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Applications in Symplectic Geometry

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

Abstract

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.

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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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