Persistence and the Sheaf-Function Correspondence
Forum of Mathematics, Sigma, Tome 11 (2023)
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The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$-vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.
@article{10_1017_fms_2023_115,
author = {Nicolas Berkouk},
title = {Persistence and the {Sheaf-Function} {Correspondence}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.115},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.115/}
}
Nicolas Berkouk. Persistence and the Sheaf-Function Correspondence. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.115
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