As some topological spaces can be metrizable, the same can be said of categories. Certain categories can be concretizable, if they are isomorphic to a concrete category. I must be making progress as I understood almost everything in that post from David Speyer.
I spent most of the day reading up on differentiable manifolds, Riemann surfaces, germs, and sheaves. Some of the concepts are extremely interesting since they tie into category theory. This led me to differential geometry. I supposed that differential geometry had more to do with Euclidean geometry, an undergrad class that I didn’t enjoy all that much³, but it’s got a lot more to do with the geometry and structure of differentiable manifolds, which interest me¹.
Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds.
The study of calculus on differentiable manifolds is known as differential geometry.
One of the marks of being a good prof is when they see that students didn’t understand something and go over it again. My algebra prof has made a habit of this, especially when he goes over stuff in class too quickly. This happened last week and left me quite furious. Someone must have mentioned something to the prof², and he went over what we saw in the last hour again. This took an hour out of the three-hour class. I appreciated. I realized that there were some undergrads in our class, who weren’t familiar with some of the more abstract concepts of category theory³. It was an issue. Class was great today. I was drinking my strong milk tea and noting stuff down. We saw direct sums and (co)universal objects. Having proofs done with commutative diagrams is so elegant and simple.
It seems to me quite logical on how math classes should proceed. The professor presents some theory, with theorems and their demonstration, as well as definitions and propositions, before venturing into a slew of examples. For some reason, this is completely absent from my classes in Taiwan. The examples. I don’t know what these profs are thinking, but examples are paramount for students to understand some of the theory. OK, my complex analysis prof is good. he gives examples and answers questions well¹.
Sunday was consumed by research on spectral sequences, category theory, homological algebra, sheaf theory, topos theory, chain complexes, homotopy theory, cohomology, and functors (tor, ext, hom). These are all concepts of advanced algebra, graduate and doctoral level math in my opinion. I didn’t understand everything, but I’m thirsting for more knowledge on these concepts. When I’m at the library on Wednesday, I’ll pop by to get a few books on these subjects. Maybe I’ll just get and introduction to homological algebra.
I’ve already sent an email to the Analysis III professor asking for class notes so that I can read them and practice for the next semester.
The American Revolution was also on my mind. Since I was watching the John Adams mini-series on HBO, I was curious about certain facts. The Boston Massacre, the Boston Tea Party, the Battle of Bunker Hill, and the Join or Die articles were read.
I’m a bit annoyed at the mini-series. Why?
Well none of the great historical battles or events are portrayed in this series, at least until now. The characters witness them, but the viewers don’t see them. It’s annoying. I find that it cuts the narrative.