Chris Kapulkin, University of Western Ontario
 
Mini-Course Title: Discrete homotopy theory

Time:    Friday, Nov 10th, 9:30am
Monday, Nov 13th, 9:30am
Wednesday, Nov 15th, 9:30am

Abstract: Discrete homotopy theory, introduced around 20 years ago by H. Barcelo and collaborators building on the work of R. Atkin from the mid seventies, is a homotopy theory of (simple) graphs. As such, it applies techniques previously employed in the "continuous" context to study discrete objects. It has found applications both within and outside mathematics, including: matroid theory, hyperplane arrangements, topological data analysis, and time series analysis.

The first part of this mini-course is an introduction to discrete homotopy theory. We define the main concepts, including homotopy of graph maps and homotopy groups of a pointed graph. We also briefly discuss some of the applications of the field.

The second part discusses the recently introduced "cubical setting" for discrete homotopy theory, which brings a whole host of new tools into the field, including the topological realization theorem (formerly a conjecture of E. Babson, H. Barcelo, R. Laubenbacher, and M. de Longueville from 2006), the long exact sequence of a fibration, and the Hurewicz theorem. It is based on joint work with D. Carranza (arXiv:2202.03516).

The third part zooms in on the notion of the fundamental group and the fundamental groupoid of a graph. Several computational tools are discussed, including the Seifert-van Kampen theorem and the theory of covering graphs. It is based on joint work with U. Mavinkurve (arXiv:2303.06029).