Related papers: On interval decomposition of persistence modules
After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define…
By utilizing domain theory, we generalize the notion of an ephemeral module to the so-called continuous posets. We investigate the quotient category of persistence modules by the Serre subcategory of ephemeral modules and show that it is…
This paper provides a method to calculate the bar codes of a PCD (point cloud data) with real coefficients in Section 3. With Dan Burghelea and Tamal Dey we developed a persistence theory which involves level sets discussed in Section 4.…
In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a $2$D persistence module…
In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under…
We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group…
We study the indivisible intervals and the monotonicity of the growth of the exponential type in the chains of de Branges subspaces in terms of the spectral measure. We prove that for spectral measures supported on $\mathbb Z$, there exist…
Understanding the structure of indecomposable $n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular…
In this paper, we prove an existence theorem of a local moduli space for geometric structures in a very general setting. Then to show the interest of this result, we apply it to the case of sasakian and Sasaki-Einstein structures.
This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric…
Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of…
We study the foundational properties of persistent homotopy groups and develop elementary computational methods for their analysis. Our main theorems are persistent analogues of the Van Kampen, excision, suspension, and Hurewicz theorems.…
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
We prove a theorem of uppersemicontinuity for the metric entropy of meromorphic maps.
We discuss a class of linear representations of the product poset of totally ordered sets $P= T_1 \times \cdots \times T_n$ which decompose into interval representations for block intervals. These can be characterised in terms of a…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…
The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…
We strengthen some results in \'etale (and real \'etale) motivic stable homotopy theory, by eliminating finiteness hypotheses, additional localizations and/or extending to spectra from HZ-modules.
We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class…
This chapter covers methodological issues related to estimation, testing and computation for models involving structural changes. Our aim is to review developments as they relate to econometric applications based on linear models.…