Niels uit de Bos will speak on

Ramified geometric Langlands: an example

Abstract: The geometric Langlands correspondence is a geometrized analogue of the Langlands correspondence for function fields. I will briefly explain and motivate this analogue and then focus on the specific case of rank 2 bundles on $\mathbb{P}^1$ with tame ramification at 4 points, where everything becomes very explicit and concrete. In particular, given a rank 2 local system on $\mathbb{P}^1 \setminus \{\infty, 0, 1, t\}$ with unipotent monodromy, I construct its corresponding Hecke eigensheaf and in addition (but this will probably not be discussed in the talk), I can explicitly describe how the Hecke operators act on a basis of the cusp forms. The existence of such an eigensheaf was already known in this case, but its explicit calculation may still lead to some interesting results and moreover, I give a new proof that seems likely to be straightforwardly generalisable to the case where the 4 distinct ramification points are replaced by any divisor of degree 4 on $\mathbb{P}^1$ --- a case where the existence of eigensheaf was not yet known.