Perhaps the most fundamental differential operator on Euclidean space 
One can only agree with Prof. Tao. This agreement only intensifies when one considers the generalizations of the Laplacian such as the Laplace-Beltrami operators that appear in the Hodge theory of Riemannian Manifolds. Or when one considers the discrete analogues of the Laplace Beltrami operator such as the Graph Laplacian which I dare say have changed the landscape of research in unsupervised machine learning in the past decade or so. For a sample consider (the related) laplacian eigenmaps, spectral clustering and diffusion maps for just three examples. I have had the chance to work on Manifold learning for a while and have been very fascinated by the Graph Laplacian and its uncanny prowess. I was thus looking for material that actually relates and talks about the laplacian from the point of view of “flow”, diffusion and the heat equation beyond a superficial sense. The idea of diffusion or flow is a very interesting way of looking at distance, which is also partly the reason why the said machine learning techniques are so successful. This paper (Semi-Supervised Learning on Riemannian Manifolds) by Belkin and Niyogi is quite beautiful from a machine learning point of view, but I was recently pointed out to this short volume by my supervisor (with the gentle warning that it might take a lot of work):
The Laplacian on a Riemannian Manifold – S. Rosenberg
Click on Image to view on Amazon
Thus, the title of the post is a little misleading, as this monograph has almost nothing to do (directly) with graph laplacians. But then I am only interested in this from the point of approach wherein I can get a better understanding of why they are so powerful. This seems like a slow read but is not inaccessible and is very well written. But the best thing about this book is that it is pointed. Some sections in between can be skipped without any problem too. In the worst case it could be considered a roadmap to what needs to be known to really understand the power of the graph laplacian.
Another book that I have been reading slowly these days (it’s actually almost like a long paper than a book and thinking of it that way has a big psychological impact), and quite enjoying is this one:
Laplacian Eigenvectors of Graphs
Click on Image to view on Amazon
Unlike the book above, this is far more accessible and I believe would be to somebody who has an interest in the graph laplacian or even spectral clustering. It mostly deals with some interesting issues related to the graph laplacian that I had never even heard of. While these books are not new, I just discovered them a month back and I think they should fascinate anybody who is fascinated with the Laplacian.
_________________________
Onionesque Reality 