Vol 21 No. 1

An approach to the topological complexity of the Klein bottle
Donald M. Davis
Recently, Cohen and Vandembroucq proved that the reduced topological complexity of the Klein bottle is 4. Simultaneously and independently we announced a proof of the same result. Mistakes were found in our argument, which was quite different than theirs. After correcting these, we found that our description of the obstruction class agreed with theirs. Our approach to showing that this obstruction is nonzero failed to do so, while theirs did not fail. Here we discuss our approach, which deals more directly with the simplicial structure of the Klein bottle.
The stability theorem of persistent homology
Adam Gardner
Persistence modules – an important tool for understanding geometric properties of data and the central objects of study in persistent homology – are collections of vector spaces indexed by the real numbers together with linear maps satisfying certain basic properties. Persistence modules appear naturally as families of homology groups associated to filtrations of topological spaces. Two equivalent ways to represent a persistence module are by its persistence diagram – a multiset of points in the Euclidean plane –and by its barcode – a multiset of real intervals. Under certain assumptions (commonly satisfied in practice), these representations exist and are unique. One of the main results in the theory of
persistence is the Stability Theorem, which asserts that small perturbations of a persistence module result in small perturbations of its persistence diagram and barcode. In this paper, we review the evolution of this theorem, with emphasis on the results appearing in The Structure and Stability of Persistence Modules (Chazal et al., 2012) and Induced Matchings and the Algebraic Stability of Persistence Barcodes (Bauer and Lesnick, 2015).
Aproximación métrica de grupos: una breve perspectiva
Luis Manuel Rivera, Nidya Monserrath, Veyna García
Los grupos sóficos y los grupos hiperlineales han generado una gran cantidad de investigación en los últimos años en diversas áreas de matemáticas tales como teoría geométrica de grupos, dinámica simbólica y álgebra de operadores. Además, los grupos sóficos han ganado interés porque se ha demostrado que cumplen varias conjeturas aún abiertas para los grupos en general. Las definiciones de ambas clases de grupos son análogas y se pueden pensar como dentro de una clase de grupos de reciente estudio que se conocen como los grupos que tienen la propiedad de aproximación métrica. En este artículo se presenta un panorama general de dicha clase de grupos.