Re: Important topics in math

On Fri, 27 May 2005, JG wrote:
> Anyways, this summer I'm planning to read up on as much math as I can,
> and I wanted to know which areas of math you guys thought would be the
> best to learn. What I'm looking for is a set of topics, which once I
> understand them, will allow me to use my knowledge to deal with as
> diverse a set of problems as possible. Obviously I'm going to be
> rereading my analysis and abstract algebra texts (actually I read
> through the analysis one already, and have gotten about 40% through the
> abstract algebra), but what else is important? Topology? I haven't
> really taken a topology course yet, but I did take differential
> geometry of curves and surfaces, which I was a big fan of, and the two
> seem related.
>
Bah, you're too goal oriented to learning. Now is the time to read up on
all those things you wanted to read about but didn't have the time. To
have fun, picking up on fun stuff. As off the top of your head, you liked
curves and surfaces, you might like algebraic topology (set theory
topology prequisite) which talks about surfaces. As for getting tangled up
with curves, take a gander at knot theory. If you really want to be
driven ecstatically crazy, gaze into dimension theory where you'll
contract dimension dementia puzzling how the different definitions of
dimension give different determinations of dimension for the same space.
Or, to consider another matter, consider a closed disk with a line coming
from the disk. What's the dimension of this space? Well, within the
disk, it's locally 2 dimensional and within the tail it's locally 1
dimensional and at the point where the tail is attached to the closed
disk, it's locally ?? dimensional. However, that snafu can be avoided
simply by making the disk an open disk.

> Also if there's any subject which is very important in current
> mathematical (or even just scientific) research, which aren't so
> complicated that I wouldn't be able to teach them to myself.
>
Relax. Find what you like, has intuitive appeal, that inspires you of
itself. With that you will excel. Anything you chose for it's
importance, at best you will become proficient. Instead of fulfillment,
you will be fulfilling chores.

> Maybe some specific type of probability would be important? Although
> it seems like a lot of probability could be covered by other courses,
> for example Lebesgue integration or some other sort of topic that I
> could cover independently of probability.
>
Doesn't seem you're very hot on probability. Perhaps the odds are against
you liking it.

> Anyways, if anyone has any ideas for topics and/or books which are good
> independent study guides for those topics, I'd appreciate them.
>
Go to a college library and browse around the isles of math topics and
shelves of math books.
.

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