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.