Ascending HNN extensions and the BNS invariant
Abstract
I will talk about random walks on groups and define the Poisson boundary of such. Studying it gives criteria for amenability or growth. I will outline how this can be used and describe recent related results and still open questions.
This talk consists of three parts. As a motivation, we are first going to introduce von Neumann algebras associated with discrete groups and briefly describe their interplay with measurable group theory. Next, we are going to consider group von Neumann algebras of general locally compact groups and highlight crucial differences between the discrete and the non-discrete case. Finally, we present some recent results on group von Neumann algebras associated with certain locally compact HNN-extensions.
We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in
the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen,
Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.
We'll present a proof of the basic Burgess bound for short character sums, following the simplified presentation of Gallagher and Montgomery.