Abstract
In this chapter, we give detailed descriptions of the concepts of derived category, persistence k-module, and singular support. These serve as preparations for the topics treated in later chapters. Sections 2.1 to 2.5 are devoted to derived category and derived functors, as well as their applications in the category of sheaves. These are basic ingredients for Tamarkin categories. Sections 2.6 and 2.7 are devoted to the theory of persistence k-module theory, which figures the interleaving distance and barcodes. Two main theorems highlight this theory: Normal Form Theorem and Isometry Theorem. Sections 2.8 and 2.9 are devoted to the definition of the singular support, its various functorial properties, and an important result, the microlocal Morse lemma, which generalizes the classical Morse lemma to a microlocal formulation. This lemma is essential in the constructions of Tamarkin categories.
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- 1.
These are based on lectures given by Yakov Varshavsky at the Hebrew University of Jerusalem in the Fall of 2017.
- 2.
These are based on lectures given by Leonid Polterovich at the Hebrew University of Jerusalem in the Fall of 2017.
- 3.
These are based on a joint work with Asaf Kislev at Tel Aviv University in the Fall of 2016.
- 4.
There is another notion, called soft, which is defined as follows: for any closed subset Z ⊂ X, the restriction map ℱ (X) every F(Z) is surjective. In particular, when X is locally compact, every flabby sheaf is soft. Moreover, soft sheaf is f !-acyclic.
- 5.
Warning on notations: in the previous section, we have defined a family of functors R iF : A→ℬ for i ≥ 0 (see (2.2)), while RF here should be regarded as a new definition/notation.
- 6.
It is possible that there exist more than one left adjoint of G. But Exercise I.2 (ii) in [32] says that all of them are isomorphic, so the left adjoint will be unique (up to isomorphisms). The same is true for the right adjoint.
- 7.
This picture is borrowed from Section 4.2 in [43].
- 8.
This is a collection of graded vector spaces. By the definition of SS(ℱ ), if (x, p) ∈ SS(ℱ ), the collection V x(ϕ) is non-zero for certain degrees.
References
Bauer, U., Lesnick, M.: Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, p. 355. ACM, New York (2014)
Bubenik, P., de Silva, V., Nanda., V.: Higher interpolation and extension for persistence modules. SIAM J. Appl. Algebra Geom. 1(1), 272–284 (2017)
Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp. 237–246. ACM, New York (2009)
Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, Cham (2016)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)
Crawley-Boevery, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(05), 1550066 (2015)
Gabriel, P.: Unzerlegbare darstellungen I. Manuscripta Math. 6(1), 71–103 (1972)
Gelfand, S., Manin, Y.: Methods of Homological Algebra. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2003)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press on Demand, Oxford (2006)
Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1990). With a chapter in French by Christian Houzel
Polterovich, L., Rosen, D., Samvelyan, K., Zhang, J.: Topological Persistence in Geometry and Analysis (2019). Preprint. arXiv: 1904.04044
Viterbo, C.: An introduction to symplectic topology through sheaf theory (2011). Preprint
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
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Zhang, J. (2020). Preliminaries. In: Quantitative Tamarkin Theory. CRM Short Courses. Springer, Cham. https://doi.org/10.1007/978-3-030-37888-2_2
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