The anyons of a (2+1)d topological order are described mathematically by a modular tensor category (MTC). Given an MTC, any n-component link becomes an n-tensor that stores some information about this MTC. For example, the 1-component unknot (twisted once) gives a vector "T" that stores the topological spins of all anyons; the 2-component Hopf link evaluates to matrix "S", from which the Verlinde formula recovers the fusion rules of the anyons. Do T and S store all the data? The answer is no: there are different MTCs with the same S and T. So the question becomes what other links to include in this list of tensors to tell MTCs apart? 
I will start by telling you how to turn any link you can drawn on a piece of paper (or a whiteboard) in to a tensor, then review the T and S story and then discuss some recent work where people considered adding the Whitehead link and the Borromean rings. There will be lots of drawings!

TBA

Junior Algebra & Representation Theory end of term social will happen in the Quillen Room at 12pm on Friday

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).