  We present an overview of the history of the heat kernel and eigenfunctions
on Riemannian manifolds and how the theory has lead to modern methods of
analyzing high dimensional data via eigenmaps and other spectral embeddings. We
begin with Varadhan's Theorem relating the heat kernel to the distance function
on a Riemannian manifold. We then review various theorems which bound the heat
kernel on classes of Riemannian manifolds. Next we turn to eigenfunctions, the
Sturm-Liouville Decomposition of the heat kernel using eigenfunctions, and
various theorems which bound eigenfunctions on classes of Riemannian manifolds.
We review various notions of convergence of Riemannian manifolds and which
classes of Riemannian manifolds are compact with respect to which notions of
convergence. We then present B\'erard-Besson-Gallot's heat kernel embeddings of
Riemannian manifolds and the truncation of those embeddings. Finally we turn to
Applications of Spectral embeddings to the Dimension Reduction of data sets
lying in high dimensional spaces reviewing, in particular, the work of
Belkin-Niyogi and Coifman-Lafon. We also review the Spectral Theory of Graphs
and the work of Dodziuk and Chung and others. We close with recent theorems of
Portegies and of the first author controlling truncated spectral embeddings
uniformly on key classes of Riemannian manifolds. Throughout we provide many
explicitly computed examples and graphics and attempt to provide as complete a
set of references as possible. We hope that this article is accessible to both
pure and applied mathematicians working in Geometric Analysis and their
doctoral students.
