We study the homology of a filtered d-dimensional simplicial complex K as a single algebraic entity and establish a correspondence that provides a simple description over fields. Our analysis enables us to derive a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Our study also implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.

/pdf/computing-persistent-homology-14gcfi5b40.pdf

Computing persistent homology

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Infinite Abelian Groups

These notes are intended as an introduction to the theory of Coxeter groups. They closely follow my talk in the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua University on May 10, 2013. They were prepared from the beamer presentation which I used during my talk. AMS classification numbers. Primary: 20F55, 20F65

/pdf/the-geometry-and-topology-of-coxeter-groups-1whqukpbxo.pdf

The geometry and topology of Coxeter groups

We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris‐Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications. 55M25, 93A15; 55N35

/pdf/coverage-in-sensor-networks-via-persistent-homology-3afut1xclm.pdf

Coverage in sensor networks via persistent homology

Persistent homology captures the topology of a filtration - a one-parameter family of increasing spaces - in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.

/pdf/the-theory-of-multidimensional-persistence-52ef3r3psf.pdf

The theory of multidimensional persistence

We prove that a δ-hyperbolic group for δ < 1 2 is a free product F ∗ G1 ∗ . . . ∗Gn where F is a free group of finite rank and each Gi is a finite group.

On hyperbolic groups

http://math.univ-lyon1.fr/~ould/Fiches-papiers/eccsa10.pdf

Existentially closed CSA-groups

We collect in this paper some remarks and observations about limit groups of equationally noetherian groups. We show in particular, that some known properties of limit groups of a free group or, more generally, of a torsion-free hyperbolic group can be seen as consequences of the fact that such groups are equationally noetherian. Especially, such properties are still true for linear groups and finitely generated abelian-by-nilpotent groups.

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Limit Groups of Equationally Noetherian Groups

http://math.univ-lyon1.fr/homes-www/ould/Fiches-papiers/embeddings.pdf

Embeddings in finitely presented groups which preserve the center

The famous Tits alternative states that a linear group either contains a non- abelian free group or is soluble-by-(locally finite). In this paper we study similar alter- natives in pseudofinite groups. We show, for instance, that an@0-saturated pseudofinite group either contains a subsemigroup of rank 2 or is nilpotent-by-(uniformly locally fi- nite). We call a class of finite groups G weakly of bounded rank if the radical rad.G/ has a bounded Prufer rank and the index of the socle of G=rad.G/ is bounded. We show that an@0-saturated pseudo-(finite weakly of bounded rank) group either contains a non- abelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between these kind of alternatives and amenability.

Abderezak Ould Houcine

Papers

On hyperbolic groups

Existentially closed CSA-groups

Limit Groups of Equationally Noetherian Groups

Embeddings in finitely presented groups which preserve the center

Alternatives for pseudofinite groups