Onedimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multidimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2dimensional persistence module structure associated with persistent cohomology, where one parameter is the cuplength
Commutative diagrams of vector spaces and linear maps over
 NSFPAR ID:
 10518635
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Journal of Applied and Computational Topology
 ISSN:
 23671726
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract and the other is the filtration parameter. This new persistence structure, called the$$\ell \ge 0$$ $\ell \ge 0$persistent cup module , is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cuplength parameter , we obtain a 1dimensional persistence module, called the persistent$$\ell $$ $\ell $ cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a$$\ell $$ $\ell $persistent invariant , which extends both therank invariant (also referred to aspersistent Betti number ), Puuska’s rank invariant induced by epimonopreserving invariants of abelian categories, and the recentlydefinedpersistent cuplength invariant , and we establish their stability. This generalized notion of persistent invariant also enables us to lift the LyusternikSchnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LScategory invariant . 
Abstract We construct an example of a group
for a finite abelian group$$G = \mathbb {Z}^2 \times G_0$$ $G={Z}^{2}\times {G}_{0}$ , a subset$$G_0$$ ${G}_{0}$E of , and two finite subsets$$G_0$$ ${G}_{0}$ of$$F_1,F_2$$ ${F}_{1},{F}_{2}$G , such that it is undecidable in ZFC whether can be tiled by translations of$$\mathbb {Z}^2\times E$$ ${Z}^{2}\times E$ . In particular, this implies that this tiling problem is$$F_1,F_2$$ ${F}_{1},{F}_{2}$aperiodic , in the sense that (in the standard universe of ZFC) there exist translational tilings ofE by the tiles , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in$$F_1,F_2$$ ${F}_{1},{F}_{2}$ ). A similar construction also applies for$$\mathbb {Z}^2$$ ${Z}^{2}$ for sufficiently large$$G=\mathbb {Z}^d$$ $G={Z}^{d}$d . If one allows the group to be nonabelian, a variant of the construction produces an undecidable translational tiling with only one tile$$G_0$$ ${G}_{0}$F . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 
Abstract We prove that the Hilbert scheme of
k points on ($${\mathbb {C}}^2$$ ${C}^{2}$ ) is selfdual under threedimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant Ktheory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^{k}\left[{C}^{2}\right]$ action. First, we find a twoparameter family$${\mathbb {C}}^\times _\hbar $$ ${C}_{\u0127}^{\times}$ of selfmirror quiver varieties of type A and study their quantum Ktheory algebras. The desired quantum Ktheory of$$X_{k,l}$$ ${X}_{k,l}$ is obtained via direct limit$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^{k}\left[{C}^{2}\right]$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (qLanglands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$l\longrightarrow \infty $$ $l\u27f6\infty $ opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsionfree rank$$\hbar $$ $\u0127$N sheaves on with the help of a different (threeparametric) family of type A quiver varieties with known mirror dual.$${\mathbb {P}}^2$$ ${P}^{2}$ 
Abstract The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval
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