Abstract: Given a group G we may consider the compact convex space of (normalized) positive definite functions on G, PD(G).
Any group of automorphisms of G acts naturally on PD(G), in particular G itself acts by conjugation, and dynamical perspective leads us to a variety of questions regarding fixed points, minimal subsystems, classification of invariant and stationary measures, etc.
A classical example is the action of GL_n(Z) on the n-torus, which may be identified as the action of Aut(G) on PD(G) in the case G=Z^n.
Here we have a rich, developed theory. What about nilpotent groups? And what about G=GL_n(Z) itself?
Note that the space PD(G) contains the space of subgroups of G, Sub(G), as well as the space of "random subgroups", namely probability measures on Sub(G).
Moreover, PD(G) captures information on the unitary representation theory of G. Accordingly, tools from operator theory are indispensable in its study.
In this talk I will try to gently introduce and survey the subject.
Along the way, I will mention some joint works with Boutonnet, Houdayer, Peterson, Vigdorovich and huji student Tal Meilin.
Join Zoom Meeting
Meeting ID: 892 6289 8399