Applications in Symplectic Geometry

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


In this chapter, we discuss various applications of Tamarkin categories in symplectic geometry. We start with a presentation of the Guillermou-Kashiwara-Schapira sheaf quantization, which associates to a homogeneous Hamiltonian diffeomorphism a complex of sheaves with a certain geometric constraint. Next, following the work of Asano and Ike, we establish a stability result with respect to the Hofer norm. Explicitly, the interleaving distance (defined in a Tamarkin category in the previous chapter) provides a lower bound of the Hofer norm. This is followed by an interesting application to displacement energies of subsets of a cotangent bundle. Further, following Chiu’s work, we study in detail a restrictive Tamarkin category associated to an open domain U of a Euclidean space. The core of this subject lies in a concept called U-projector. There is a section, from a joint work with Leonid Polterovich, devoted to a geometric interpretation of a U-projector. Finally, after defining a sheaf invariant of a domain, we give a quick proof of Gromov’s non-squeezing theorem.


  1. 1.
    Asano, T., Ike, Y.: Persistence-like distance on Tamarkin’s category and symplectic displacement energy (2017). Preprint. arXiv: 1712.06847Google Scholar
  2. 2.
    Audin, M., Damian, M.: Morse Theory and Floer Homology. Springer, London (2014)CrossRefGoogle Scholar
  3. 4.
    Biran, P., Polterovich, L., Salamon, D.: Propagation in Hamiltonian dynamics and relative symplectic homology. Duke Math. J. 119(1), 65–118 (2003)MathSciNetCrossRefGoogle Scholar
  4. 10.
    Chiu, S.F.: Nonsqueezing property of contact balls. Duke Math. J. 166(4), 605–655 (2017)MathSciNetCrossRefGoogle Scholar
  5. 16.
    Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120(4), 575–611 (1989)MathSciNetCrossRefGoogle Scholar
  6. 17.
    Floer, A., Hofer, H.: Symplectic homology I open sets in \(\mathbb C^n\). Math. Z. 215(1), 37–88 (1994)Google Scholar
  7. 19.
    Frauenfelder, U., Ginzburg, V., Schlenk, F.: Energy-capacity inequalities via an action selector. In: Geometry, Spectral Theory, Groups, and Dynamics. Contemporary Mathematics, vol. 387, pp. 129–152. AMS, Providence (2005)Google Scholar
  8. 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
  9. 29.
    Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel (1994)CrossRefGoogle Scholar
  10. 32.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1990). With a chapter in French by Christian HouzelGoogle Scholar
  11. 35.
    Lalonde, F., McDuff, D.: The geometry of symplectic energy. Ann. Math. 141, 349–371 (1995)MathSciNetCrossRefGoogle Scholar
  12. 36.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 2nd edn. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  13. 42.
    Polterovich, L.: Symplectic displacement energy for Lagrangian submanifolds. Ergodic Theory Dynam. Syst. 13(2), 357–367 (1993)MathSciNetCrossRefGoogle Scholar
  14. 43.
    Polterovich, L., Rosen, D., Samvelyan, K., Zhang, J.: Topological Persistence in Geometry and Analysis (2019). Preprint. arXiv: 1904.04044Google Scholar
  15. 47.
    Salamon, D.: Lectures on Floer homology. In: Symplectic Geometry and Topology (Park City, UT, 1997), vol. 7, pp. 143–229. American Mathematical Society, Providence (1999)Google Scholar
  16. 50.
    Schlenk, F.: Applications of Hofer’s geometry to Hamiltonian dynamics. Comment. Math. Helv. 81(1), 105–121 (2006)MathSciNetCrossRefGoogle Scholar
  17. 56.
    Usher, M.: The sharp energy-capacity inequality. Commun. Contemp. Math. 12(03), 457–473 (2010)MathSciNetCrossRefGoogle Scholar
  18. 57.
    Usher, M.: Hofer’s metrics and boundary depth. Annales scientifiques de l’École Normale Supérieure 46(1), 57–129 (2013)MathSciNetCrossRefGoogle Scholar
  19. 60.
    Viterbo, C.: Functors and computations in Floer homology with applications, I. Geom. Funct. Anal. GAFA 9(5), 985–1033 (1999)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

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

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