Geometric methods proved to be useful in the study of some groups. However the geometry of the Cayley graph of a group is rather different from the geometry of classical geometric objects such as homogeneous spaces of Lie groups. The similarity between these two geometries grows as the scale of observation increases. And the asymptototic behavior of them shows surprising similarity. Random walks is an essential tool in studying large-scale geometry of groups. On the other hand it is an interesting object for probabilists since many properties of general stochastic processes are manifested here in a rather simple form. In my talk, I will provide an elementary introduction to this vast area. No special knowledge beyond the usual university mathematics is required.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
