It occurred to me this morning that the state of affairs in mathematics is very strange. A famous problem that goes unsolved for a long time often becomes a simple exercise to a future generation. The three Greek problems (squaring the circle, trisecting an angle, and duplicating a cube) are now simple exercises in Galois theory even though they stumped the greatest mathematicians for two millenia. The sum of reciprocal squares and some variants are problems frequently given in courses on complex variables and Fourier series. e^(i*theta)=cos(theta)+i*sin(theta) is frequently stated as mere tautology in many introductions to complex variables.
Goodearl frequently points out after presenting a proof in class that the theorem was first proved 100 years ago, and then a bunch of other people proved it as well, and the proof he presents is the simplest one known. It seems like an artificial way to understand mathematics to me, but it’s necessary to study mathematics in that way nowadays since there is so much material to be covered in such a short amount of time. On a similar note, doing algebra problems seems very strange to me. All I can do is try a bunch of random ideas that don’t work until I get one idea that does. Even then, I have no idea why such an approach should work. Yet I think that most problems I solve about finite groups can be solved fairly mechanically with a bit of representation theory. So why don’t people teach representation theory earlier on in algebra classes? Speaking of representation theory, I should take advantage of the yellow sale to learn about representation theory, so please feel free to recommend a book (preferably published by Springer) on the topic.
I suppose a lot of old papers are quite difficult to stomach. Putinar gave a talk on Takagi’s paper in the functional analysis seminar last week, and I didn’t understand it. I don’t think I can understand the paper either.
On a rather different note, the <=3 of you who read this journal and take 220 may be amused to learn that double cosets are mentioned in Lang.