245 (2012)MathSciNetCrossRefGoogle Scholar
  10. 29.
    Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel (1994)CrossRefGoogle Scholar
  11. 31.
    Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 20(2), 319–365 (1984)MathSciNetCrossRefGoogle Scholar
  12. 34.
    Kashiwara, M., Schapira, P.: Persistent homology and microlocal sheaf theory. J. Appl. Comput. Topol. 2(1–2), 83–113 (2018)MathSciNetCrossRefGoogle Scholar
  13. 36.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 2nd edn. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  14. 37.
    McDuff, D., Salamon, D.: J-holomorphic Curves and Symplectic Topology, vol. 52. American Mathematical Society, Providence (2012)zbMATHGoogle Scholar
  15. 38.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22(1), 233–286 (2009)MathSciNetCrossRefGoogle Scholar
  16. 39.
    Ng, L., Rutherford, D., Shende, V., Sivek, S., Zaslow, E.: Augmentations are sheaves (2015). Preprint. arXiv: 1502.04939Google Scholar
  17. 40.
    Oancea, A.: A survey of Floer homology for manifolds with contact type boundary or symplectic homology. In: Symplectic geometry and Floer homology. A survey of Floer homology for manifolds with contact type boundary or symplectic homology. Ensaios Matemáticos, vol. 7, pp. 51–91. Sociedade Brasileira de Matemática, Rio de Janeiro (2004)Google Scholar
  18. 42.
    Polterovich, L.: Symplectic displacement energy for Lagrangian submanifolds. Ergodic Theory Dynam. Syst. 13(2), 357–367 (1993)MathSciNetCrossRefGoogle Scholar
  19. 44.
    Polterovich, L., Shelukhin, E.: Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Selecta Math. (N.S.) 22(1), 227–296 (2016)Google Scholar
  20. 48.
    Sandon, S.: Contact homology, capacity and non-squeezing in \(\mathbb {R}^{2n}\times S^{1} \) via generating functions. Ann. Inst. Fourier 61(1), 145–185 (2011)Google Scholar
  21. 49.
    Sandon, S.: Generating functions in symplectic topology. Lecture notes for the CIMPA research school on geometric methods in classical dynamical systems, Santiago (2014)Google Scholar
  22. 51.
    Shende, V., Treumann, D., Zaslow, E.: Legendrian knots and constructible sheaves. Invent. Math. 207(3), 1031–1133 (2017)MathSciNetCrossRefGoogle Scholar
  23. 52.
    Tamarkin, D.: Microlocal condition for non-displaceability. In: Algebraic and Analytic Microlocal Analysis, pp. 99–223. Springer, Cham (2013)Google Scholar
  24. 53.
    Tamarkin, D.: Microlocal category (2015). Preprint. arXiv: 1511.08961Google Scholar
  25. 54.
    Traynor, L.: Symplectic homology via generating functions. Geom. Funct. Anal. 4(6), 718–748 (1994)MathSciNetCrossRefGoogle Scholar
  26. 55.
    Tsygan, B.: A microlocal category associated to a symplectic manifold. In: Algebraic and Analytic Microlocal Analysis, pp. 225–337. Springer, Cham (2013)Google Scholar
  27. 57.
    Usher, M.: Hofer’s metrics and boundary depth. Annales scientifiques de l’École Normale Supérieure 46(1), 57–129 (2013)MathSciNetCrossRefGoogle Scholar
  28. 59.
    Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292(1), 685–710 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Jun Zhang
    • 1
  1. 1.Département de Mathématiques et StatistiqueUniversity of MontrealMontréalCanada

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