TITLE: Atiyah-Singer theorem-a big picture
QUESTION [61 upvotes]: So far I made several attempts to really learn Atiyah-Singer theorem. In order 
to really understand this result a rather broad background is required: you need
to know analysis (pseudodifferential operators), algebra (Clifford algebras, 
spin groups) and algebraic topology (characteristic classes, K-theory, Chern 
character). So every time I have stuck in some place and have too many doubts to 
continue my journey into the realm of index theory. Finally I came to conclusion that what I'm lacking is some big picture about various approaches to index theorem: there are many (excellent) books that treat the subject but before you arrive at the proof of index theorem itself you have to go through about 150 pages of background material (which usually you partially know). 
Some issues which I already figured
out:

There are at least two approaches: the one which uses K-theory and the second
so called heat kernel approach. As I understood for the first one good knowledge of algebraic topology is required while for the second you have to know differential  geometry stuff (connections, curvatures etc.)  
As far as I know these approaches have different range of applicability: the
heat kernel approach is more restrictive and do not include all differential elliptic operators.    
There are some other proofs (using K homology - see the book of Higson and Roe, via tangent groupoid, there is also quite recent proof due to Paul Baum)

So what answer(s) I would be very pleased to see? 


I would like to gather in one place 
    various possible approaches to the proof of index theorem together with discussion of
    the main ingredients needed to complete such proof (for example: "you need good
    understanding of K-theory not only for compact spaces, but also for locally compact. 
    You also have to understand Bott periodicity and Thom isomorphism theorem), the 
    formulation of the index formula together with comparison and relation to other 
    index formula (for example: you obtain topological index as a K-theory class- 
    in order to obtain the famous cohomological formula (involving Todd class) you apply 
    Chern character) and finally the level of generality of formulation of index theorem
    (for example: this approach works for all elliptic differential operators).   


Maybe this question is broad but I have an impression that I'm not the only person who
will find such a discussion useful.

REPLY [16 votes]: Just for completeness let me add a few words on the heat kernel proof. It is true that some analysis is needed. But one should not forget that even the $K$-theoretic proof depends on the notion of pseudo-differential operators and therefore cannot avoid analysis completely.
It is also true that the proof really only works well for Dirac operators. 
But then, Dirac operators generate $K$-homology, and most applications of the index theorem are formulated in terms of (0-order perturbations of) Dirac operators. Contradicting comments are welcome! The most prominent examples are mentioned in Coudy's answer, but one should add the signature operator (which is the Euler operator with respect to a different grading), and the so-called untwisted Dirac operator.
The only real drawback is that one only sees the image of the index in rational cohomology. But then, using family index techniques or $\eta$-invariants, one can sometimes recover a bit more of the topological content of the index.
The proof consists of the following main ideas. 

The McKean-Singer trick. This still works for all elliptic differential operators. If $P\colon\Gamma(E)\to\Gamma(F)$ is an elliptic differential operator on a manifold $M$, consider its adjoint $P^*\colon\Gamma(F)\to\Gamma(E)$. Then it makes sense to consider
$$\operatorname{ind}(P)=\operatorname{tr}\bigl(e^{-tP^*P}\bigr)-
\operatorname{tr}\bigl(e^{-tPP^*}\bigr)\;.$$
In the limit $t\to 0$, the traces above become local, that is, they can be represented by an integral over $M$ that depends only on the coefficients of $P$ in local coordinates. 

From here on, one could try to prove that only finitely many geometric terms can contribute to the index. This was the idea of Gilkey and Patodi. One difficulty comes from the fact that the relevant contribution to the index sits in the $t^0$-term of the asymptotic expansion, which is by far not the leading term. However,
in the special case of Dirac operators, one gets around this problem.

Getzler rescaling. If $P=D$ is a Dirac operator, then $\Delta=D^*D\oplus DD^*$ is a Laplace operator, and $e^{-t\Delta}$ is the standard heat kernel. One performs a standard parabolic rescaling of space and time together with Getzler's rescaling of the Clifford algebra to show that as $t\to 0$, $\Delta$ converges against a model operator that looks like a harmonic oscillator with coefficients in $\Lambda^{\mathrm{even}}T^*M$. The relevant contribution to the index now appears in the leading order term thanks to Getzler's trick. This only works if $D$ is compatible with a Riemannian metric (here one makes use of the homotopy invariance of the index to achieve this), and one has to chose nice local coordinates.
Mehler's formula. The heat kernel for the model operator has an easy explicit formula in terms of the Riemannian curvature and the coefficients of the Dirac operator.
Chern-Weil theory. The output of step 3 is an integral over $M$ of an invariant polynomial, applied to the Riemannian curvature and the so-called twisting curvature of the Dirac bundle on which $D$ acts. Hence it can be interpreted as the evaluation on the fundamental cycle of $M$ of an expression in characteristic classes.

Indeed, as pointed out in the comments to Paul's answer, one does not need any advanced analysis. Some notions from a first course in Riemannian geometry are needed. The most tricky part seems to be Getzler's rescaling, which is mostly algebraic.