Heat kernel

The heat kernel is the fundamental solution to the heat equation ∂u/∂t = Δu on a Riemannian manifold, representing the diffusion of heat from a point source. In spectral geometry, the trace of the heat kernel e^{-tΔ} encodes the spectrum of the Laplacian: as t → 0, the trace has an asymptotic expansion whose coefficients are integrals of local curvature invariants.

The heat kernel provides one of the most illuminating proofs of the Atiyah-Singer index theorem. For an elliptic operator D, the supertrace of e^{-tD²} interpolates between the topological index (as t → 0) and the analytical index (as t → ∞), revealing that the index theorem is a consequence of the local-global duality of diffusion. This method, developed by Atiyah, Bott, and Patodi, transforms a global topological statement into a local calculation involving the asymptotics of a parabolic partial differential equation.

Beyond index theory, the heat kernel is a central tool in geometric analysis, where it controls the smoothing properties of diffusion processes and provides probabilistic representations of solutions to parabolic equations.