On Zimmer's conjecture

The group $\mathrm{SL}_n(\mathbb Z)$ (when $n > 2$) is very rigid, for example, Margulis proved all its linear representations come from representations of $\mathrm{SL}_n(\mathbb R)$ and are as simple as one can imagine. Zimmer's conjecture states that certain "non-linear" representations ( group actions by diffeomorphisms on a closed manifold) come also from simple algebraic constructions.

For example, conjecturally the only action on $\mathrm{SL}_n(\mathbb Z)$ on an $(n-1)$ dimensional manifold (up to some trivialities) is the one on the $(n-1)$ sphere coming projectivizing natural action of $\mathrm{SL}_n(\mathbb R)$ on $\mathbb R^n$. I'll describe some recent progress on these questions due to A. Brown, D. Fisher and myself.