Francesco Vaccarino: Multi - Persistent homology, complex networks and the brain functional connectome

First Hour is an introduction to persistent homology and its natural extension, multi-persistent homology. The relationships of these objects with geometry and representation theory will be highlighted.  In particular we will illustrate some open problems arising in shifting from the persistent to the multipersistent case. 

 During the second hour we will show how to extend in a meaningful way persistent homology to weighted graphs. We give some striking applications to complex network classification and to the brain functional connectome, which motivated the approach via weighted graphs.