Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau theorem, the positive mass theorem, the Yamabe theorem, Donaldson's theory of smooth 4-manifolds, nonlinear stability of the Minkowski space-time, the Riemannian Penrose inequality, the Poincaré conjecture in 3D, and the differentiable sphere theorem. These examples all come from geometry and topology, and I was trying to find similar examples in other branches of mathematics without luck. I can sort of imagine why geometry and topology maybe amenable to PDE but this does not mean PDE cannot find applications in other branches. I asked probabilists and was told that most of the examples they think of seem to be the other way around, i.e., using probability theory to say something about PDE. Can you provide an example, or give a reason why such examples must be confined to geometry and topology.
The reason I am asking this question is that majority of "pure math" students don't seem to like PDE courses, thinking it as an "applied" subject so it has nothing to do with them. My impression is that for instance students in algebraic or differential geometry somehow get their "own version" of PDE theory from specialized books in their subject, specifically tailored for the problem at hand. It would be much easier and methodical if the student had taken a general PDE course before. So I thought this kind of list maybe helpful in convincing the beginning student to take PDE classes. As the list stands now, we have enough for geometry/topology and perhaps mathematical physics students, but it would be great for instance to have something for probability, number theory, analysis, and algebra students.
Best Answer
As alluded-to by Qiaochu Y. above, and as I can personally attest, PDE arise in the modern theory of automorphic forms. Superficially/historically, this might be viewed as a formal generalization of "holomorphic" to "eigenfunction for Laplace-Beltrami operator". Indeed, already c. 1947, Maass showed that real quadratic fields' grossencharacter L-functions arose as Mellin transforms of "waveforms", Laplace-Beltrami eigenfunctions on $\Gamma\backslash H$, a complementary result to his advisor Hecke's result that $L$-functions for complex quadratic extensions of $\mathbb Q$ arose from holomorphic modular forms.
The spectral theory of automorphic forms, from Avakumovic, Roelcke, and Selberg c. 1956, in effect decomposes $L^2(\Gamma\backslash H)$ with respect to the invariant Laplacian, descended from the Casimir operator on the group $SL_2(\mathbb R)$, which (anticipating theorems of Harish-Chandra) almost exactly corresponds to decomposition into irreducible unitary representations.
The Selberg trace formula, and Langlands' and Arthur's, as well as Jacquet's "relative" trace formula, do afford an interpretation as spectral decompositions of various integral operators, rather than differential operators. Nevertheless, or "however", some aspects of the situation that are clumsy, because of their "extreme" features, but interesting for applications for the same reason, from that viewpoint are amenable to thinking about solutions of (invariant) inhomogeneous PDEs with distributional "targets". A typical scenario is a "Helmholtz" equation (a wave equation Fourier-transformed in the time parameter), $(\Delta-\lambda)u=f$. Among other cases of interest, the case that $f$ is an (automorphic) delta is very useful in various number-theoretic applications, such as proving "subconvex" bounds: Anton Good sketched this application already in 1983 (and Diaconu and I treated $GL_2$ over number fields recently... implicitly using this idea, although reference to classical special functions gave a shorter argument for the official version).
Philosophizing a bit, such experiences, and continuing ones of a related sort, indicate to me that geometrically meaningful, that is, group-invariant, "PDE" are a natural/obvious extension of "calculus"... so that, in particular, their natural solutions in Sobolev spaces (etc) are "natural objects", whether or not they are classical special functions, or entirely elementary.
(One can't help but note that there is an understandable, if unfortunate, human tendency to declare and understand "turf", so that one chooses one's own, and stays away from others'. Similarly, "experts" on subject X do not favor outsiders' appropriating bits of it "for applications", as though anything other than a life-long dedication could penetrate the mysteries... One may read about medieval European "guilds" and their protection of their "secrets".)
As a methodological philosophizing: my own experience tells me that means of description are useful. That is, structural, meaningful characterization of objects is good. Saying that something is a solution of a natural (group-invariant?...) PDE is a strong, meaningful constraint. Ergo, helpful/good.
The small rant at the end: the usual style of seemingly-turf-respecting narrowness is not so good for genuine progress, nor even for individual understanding.