Extensions of smooth representations of $\operatorname{SL}_2(\mathbb{Q}_p)$ in natural characteristic

Representation theory of $p$-adic groups in characteristic $p$ is something which is still not understood too well. One issue, which we already see for modular representations of finite groups, is that things are not semisimple anymore, and so understanding irreducible representations is not quite enough to have full information. During the talk I will present some of my attempts at creating homological methods to understand extensions of representations of $\operatorname{SL}_2(\mathbb{Q}_p)$, and what those results imply about injective smooth representations (that are my main focus of research).