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Talbot Workshop 2010 Talk 2: K-Theory and Index Theory

These are notes from a talk at the 2010 Talbot Workshop on Twisted K-theory and Loop Groups. This particular talk is an overview of index theory from the point of view of topological K-theory. Assuming little background in analysis, but some familiarity with complex K-theory, the talk covers the (families) Atiyah-Singer index theorem from the point of view of Gysin maps for fibrations, and orientations for complex K-theory in terms of spin^c structures, Clifford algebras and Dirac operators. Higher index theory is also discussed, in terms of Cl_k modules and Cl_k-linear operators. It is meant to be a readable introduction to the subject, with references to the literature.