Introduction to local geometric Langlands

Local Langlands program predicts a close relationship between
representations of the absolute Galois group of a
local field F and irreducible representations of reductive groups over F. A
geometric version of this program
attempts to relate vector bundles equipped with (flat) connections on the
complex puncture disk to
representations of the affine Kac-Moody algebras. After laying out the
process of going from classical
statements to geometric predictions, I will explain Frenkel-Gaitsgory's
picture in the unramified case. I will
then explain what we expect for certain ramified principal series
representations. The last part is joint work
with Travis Schedler.