The heat kernel is one of the most beautiful objects in geometric analysis. It connects partial differential equations, probability, and the shape of spaces in a single, elegant package. In this post, I want to build intuition for what the heat kernel is and why geometers care about it so deeply.
The Setup
Imagine placing a point source of heat on a curved surface — say, the north pole of a sphere. As time progresses, the heat diffuses across the surface. The heat kernel $K(t, x, y)$ tells you exactly how much heat arrives at point $y$ after time $t$, given that the source was at point $x$.
On flat Euclidean space $\mathbb{R}^n$, this is the familiar Gaussian:
$$K(t, x, y) = \frac{1}{(4\pi t)^{n/2}} \exp!\left(-\frac{|x - y|^2}{4t}\right)$$
But on a curved manifold, the geometry bends the heat flow. Curvature acts like a lens — positive curvature focuses the heat, while negative curvature disperses it.
Why It Matters
The heat kernel encodes an astonishing amount of geometric and topological information:
- Short-time asymptotics reveal local geometry (scalar curvature, Ricci tensor).
- The trace $\int_M K(t, x, x), dx$ connects to the spectrum of the Laplacian via the heat trace expansion.
- Long-time behaviour detects topology — the number of connected components, Betti numbers, and $L^2$-invariants.
This last point was the starting point for my Master’s thesis on the analytic aspects of $L^2$-invariants.
A Probabilistic Perspective
There’s a beautiful probabilistic interpretation: $K(t, x, y)$ is the transition density of Brownian motion on the manifold. In other words, if you release a random walker at $x$, the heat kernel tells you the probability density of finding them at $y$ after time $t$.
This connection between probability and geometry is one of the deepest themes in modern mathematics, and it continues to inspire new research directions — from stochastic analysis on singular spaces to applications in machine learning and graph neural networks.
In a future post, I’ll explore how these ideas extend to non-compact manifolds and how the heat kernel is used to define $L^2$-Betti numbers.