AAA88

Contributions of lattice theory to the study of computational topology

AAA88 Conference, Warsaw, 2014

Joao Pita Costa

Institut Jozef Stefan, Slovenia

Coauthors: Primoz Skraba and Mikael Vejdemo-Johansson

Persistent homology is a central tool in topological data analysis, which examines the structure of data through topological structure. Essentially it applies the qualitative methods of topology to problems of machine learning, data mining or computer vision. It is an area of mathematics interested in identifying a global structure by inferring high-dimensional structure from low-dimensional representations and studying properties of a often continuous space by the analysis of a discrete sample of it, assembling discrete points into global structure. The basic technique is extended in many different directions, permuting the encoding of topological features by barcodes and correspondent persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between correspondent persistence bars and provide a global perspective over this approach. In this talk we shall look at the topos of sheaves over such algebra, discuss its algebraic construction and the potential for a generalized simplicial homology over it.

For more information see: http://at.yorku.ca/cgi-bin/abstract/cbix-26