**The Atiyah-Singer index theorem and its applications
Steven Rosenberg
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*Abstract: *The Atiyah-Singer index theorem generalizes the Gauss-Bonnet theorem, the Hirzebruch signature theorem, and the Hirzebruch-Riemann-Roch theorem. The original proofs are a mixture of analysis and topology, but the more recent heat equation proof uses more differential geometry. The main application of the index theorem at present is the computation of dimensions of moduli spaces of e.g. instantons, Seiberg-Witten solutions, and Gromov-Witten moduli spaces. I will discuss the statement and heat equation proof of the index theorem and the moduli space computations.