Activities This Week

Ribet’s lemma is an algebraic statement that Ribet used in his proof of the converse of Herbrand’s theorem. Since then various generalisations of Ribet’s lemma have been found, with arithmetic applications. In this talk I will discuss a joint work with Ariel Weiss in which we show that two measures of reducibility for two dimensional representations over a DVR are the same, thus answering a question of Bellaiche and Cheneveier, and deducing from it a particular generalisation of Ribet’s lemma. An interesting feature of the proof is that it applies to both the residually multiplicity-free and the residually non-multiplicity-free cases. I will discuss an application to a local-global principle for isogenies of elliptic curves.

We study the character theory of infinite solvable groups, focusing on the metabelian and polycyclic cases. This theory has applications towards the Hilbert-Schmidt stability of such groups - whether almost-homomorphisms into the unitary groups U(n) are nearby honest homomorphisms? We explore an interesting link between stability and topological dynamics via a notion of “sofic dynamical systems”. I will introduce all relevant notions.

The talk is based on a joint work with Itamar Vigdorovich.

Consider a dynamical system, and let us study the structure of the corresponding crossed product $C^*$-algebra, in particular on the classifiability, comparison, and stable rank. More precisely, let us introduce a uniform Rokhlin property and a relative comparison property (these two properties hold for all free and minimal $Z^d$ actions). With these two properties, the crossed product $C^*$-algebra is shown to always have stable rank one, to satisfy the Toms-Winter conjecture, and that the comparison radius is dominated by half of the mean dimension of the dynamical system.

In the study of infinite discrete groups it is useful to consider imperfect approximations by finitary models (either permutations or matrices). I will talk about the stability of such approximations, i.e. can it always be corrected to a perfect approximation, focusing mostly on amenable groups. The involved techniques include ergodic theory and dynamics as well as character theory of infinite groups. Some directions and open problems will be presented.

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