What are the connections between representation theory and differential equations? I have just finished my undergrad and while I haven't studied much in representation theory I find it a very fascinating subject. My current interest is in differential equations, and I am wondering is there any ongoing research that combines these two areas?
 A: There are strong connections. For example look at this question in MO and its answers.
There are also older books and papers about connections between these subjects. For example, "Theory of Differential Equations : Representation Theory and Automorphic Functions" by I.M. Gelfand, M. I. Graev, and I. I. Pyatetskii-Shapiro.
A: I'll offer another view: physics, especially quantum mechanics, is essentially about the interplay between representation theory and differential equations. Representations of groups like the unitary groups and the Heisenberg group(s) encode information about symmetries of physical systems, invariances under various sort of coordinate changes. These representations are often not on finite dimensional spaces, but infinite dimensional function spaces, and the group elements are often represented as differential operators, giving rise to equations of motion for physical systems. One can study these equations both on the representation theory of Lie groups and algebras side of things, and also with the techniques of PDEs, and both have interesting things to say about the systems that are governed by these representations.
I am not a physicist, but I helped a very talented undergraduate in physics learn the mathematics underlying the physics based on the book by Peter Woit, "Quantum Theory, Groups, and Representations." This book is not really a PDEs book, but touches on many topics adjacent to PDEs, including some harmonic analysis, some symplectic topology and geometry, some complex geometry and the infamous path integral. It's pretty reasonable to me that someone with a background in the analysis of these topics would have quite a bit to say about PDEs.
A: The wide variation in interpretation of your question should be a convincing indicator that your question is not going to get a useful answer. At extremes, the answer is "obviously yes" or "obviously no", depending on interpretations of many of the words... :)
That is, on one hand, if people in one field, with a self-chosen label, want to de-legitimize other people who claim the same label... well, does that really mean anything about the mathematics itself? So the answer to your question can easily be made to be "no", by fiat. :)
On another hand, especially in my experience in "number theory in the broad sense", almost every bit of mathematics has some significant (if not profound) connection to every other. So, the answer to your question is trivially "yes". :)
Still, I would tend to be sympathetic to @markvs's answer: the modern theory of automorphic forms uses a lot of representation theory. The representation theory itself uses some very serious PDE business: in the rank-one case, it's really just the classical theory of asymptotics of ODEs. But in the higher-rank case, the corresponding subquotient theorem of Harish-Chandra, and then the subrepresentation theorem of Casselman (and Milicic), are significantly subtler. (The Casselman-Milicic paper has a wonderful appendix expanding a paper of Deligne's on PDE with regular singular points...)
Not so elementary, though.
