Week 1

So, I started this journey by trying to remember what got me interested in manifold topology with which to begin. I took a physics course in mechanics, and we went over Lagrangian mechanics, and I was totally blown away by it: I had no idea what these "generalized coordinates" about which the professor was talking were. So, I began to do some digging and wound up taking some more advanced math courses (as I was a math major who was doing a physics major just to find out about math's major consumer), and I discovered these things called "smooth manifolds" and "coordinate patches" on them, and I learned that "generalized coordinates" are just coordinate patches on smooth manifolds (Why couldn't the physics instructor just have come right out and said that?).

So, I remembered taking my first Ph.D. course in smooth manifolds, and we learned about smooth maps between smooth manifolds, which I thought were the most beautiful thing about which I had ever learned in my life, and about derivatives of smooth maps between smooth manifolds, Df: TM ---> TN, which are vector bundle morphisms between the tangent bundles of manifolds, which seemed even more beautiful to me than the smooth maps themselves.

So, I wanted to do something with tangent bundles, but the only thing of which I could think to do with them was have a tangent vector field on them aka a first-order ordinary differential equation on them. I knew ODEs on manifolds are a well-studied phenomena, and that I wasn't going to make many contributions there, so I started to think about other things, and I came up with second-order ODEs on manifolds. It turns out, there appears to be a much smaller literature on them, and there appears to be some confusion in the literature on whether you need a Riemannian metric, or at least an affine connection, to even define the acceleration of a curve in a smooth manifold or if you can just do it with the double tangent bundle TTM.

So, that was nice to see a possible niche in the literature, but then I remembered one of the things I found interesting about smooth manifolds and smooth vector bundles was characteristic classes. So, I did some Googling about those and found that they're really defined for principle bundles of smooth manifolds with topological or Lie groups as the fiber (On or SOn in the case of principle bundles associated to vector bundles), and that one then transferred the characteristic classes of the principle bundle back to the associated vector bundle with which one started.

So, I searched YouTube for videos on characteristic classes, and I came across this video, evidently by a CalTech instructor who was giving a simultaneous lecture/class with graduate students at the University of Tokyo via the videos that I was currently watching on YouTube. I jumped in in medias res with this video



and discovered I could hardly follow a thing. He appeared to be defining differential forms (a generalization of vector fields with which I am fluently familiar) and deRham cohomology (another field on which I can practically give lectures) by defining them to be the traces or determinants of powers of differential forms, a formalism that didn't even make sense to me on the surface of it, much less a formalism in which I could do computations - back to "generalized coordinates". So, that was Lecture 7 evidently in a lecture series, so I went back and started with Lecture 1, and I'm not back up to Lecture 7 yet, so I'm not sure how much sense Lecture 7 will make to me yet, but I have learned a lot. I learned about the Laplacian on arbitrary closed Riemannian smooth manifolds and how its solutions are sort-of unique or "our favorite" differential forms representing generators of deRham cohomology; as I had said, I have almost given lectures on deRham cohomology and its relation to integral or group integral handlebody cohomology - they're certainly in my dissertation - but this notion of harmonic representatives - differential forms that satisfy Laplace's equation  - of cohomology was news to me.

And, it was beautiful news.

So, next time, I'll outline what I've learned about harmonic differential forms and some examples of them along with the Hodge star (again, news to me and very beautiful mathematics) and normalized harmonic representatives of generators of deRham cohomology. That's all for this week.