Speaker
Description
The Dowker complex of a relation was introduced by C.H. Dowker in 1952. Given a relation R on X times Y, the Dowker complex D(X, Y, R) is a simplical complex with vertex set X, and a subset sigma of X forms a simplex, if there is a y in Y with which every element of sigma is in relation. The number of such elements y is called the `weight' of the simplex. Although the construction has attracted a steady interest over the few decades following its introduction, this interest has vastly increased with the appearance of Topological Data Analysis at the beginning of this century. This follows from the fact that the Dowker construction is a very good mechanism for turning data sets (aka point clouds) into filtered complexes.
I will start by introducing the Dowker construction via a sequence of examples. Next, I will recall a number of results in this area, especially those concerned with extending Dowker duality to the filtered setting. Given a measure on the space Y, I will present a bifiltration on D(X, Y, R), obtained by requiring that the intersection of balls of appropriate size around the points of sigma has sufficiently large measure. The stability of this construction will be discussed. Finally, I will present a few applications.
This is joint work with Niklas Hellmer.
A. Blumberg and M. Lesnick, Stability of 2-parameter persistent homology, Found. Comput. Math. (2022).
M. Brun and L. Salbu, The Rectangle Complex of a Relation, Mediterranean Journal of Mathematics 20.1, (2023).
M. Brun, B. Garcia Pascual, and L. Salbu, Determining Homology of an Unknown Space from a Sample, European Journal of Mathematics 9.4, (2023).
C. H. Dowker, Homology Groups of Relations, Annals of Mathematics, 56.1 (1952).
N. Hellmer and J. Spaliński, Density Sensitive Bifiltered Dowker Complexes via Total Weight, ArXiv: 2405.15592.
M. Robinson, Cosheaf representations of relations and Dowker complexes, J. Appl. and Comput. Topology, 6.1 (2022).
D. R. Sheehy, A Multicover Nerve for Geometric Inference, CCCG: Canadian Conference in Computational Geometry, (2012).
