I will discuss an extension of the so-called "Routh reduction" in mechanics to first-order field theories. Roughly speaking, this is a classical technique to reduce the number of unknowns in a Lagrangian system in the presence of symmetry.

In the first part of the talk, aimed at a general audience, I will give an outline of what reduction is in mechanics. I will also make an introduction to polysympectic geometry and the polysymplectic formalism of field theory. In the second part of the talk I will review some of the geometric mechanics literature on Routh reduction, and build on some of these ideas to extend Routh reduction to first-order field theories. We will describe two possible ways of doing this: one based on the polysymplectic reduction theorem, and (time permitting) another one based on variational calculus.