Algebraic invariants for filtered data and their computation

Abstract

This thesis explores the construction and computation of algebraic invariants for filtered topological spaces. These invariants are derived from the classical algebraic object of homology, which is invariant under homeomorphisms, i.e., continuous transformations of topological spaces with continuous inverses. Filtered spaces, however, are richer structures that often encode geometric information, and thus our algebraic constructions are invariant with respect to isometries and scaling operations. We are also interested in the stability of invariants with respect to small perturbations of the input data, including the addition of noise.

Given a space with a poset-valued filtration function, the homology (with field coefficients) of sublevel sets forms a functor from the poset to vector spaces: this functor is called a persistence module, and is a central object in the field of topological data analysis (TDA). In Paper A, we develop the theory of relative homological algebra for persistence modules, including the use of local Koszul complexes as a tool for computing relative Betti diagrams, which are a numerical invariant of minimal relative projective resolutions. In Papers B and C, we study the case where the indexing poset is a total order. In this case, persistence modules decompose as sums of simple bar modules, and we introduce the notion of bar-to-bar morphisms between persistence modules as an algebraic version of bar matchings. Moreover, we develop algorithms for computing matchings induced by morphisms, passing through bar-to-bar morphisms.

Papers D and E generalize a more classical invariant, the critical points of a Morse function, to the setting of metric algebraic geometry, an emerging field that combines (or reunites) algebraic and differential geometry. Specifically, we develop a Morse theory for distance functions from an algebraic variety, restricted to an algebraic variety, and show that, generically, such a distance function is Morse. We define both geometric and algebraic notions of critical points and provide upper bounds on their number. In this sense, we construct and compute invariants for filtered algebraic spaces.

Link to DiVA