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I'm a lot earlier in my math education that most of the people on this site. Currently I'm studying computer science, and I'm interested in looking into statistical and optimization applications, as well as theory (yeah, I know that sound broad, but I'm quite early in my education!). Anyways, I want to get deep into the math behind these things - statistics, combinatorics, linear and integer programming, maybe some analysis. Algebra also interests me.

I've found that math studies go okay, but from time to time I have trouble. Understanding delta-epsilon proofs was a big obstacle for calculus back in high school, and more recently generating functions have been giving me serious trouble.

If I pursue a path towards graduate level studies, what sort of things should I watch out for? More generally, since I don't know where I'm going, what concepts did you find hardest to grasp during your undergrad/early graduate studies?

Thanks.

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"soft questions" which don't really have a right answer should generally be made community wiki. You can do this by clicking "edit" and checking the "community wiki" box at the lower right corner of the input field. –  Anton Geraschenko Oct 25 '09 at 17:23

3 Answers 3

One general advice about early maths education is to not decide what you like too early!

Keep learning new fields, keep reading books about things you don't know yet. Most mathematicians I know aren't working on problems they would have expected when they arrived in grad school. It takes a surprisingly long time for most people to reach the level of mathematical maturity where they're reading to work on new problems, and until you have an inkling that you're ready for this, don't overspecialise. There's plenty of time for that later!

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I have advice, but it is dependent on the size of your university and/or the mathematics department therein.

A large part of my undergraduate process was to tutor others in the class. A good test of your understanding is if you assist someone else in coming to the same understanding. Unfortunately, it may be that your class sizes are epsilon. Thus it will be more difficult to form a study group. Applied Functional Analysis is a topic of great importance for a young computer scientist, especially one who is aspiring for a PhD, but it can be slow to grasp if you are taking it as an independent reading course.

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I go to a pretty big university, though the combinatorics class I'm in is TINY. I'll probably have 60 people in my analysis class, which is plenty, and math faculty is routinely excellent from what I've seen. I'll do the study group thing, I think, thanks. I may end up being the assisted one though. –  DoubleJay Oct 25 '09 at 17:48

My gut response is to say there are a limitless supply of these plateaus. There is so much out there that even the best mathematicians are limited in what they can understand well. So in terms of specific concept plateaus, well if you're like most of us you'll probably have lots of them, and that's a good thing.

In terms of concepts, I think what I found tough was often much clearer after I lost an early misconception. E.g. for a long time I thought the Killing form in Lie algebra was using "killing" as a synonym for "erasing"... I tried to build my understanding around that conception and it didn't work very well (Killing is a name). A lot of "simple" mathematical ideas are known by proper names rather than descriptive terms, so as more of these accumulate you have to rely more on memorization than intuition.

Outside of concepts, here's what I found tough:

  • Transition from coursework to research. Some people are very good at getting the A when the material is put in front of them, and most textbooks are good at giving you the necessary tools to solve the problems they present. I found the transition to more open-ended problems a significant challenge.
  • Understanding the frontier of a field. As stated in a previous response, it's tough to get to the frontier of a field. It takes a lot of work, and a lot of time. So graduate school requires a lot of perseverance.
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