I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic topology ( fundamental group and singular homology ) some familiarity with manifolds and vector bundles but no characteristic classes or K-theory or cobordism. I know some distributions but nothing about psuedo-differential operators.
I am willing to spend a few weeks getting to know the definitions and main examples and theorems. I am okay with not knowing the proofs but I want to be able to convince myself with lots and lots of examples and applications of these results.
Any suggestions will be greatly appreciated.