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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.

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Concerning the K-theory version of Atiyah-Singer and PDE, you may be interested in my recent answer at math.stackexchange.com/a/252225/4058 –  Javier Álvarez Dec 6 '12 at 15:18

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up vote 12 down vote accepted

I would suggest the Atiyah–Singer index theorem if this counts as a PDE-related topic.

This is a theorem about elliptic differential operators on compact manifolds. The original proof uses K-theory.

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Thank you, Rasmus! This seems to be a very good example. Still I wonder whether there are other good examples out there. –  lvb Sep 29 '10 at 17:31
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If I understand correctly, this is perhaps the example -- most other results of a similar flavor will just be special cases (or mild generalizations). In any case, there's a decent bit of K-theory that goes into the proof, which is really fun and draws on a lot of different parts of geometry, topology, and PDEs. If I didn't know better, I'd almost be suspicious that you were fishing for this answer in the first place! –  Aaron Mazel-Gee Apr 18 '11 at 16:18

As mentioned in the previous answer, the Atiyah-Singer index theorem is an excellent answer to your question. I would like to convince you that, in a sense, it is probably the only answer to your question. Fortunately that one theorem admits so many applications, generalizations, and elaborations that it almost becomes an area of mathematics unto itself (particularly when infused with the tools of C*-algebra theory).

My first remark is that K-theory is an inherently global tool - its power lies in the fact that it is built from but insensitive to the details of local geometry. From what I understand about PDE theory many of the interesting questions live on open balls in Euclidean space, about which algebraic topology in general has little new to contribute. Even when one considers boundary value problems where the geometry gets a little more interesting, the challenges are usually local on the boundary (i.e. the concern is with smoothness, not interesting global structure).

Once we have accepted that we are looking for a global answer to your question, it is natural to ask: is there a sense in which PDE's organize themselves into a full-fledged (co)homology theory? This, after all, is the way that topology usually interacts with other parts of mathematics: one begins with objects whose structure one wants to globalize (e.g. embedded loops, differential forms, vector bundles...) and one aims to build algebraic invariants out of those objects. In the case of PDE's the answer is K-homology, a generalized homology theory for the category of manifolds with the property that every first order linear elliptic operator $D$ on a manifold $M$ gives rise to a class $[D]$ in $K_*(M)$. K-homology, as the name suggests, is the homology theory which is naturally dual to K-theory, regarded as a (generalized) cohomology theory on the category of manifolds.

So the question is: what can we do with K-homology? The answer is that one can do quite a lot, but as with many constructions in algebraic topology many of the most interesting results involve pairings between homology and cohomology. The most fundamental pairing between K-homology and K-theory is the so-called index pairing, which takes an elliptic operator and a vector bundle and spits out the Fredholm index of the operator "twisted" by the bundle. The Atiyah-Singer index theorem is really a theorem about the topological properties of this pairing, and consequently it plays a central role in applications of K-homology.

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+1 This is a really nice answer! –  Raeder Apr 18 '11 at 22:35

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.

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