The Atiyah-Singer index theorem is one of the cornerstones of modern geometry. It relates the number of solutions
of elliptic linear differential equations to the topology of the underlying manifold. For a Dirac operator
on a spin manifold $M$ acting on $E$-valued spinors the index formula states
This first half of the lectures will be devoted to explaining the components of this formula, developing the required functional analysis for elliptic differential operators, and providing the topological and geometric background. Then we will sketch a proof and finally discuss some of its many applications,
in particular to positive scalar curvature metrics.
Vector bundles, Chern classes, covariant derivatives, Chern-Weil theory; Hirzebruch theory of multiplicative sequences; complex representations of Clifford algebras, spinors; globalization to Riemannian manifolds, spin structures; Dirac operators, other fundamental elliptic operators; pseudo-differential operators and elliptic theory, Fredholm operators; excision proof of the Atiyah-Singer index theorem; applications to positive scalar curvature metrics, generalizations of Rokhlin's theorem.
Basic knowledge in Differentiable Manifolds, Elementary Topology, Functional Analysis
Introduction and overview
Motivation for index theory by explicit examples (torus, circle, projective space); Historical introduction; Dirac's square root of the Laplacian; proof sketch of Hirzebruch signature theorem; applications (Rokhlin, positive scalar curvature)
[3, Introduction], [4, Introduction], [5, p.120]
Vector bundles, characteristic classes, and connections