Connections between K-Theory and PDEs? I've recently spent some time learning (the very basics of) K-theory for $C^*$-algebras and topological K-theory. Actually, my main fields of interest are PDEs and related topics, in particular functional calculus for unbounded operators, Sobolev/Bessel potential/Besov spaces, interpolation theory, semigroups and so on. You get the idea.
Now I would like to deepen (and broaden) my knowledge about the machinery of K-theory, ideally by learning some interesting connection to said PDE-related topics such as for example some PDE-relevant result admitting a proof with a k-theoretic flavour. Or some general idea how K-theory might provide insight into (or an interesting point of view on) some PDE-related results or concepts.
So I'd be very thankful, and I hope this request is not too broad, if you could provide me with some examples of interesting relations, if they exist, between K-theory and mentioned PDE-related topics.

Here is another way that K-theory will show up when studying PDEs. Say you have an elliptic operator $D$ on a manifold with boundary $M$ and you want to know if you can impose local boundary conditions $B$ so that the boundary value problem
$Du=f, Bu\vert_{\partial M}=0$ is well behaved.
Here well-behaved means that: there are finitely many linear conditions on $f$ that guarantee the existence of a solution and, when there is a solution, the solutions form a finite dimensional space. In short, you want the operator with boundary conditions to be Fredholm.
A necessary and sufficient condition for the existence of such a $B$ is given by a triviality condition on the K-theory class of the symbol of $D$ restricted to the boundary. (It should be a pull-back from the K-theory of the boundary.) This was proven by Atiyah and Bott in their paper on the index theorem on manifolds with boundary.
