Re: advice on how to study math in grad school
From: Tim Ball (tb_at_timball.net)
Date: 08/30/04
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Date: Mon, 30 Aug 2004 10:08:07 -0400
Mike I. Thompson wrote:
>Hi,
>
>I was wondering how to approach studying math in graduate school. I'm
>starting my second year in math grad school, I'm in what is called the
>pure math program, and I've passed 2 of the 3 qualifying exams (the
>remaining exam is analysis, and i'll be taking it in a few weeks) but
>I feel there's a lot of basic math, especially analysis, that I don't
>know. I was wondering if a good approach to studying math is to set
>the goal of trying to learn all of "pure" mathematics. (Obviously,
>this is an impossible goal, but one can set impossible goals, and
>while they won't reach them, they would still get far.) So with this
>goal, I'd take time to thoroughly learn analysis (e.g. via various
>comprehensive texts in analysis), group theory, general topology,
>algebra, combinatorics, etc. all by working through various
>comprehensive texts on these subjects. I would eventually like to do
>research involving algebraic geometry, but I feel that maybe when I'm
>at that point, I'll come across a problem which I'll be able to solve
>only by knowing some trick I learned because I thoroughly studied some
>more basic math. Comments?
>
>
>Mike
>
You touch on the basic problem faced by mathematicians in this age of
specialization: There is so much to know, and life is so short.
Eventually you will die, and even if you spend every waking moment
studying mathematics, you will not be able to understand all worthwhile
ideas before it's all over.
For what you want, working through comprehensive texts on all the
various subjects, with all the proofs and exercises, doesn't seem a good
idea; it will take way too long to cover them all.
Unfortunately, mathematicians are generally not very good at explaining
their great ideas to other mathematicians from outside of their narrow
specializations. But there do exists surveys that describe the basic
ideas of a mathematics subject and pointers to more detail if you ever
need them. Bulletin of the AMS has some very good surveys like that.
Attending survey lectures, or finding them on the internet, will also
help you understand what the great themes are.
And in a few years you will get your PhD and start teaching
undergraduates, and that will be a good chance to learn various subjects
in more detail. The best way to learn something is to teach it to others.
TB
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