Thursday, April 8, 2021 - 3:00pm
University of Florida
Zoom link: https://upenn.zoom.us/j/91064953631?pwd=UVUvUDc3eDRVNmp1WVNQa0hLeVU4QT09 We use Cech's closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses persistent homology, discrete homology of metric spaces, singular homology of topological spaces, homology of (directed) clique complexes, along with their respective accompanying homotopy theories. We have six different homology and homotopy theories of closure spaces, which we show satisfy analogues of classical results in algebraic topology. Our framework also allows us to construct a bridge between continuous and discrete aspects of applied topology. We reintroduce sheaf theory for closure spaces. As an application, we define sheaves on directed graphs. This is joint work with Peter Bubenik.