A combinatorial non-positive curvature I: weak systolicity

We introduce the notion of weakly systolic complexes and groups, and initiate regular studies of them. Those are simplicial complexes with nonpositive-curvature-like properties and groups acting on them geometrically. We characterize weakly systolic complexes as simply connected simplicial complexes satisfying some local combinatorial conditions. We provide several classes of examples --- in particular systolic groups and CAT(-1) cubical groups are weakly systolic. We present applications of the theory, concerning Gromov hyperbolic groups, Coxeter groups and systolic groups.Comment: 35 pages, 1 figur

Metric systolicity and two-dimensional Artin groups

We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.Comment: final preprint version, to appear in Math. An

HomologyBasis: Fast Computation of Persistent Homology

Simplicial complexes are used in topological data analysis (TDA) to extract topological features of the data. The HomologyBasis algorithm is proposed as an efficient method for the computation of the topological features of a finite filtered simplicial complex. We build up the implementation and intuition of this algorithm from its theoretical foundation ensuring this schema produces the desired simplicial homlogy groups as claimed. HomlogyBasis implemented and compared with the GUHDI algorithm to determine the HomologyBasis' efficiency at computing persistence pairs for finite filtered simplicial complexes. We find the HomologyBasis algorithm performs much better than GUHDI on large low-dimensional simplicial complexes but needs further refinement before it can more efficiently work with high-dimensional complexes.Master's Thesis in MathematicsMAT399MAMN-MA