Third Year Candidacy Requirement: "An introduction to the Atiyah-Singer index theorem for the chiral Dirac operator"

Abstract: Atiyah and Singer proved the Atiyah-Singer index theorem in 1963, which states that for an elliptic differential operator on a compact manifold, the analytical index(related to the dimension of the space of solutions) is equal to the topological  index(defined in terms of some topological data). My presentation will give an introduction to the Atiyah-Singer index theorem for the chiral Dirac operator. For introducing the theorem, I will introduce Clifford algebra, spin structure and spinor  bundle, then give the definition of the Dirac  operator on a spinor bundle and the associated chiral Dirac  operator. In the last of my presentation, I will give an application of the Atiyah-Singer index theorm for the chiral Dirac operator, which proves the Lichnerowicz theorem. The Lichnerowicz theorem states that on a connected closed Riemannian manifold with a spin structure, if its scalar curvature is non-negative and not zero everywhere, then its A-genus, which is a characteristic number of M, is 0.

Host: Renato Feres