The departmental colloquium today was about certain aspects of the Ricci flow. Although the wikipedia page is quite intimidating, the geometric ideas behind it are quite intuitive. Several parts of the talk reminded me of the style of problems in our class. Since Aaron and I were the only students that went, I will try to explain some of it as well as I can (I really do not know a lot about this and perhaps someone will correct me if I make mistakes).
The Ricci flow is a generalization of the curve-shortening flow and mean curvature flow, which are themselves based on the heat equation. The heat equation tells us in good cases the heat will dissipate in finite time until it equal throughout. In simple cases, the other flows can be interpreted similarly, but their behavior is determined by the curvature of the geometric object. The curve-shortening flow shockingly enough shortens the lengths of curves in the plane and decreases the area at a constant rate so that it will go to zero in finite time. Also, it pushes out the curve where the curvature is negative and pushes in areas of positive curvature. That is, a loop will become more circular. The mean curvature flow is slightly more complicated but essentially acts the same on surfaces. The Ricci flow is a differential equation involving the curvature of a Riemannian manifold (smooth manifold with inner product on tangent spaces) that basically generalizes these ideas. One key difference is that the Ricci flow runs on the manifold itself while the other two run on embedding or immersions in \( \mathbb{R}^2 \) and \( \mathbb{R}^3 \), respectively.
A large part of the talk and the part that reminded me of our class was discussing the singularities that arise under these flows for embedded loops, embedded spheres, and simply connected manifolds. As I mentioned above, the curve-shortening flow asymptotically sends loops to spheres and this happens to be the only singularity for embeddings of \( S^1 \), i.e., Jordan curves. Things are more interesting for surfaces. The mean curvature flow has three types of singularities for embedded spheres: the round sphere, a cylinder, and a sort of half cylinder (capped at one end). As I understand it, a big part in the Perelman proof is showing that these same types of singularities are the only ones that appear in the Ricci flow of a simply connected 3-manifold and that theses singularities are precisely the ones that can be removed via the connected sum without changing the topology of the manifold.
In our class, we proved the fundamental theorem of algebra by looking at what happens to the winding number as we passed through singularities (roots) and saw that the winding number changed. Similarly, a key part of the proof of the Jordan curve theorem is how the winding number changes passing through boundaries of connected components. This is the opposite of what happens with the Ricci flow on simply connected 3-manifolds: the topology does not change. I think this is quite interesting. Another idea that appears to be very close to these (sort of in between the two) is that the Sturm-Hurwitz theorem (analogue of fundamental theorem of algebra for trigonometric polynomials) can be proved using the heat equation. This proof was presented by Professor Tabachnikov in his MASS course last year, and if anyone is interested, I could show you my notes.
This is a lot of stuff at once, but hopefully you all find it interesting.