Partial differential equations in "pure mathematics"

One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in Wikipedia follow this pattern. I can't help wondering whether there are PDE's arising "naturally" in "pure" mathematics like Geometry, Topology, etc.? Of course, I can always write an arbitrary surface as one or more PDE's, but they don't seem to drive much research in the subject, afaik. I understand that PDE's originated in Physics, but hasn't the subject grown away from physical models into more abstract examples?

I believe that one of the examples can be Ricci flow which are parabolic PDEs describing the deformation of metric for Riemann manifold. The new result is e.g. the solution of Thurston's gometrization conjecture which Perelman did using Ricci flow.

In general as you have mentioned, many PDEs appears from the applied science: physics, finance etc. though I am not sure that PDEs related to Markov processes appeared only after considerations of applied problems.

Finally, there is a theory of Harmonic functions which was based on the Laplace PDE $\Delta u = 0$. These functions are generalizations of linear functions on the real line for the multidimensional case and they inherit many of nice properties of linear 1-dim functions. Although this equation appeared from the physical problem, it has been also important for the needs of the pure mathematics.

Some more examples from geometry and topology:

The Atiyah-Singer Index Theorem connects analysis of elliptic differential operators, with the topology of smooth compact manifolds. There are many approaches to proving the Index Theorem, but there is one in particular which does so via studying the heat equation on the compact manifold.

On non-compact Riemannian manifolds, an uniform geometric structure at infinity has deep connections to the spectrum of the associated (conformal) Laplacian, this leads to deep connections with scattering properties of wave equations evolving on such manifolds.

The topology of four dimensional manifolds is connected to their geometries manifesting in Yang-Mills theory, which gives rise to very interesting non-linear elliptic partial differential equations. There's quite a nice book about this whole field.

The study of over-determined systems of partial differential equations (such as this one) can trace a not-insignificant-part of its roots to E. Cartan's program to understand differential geometry using the moving frames method.

On the other direction, PDEs can give rise to interesting problems in other fields. For example, the fundamental solution of a linear constant coefficient hyperbolic partial differential equation has deep connections to topology and geometry of algebraic varieties. And the so-called strong Huygen's principle for the wave equation is closely connected to the geometry of homogeneous spaces.