Omar Leon Sanchez, Pure Mathematics, University of Waterloo
One of the most interesting applications of (algebraic) Galois theory, and perhaps why it all started, is that it translates the problem of solving polynomials by radicals to group-theoretic questions (in which we sometimes have an easier way to find an answer, e.g. finite groups).
Differential Galois theory (whose founding fathers are Picard, Vessiot and mostly Kolchin) aims to understand the solutions of differential equations by means of the group of differential automorphisms (which has the nice structure of an algebraic group). The Galois correspondence between intermediate differential fields and algebraic subgroups is, as in the algebraic case, the fundamental theorem.
elementary functions and integration.