AREAS OF MATHEMATICS THAT I LIKE THE MOST
This will probably sound boring to you. sorry.
1. topology in general, and topological groups where I convinced my pal famous topologist James Dugundji who I was auditing in another course to also present another course in my college dorm the next semester for credit. The students did the lecturing.
2. homotopy theory, part of algebraic topology, which is of interest in economics high class work also.
3. surgery theory and obstructions, part of algebraic topology
4. fibre bundles, where I gave a seminar in graduate school after studying one of the 2 only books on it for 2 years
5. differential topology and singularity theory, where I am researching to entend the work of Rene Thom and the genius Russian V.I Arnold. This is also known as catastrophe theory
6. category theory in the foundations of math, which is partially a revised set theory, and also a way to say "hey this is the same situation in another area, so the same proof applies by analogy that we already have"
7. elementary geometric transformations and their
combinations, taught now in high school, for intuition and confident visualization purposes. It gets embarrassing when your mind goes blank when asked a question related to this. Of course, if your professor will let you doodle a bit on the blackboard, you can probably work it out...
8. general algebraic topology, not really, because like calculus it needs another 300 years to standardize the exposition and put the rubbish in the trash or as a minor footnote for
picky people. There are just too many variations of the expositions currently. They didn't get good until 1974 when the 1st good example exposition came out, based on one good approach. Either we need to select the perfect exposition or find a way, not just by category theory, to transform one exposition into another. Perhaps
a computer can do it..., so everybody can be happy about seeing the proofs in the terms that are most congenial to their particular way of thinking. We also need more examples to be presented in the books and exercizes to be worked out by students ( and teacher LOL!). Let's hope the resulting work that a person gets is not over 500 pages long for the essentials,
which it is tending to exceed these days.
The Russians seem to do the most advanced research in math, while giving skimpy proofs. A lot of their works are translated into English. They are about 100 years ahead of everyone else. The French also research a lot, but their
works are not translated, except the famous anonymoous "Bourbaki" works. The Americans and British come along and digest it all and are the greatest expositors.
The most famous Russians that I am somewhat familiar with are Pontryagin, Alexandrov both dead, and V.I. Arnold now old or dead. Arnold has many associates.
I wish to discuss further item 5 above, on catastrophy theory.
In our world, don't we need some help in this area, and I hope the rich people haven't sponsored secret research and kept it to themselves to just make their stranglehold on the world worse. I am not being a paranoid political, but it seems possible, if the poor mathematician can't otherwise make a living.
Like me!!! No just joshing. But hey rich people, give me your proposals and it has to be at least 5 million a year or I am not interested. Of course, like the scientists, I cannot be held
responsible that I will ever come up with anything, and I might put in hard to find gaps in the proofs which would backfire on you.lol.
That's just enjoying writing, nothing else.
Back to real mathematical catastrophy theory or la la land.
Rene Thom made it interesting to people outside of mathematics by proving in 1955, almost without gaps in his proofs, that in 4 dimensions
as in time/space, there are just 7 kinds of catastrophes, each of which has a cute name and very common physical examples.
I forget the cute names but the situations are:
1. volcano erupting
2. landslide of big boulders
3. drowning or huge rains and flooding
4. desert
5. earthquake, also female orgasm
6. tornado, hurricane
7. wild fires
Perhaps I should study 5 more carefully...?!
Rene's theorem really astounded the world. His interests were more philosophical and about biological metamorphosis. Anyway, this was what we call a smoke and mirrors proof, because the cute names were math names well known. His remarks were very easy to understand while the Russsians remarks later and earlier are much more difficult and obscure. Rene's little ditty was expanded to several documents of about 400 pages to convince the skeptics over a period of about 20 years by another quite talented man. I guess he made a lifetime career of proving
just that one theorem.
V.I. Arnold extended Rene's work to many higher dimensions, and I want to extend their works to even higher dimensions. This is based on a
study of Arnold's 1974 English translation called "Singularity Theory". I should at least be able to exposit the thing and fill in the gaps. After I have gotten further along I will give a seminar to at least do that. This book is only "real dimensions". They switched to "complex
dimensions" after that and I am not so interested there. They really haven't gotten very far in the English translations at least in those studies.
Well, that is about it, except to mention at by adding just one more Star Wars dimension to make it 5, there are 4 more catastrophes possible, generically.
If it weren't for Star Wars, I might have lost interest in continuing my math studies for PhD. I basically only need to take one more course,
give the seminar, and write/defend the thesis. My wife had brain surgery on her pituitary and hypothalamus and I got rattled, plus my mind was
too immature to write the thesis then. So I switched to computers and by diligently working had a good career. See Hebrews 11:6.
At about 40, everything suddently clicked, but with quirks. Now being retired I can explore it again, if I can get a control on this amongst my
other interests. I'll be totally happy if I just give the seminar actually, as payback for my mentors gone before and all the good breaks that
I got in life. Gee, what if I live another 30 years? When young I thought I would be dead by now.
