With D. Nadler. **Geometric Base Change**.

We propose a geometric version of Langlands' base change principle.
This describes the behavior of categories of sheaves on moduli of bundles
under the passage from a curve to its covering spaces.
As evidence for the conjecture we prove its spectral version
(describing coherent sheaves on moduli of local systems
on a curve and its coverings) and the special case of abelian groups.
We then show how a twisted (or "orientifold") version of geometric
base change for the covering P^1 ---> RP^2 implies Soergel's conjecture,
a categorical form of the real local Langlands program. We also discuss
other applications and relations with topological field theory.