Pure math is awesome! It’s so beautiful and g-loaded, enough that most of those mediocrities in tech cannot even touch it. Sadly, it’s quite a declining career, in terms of career prospects, and chances are I will say goodbye to it, maybe dabble in it a bit on a side, or help my kid become a successful mathematician. I’ve written a fair bit of pure math on this blog, though none of it I consider advanced. I don’t actually know algebraic topology or algebraic geometry, though I would say I know the basic ideas. I certainly wish I had learned more pure math earlier on. Unfortunately, it seems that the formal school environment (and this is college) is increasingly dull in terms of the pure math education and expectation and overall ethos, which has been increasingly tailored to mediocrity (especially as the career no longer attracts far tail IQ people as much as it used to), at least at the non-top places. The words of a top school math PhD who appears to be far more knowledgeable about the whole of mathematics (and physics) than most top school math PhDs on the career and community can be summed up in the following:

The only sure-fire way to become a tenured professor at a good research universitiy is to have a genuine breakthrough in your work. But it is insane to expect to achieve any such thing. A huge amount of luck typically goes into getting something really significant done. Expecting to be so lucky is lunatic. You are surely quite right that most people doing math in grad school ought to be doing something else, somewhere else. Certainly this is true of all people doing math in grad school without the slightest idea of why they’re there, or of what they hope to learn and to achieve. I would say that anyone pursuing mathematics for the right reasons and in the right way has about as much chance of achieving professional success as does the typical would-be poet or sculptor. Or perhaps I ought not to have said “professional success,” when what I meant was “academic success,” or “success in the form of a tenured professorship at a major research university.” To become a professor at a major research university as a mathematician these days is either the result of a tremendous fluke, or of a systematic program of professional advancement (i.e., of social climbing within the mathematics community) that is probably incompatible with serious research activity.

I got a similar impression. It sometimes makes me not want to do math ever again. Even if that’s the case, I might want to look back at what I’ve written mathematically, so I’ll keep a catalog here. And of course, this is also convenient for whoever reader of this blog wants a “higher level view” of my mathematical writings here. Of course, the reader is free to refer to the WordPress category feature which I’ve taken advantage of for partitioning by mathematics and within mathematics

- Big Picard theorem
- Elliptic functions
- Lie derivative
- Sheaves of holomorphic functions
- Construction of Riemann surfaces as quotients
- Weierstrass products
- Cayley-Hamilton theorem and Nakayama’s lemma
- Implicit function theorem and its multivariate generalization
- Urysohn metrization theorem
- Riemann mapping theorem
- Arzela-Ascoli theorem
- Path lifting lemma and fundamental group of circle
- Universal entire function
- Jordan normal form
- Hahn-Banach theorem
- Riesz-Thorin interpolation theorem
- Hilbert basis theorem
- Primitive element theorem and fundamental theorem of Galois theory
- Convergence in measure
- Second Sylow theorem
- Derivation of a (non-trivial) Riemann zeta function identity
- Composition series and Jordan-Hölder theorem
- Automorphisms of the quaternion group
- More to come later, this list is not inclusive