Related papers: Level-sets persistence and sheaf theory

We show that stable derivators, like stable model categories, admit Mayer-Vietoris sequences arising from cocartesian squares. Along the way we characterize homotopy exact squares, and give a detection result for colimiting diagrams in…

We develop a general algebraic framework involving "Poincar\'e--Novikov structures" and "filtered matched pairs" to provide an abstract approach to the barcodes associated to the homologies of interlevel sets of $\mathbb{R}$- or…

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified…

In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered…

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander--Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence…

We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another…

As well-known, inner functions play an important role in the study of bounded analytic function theory. In recent years, persistence module theory, as a main tool applied to Topological Data Analysis, has received widespread attention. In…

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of…

The Fourier-Mukai transform is lifted to the derived category of sheaves with connection on abelian varieties. The case of flat connections (D-modules) is discussed in detail.

Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…

We present a detailed introduction of the theory of constructible sheaf complexes in the complex algebraic and analytic setting. All concepts are illustrated by many interesting examples and relevant applications, while some important…

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$.…

We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to…

Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix.These matrices, a generalization…

Using sheaf theory, I introduce a continuous theory of persistence for mappings between compact manifolds. In the case both manifolds are orientable, the theory holds for integer coefficients. The sheaf introduced here is stable to…

We develop the theory of geometric Eisenstein series and constant term functors for $\ell$-adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors,…

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbert schemes of points -- can be used to define intersection…

Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from…

Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of…

Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants…