I learnt how to do proofs and what proofs really mean in my abstract algebra class. I still use abstract algebra to refresh my math skills.
Re:Interview with Terence Tao
by yummyfajitas on Monday September 29, 2014 @11:45AM
(#8383416)
I learned proofs in a numerical analysis class. That doesn't mean a budding algebraist should study numerical analysis, it means you will learn how to do proofs in any rigorous math class.
Sorry didn't mean to imply that one needs to learn abstract algebra to do proofs. Was just mentioning my experience as it was course where I really enjoyed doing proofs.
Wondering why this is voted down. He is quite right. Several math professors at my school who teach applied math do not know even the utmost basics...say things like computing kernel of a quotient group. Am sure they must have known it at some point during their student years, or maybe during the 60s-70s, you could get away without knowing these things, I dunno.
otoh, my abstract algebra professor was able to learn and then teach me Ito's lemma all the way up to semimartingales, topics he had never set eyes upon in his entire life! So I'd say the abstract stuff makes you much more stronger - gives you tools to understand from scratch material you've never seen before. The applied people generally glaze over when you mention math topics out of their competence zone, especially topics that are "useless" for some definition of "use".
Do you think this actually reflects the subjects validity? I'm sure that applications can be found, but is it because people cannot do it?
Re:Interview with Terence Tao
by yummyfajitas on Tuesday September 30, 2014 @12:06PM
(#8388981)
I don't know what you mean by "validity". I am absolutely certain that applications of all the "standard" fields (real, complex, algebra) can be found. They aren't bad things to learn.
I'm just suggesting that from what I've seen, the other fields I listed are more commonly useful. Everyone I know, across pure and especially applied math has used probability and linear algebra. Lots of people haven't used complex analysis or algebra ever, except on the qual.
In modern mathematics, particularly applied mathematics, these are NOT the most useful topics. That's just how things have always been done, and no one wants to argue about changing them in faculty meetings. See also the language exam: http://web.math.princeton.edu/~templier/language-exam.txt
In my former career I never once used topics covered in "abstract algebra" (note that I'm distinguishing linear algebra from abstract), a common situation among analysts. Most people outside of analysis never use complex variables.
The only common core of topics I can identify that nearly every mathematician I know uses is:
Measure theoretic probability (this intersects with real analysis)
Linear Algebra
Algorithms and optimization (this is less common than the above two).