Atiyah-Singer Index Theorem & Heat Kernel

The index theorem connects global topology to local geometry: ind(D) = dim ker(D) − dim ker(D†) equals a topological invariant computable from curvature. The heat kernel method: ind(D) = Tr(e^{−tD†D}) − Tr(e^{−tDD†}), independent of t, with small-t asymptotics giving the Â-genus.