I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing pseudodifferential operators), I am wondering how practical this theorem is. The following question is general in nature:
Suppose we have a PDE given by certain elliptic differential operator, how computable is the topological index of this differential operator? If we give certain boundary conditions on the domain (for example, the unit circle with a point removed, a triangle, a square, etc), can we extend the $K$-theory proof to this case? I know the $K$-theory rhetoric proof in literature, but to my knowledge this proof is highly abstract and does not seem to be directly computable. Now if we are interested in the analytical side of things, but cannot compute the analytical index directly because of analytical difficulties, how difficult is it to compute the topological index instead? It does not appear obvious to me how one may compute the Todd class or the chern character in practical cases.
The question is motiviated by the following observation: Given additional algebraic structure (for example, if $M$ is a homogeneous space, $E$ is a bundle with fibre isomorphic to $H$) we can show that Atiyah-Singer can be reduced to direct algebraic computations. However, what if the underlying manifold is really bad? What if it has boundaries of codimension 1 or higher?How computable is the index if we encounter an analytical/geometrical singularity?(which appears quite often in PDE).
On the other hand, suppose we have a manifold with corners and we know a certain operator's topological index. How much hope do we have in recovering the associated operator by recovering its principal symbol? Can we use this to put certain analytical limits on the manifolds(like how bad an operator on it could be if the index is given)?