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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Notes
- 1.
Here, “compactly supported” means \(\mathrm {supp}(\phi _t)/\mathbb {R}_{+}\) is compact.
- 2.
This version is based on Leonid Polterovich’s lecture at the Kazhdan seminar held at the Hebrew University of Jerusalem in the Fall of 2017.
- 3.
This is based on conversations with Leonid Polterovich and Yakov Varshavsky.
- 4.
This comes from the following standard fact concerning the derived category: Let A be an abelian category. If G• ∈D(A) has non-zero cohomology only in degree k, then G• is quasi-isomorphic to the single term complex (0 → h k(G•) → 0) (Exercise).
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Zhang, J. (2020). Applications in Symplectic Geometry. In: Quantitative Tamarkin Theory. CRM Short Courses. Springer, Cham. https://doi.org/10.1007/978-3-030-37888-2_4
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DOI: https://doi.org/10.1007/978-3-030-37888-2_4
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