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).

Junior Algebra & Representation Theory starts with a week 1 social. Join us in the Quillen Room!

TBC

Using the techniques of arc presentations and open book foliations, we prove new lower bounds on the crossing number and arc index of a satellite knot in terms of it's companion, pattern, and wrapping number.

In this talk I will discuss homological and homotopical isoperimetric inequalities, including those of higher rank.  I will emphasise what is known in the non-positive curvature case.