We describe new classes of noetherian local rings R whose finitely generated modules M have the property that ToriR(M,M)=0 for i≫0 implies has finite projective dimension, or ExtRi(M,M)=0 dimension injective dimension.

Persistent homology enables fast and computable comparison of topological objects. We give some instances a recent extension the theory persistence, guaranteeing robustness computability for relevant data types, like simple graphs digraphs. focus on categorical persistence functions that allow us to study in full generality strong kinds connectedness—clique communities, k-vertex, k-edge connect...

The ?ech and Rips constructions of persistent homology are stable with respect to perturbations the input data. However, neither is robust outliers, both can be insensitive topological structure high-density regions A natural solution consider 2-parameter persistence. This paper studies stability homology: we show that several related density-sensitive bifiltrations from data satisfy properties...

Abstract The combination of persistent homology and discrete Morse theory has proven very effective in visualizing analyzing big heterogeneous data. Indeed, topology provides computable coarse summaries data independently from specific coordinate systems does so robustly to noise. Moreover, the geometric content a gradient vector field is useful for visualization purposes. case multivariate sti...

Computational topology has recently seen an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be natur...

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be natu...

Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them.

In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is weighted and signed count bars module, in which bar form [a,b) degree d counted with weight (e−a−e−b) sign (−1)d. Persistent magnitude has good formal properties, such as additivity respect to exact sequences compatibility tensor products, interpretations terms...

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We extend a construction of homology of digraphs due to Grigoryan, Lin, Muranov and Yau to the persistent framework. The result, which we call persistent path homology or PPH,...