Abstract:

Pseudotopologies are a generalization of topologies which allow one to capture the idea of a "neighborhood" of a point, even when the intersection of two neighborhoods may not be a neighborhood of the points it contains. Such a situation is typical in graphs, where we would like to define the neighborhood of a vertex to be the vertices in its star, and it is easy to construct graphs where the intersection of the stars of two vertices is not the star of any vertex. Pseudotopological spaces give a way of managing such pathologies, and we show that a rich algebraic topology may be constructed in this category. We outline the construction of homotopy, homology, persistence, and sheaf theory in pseudotopological spaces, and show how these may be applied to both discrete homotopy theory and topological data analysis.

Please email Sergio Zamora ([email protected]) for the zoom link.