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What are the different subsections in applied and theoretical mathematics?

I want to pursue a career in math, and I know that I would have to decide on either theoretical or applied mathematics, but I was wondering what choices I would have to make further down the line as a math-major.
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Greg’s Answer

I'm answering this because there are no other answers so far and because one of my majors 35 years ago (ugh!) was mathematics. But I'm sure a current, practicing mathematician could give you a much more complete answer.


At that time, the big courses included abstract algebra (group theory and its relatives), classical/real analysis, and topology, with smaller siblings complex analysis, number theory, set theory, combinatorics, probability, statistics, and so forth. The latter pair are closer to applied. There were professors specializing in various of those areas, but I couldn't give you the names of the individual specializations themselves.


More recently, one of the folks I follow (John Baez, http://math.ucr.edu/home/baez/ , https://plus.google.com/u/0/+johncbaez999) has been working on and blogging about something called category theory (which frequently sounds very theoretical but has real applications, and not just to physics). A grad-student friend of mine, Curtis Bennett (now at CSU Long Beach), worked in an abstract area of geometry and topology called "buildings" (https://en.wikipedia.org/wiki/Building_(mathematics)), but that's the limit of my knowledge there.


As I said, a real, practicing mathematician could give you a much better answer, but perhaps this will serve as a starting point. (Note, however, that you'll have advisors in college who can help directly; you'll also have access to all of the professors, postdocs, and grad students, most of whom will be doing active research in various areas and some of whom may employ undergrads as research assistants.)

Greg recommends the following next steps:

Take a look at the links above and see what strikes your fancy. Baez's home page has links to quite a few interesting areas.
Skim through the list of Millenium Prize Problems (http://www.claymath.org/millennium-problems , https://en.wikipedia.org/wiki/Millennium_Prize_Problems) just to get a flavor of some longstanding, very difficult challenges in mathematics, mathematical physics, and computer science.
Dig into the field of quantum computing for some very applied but still very challenging areas of research (particularly algorithm design, which is mathematical, though there are many challenges on the hardware/implementation end of things, too).
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Mathew’s Answer

I am not a practicing Mathematician, but do know a few very well.

I would say Theoretical or Pure Mathematics needs a real love for the field and it is best to explore and experience it a bit before deciding on choosing it as a career. The main sub-areas are listed here
https://math.mit.edu/research/pure/index.php

To a large degree, Applied Mathematics is what most of us use in STEM related and other fields.
https://math.mit.edu/research/applied/index.php
https://en.wikipedia.org/wiki/Category:Applied_mathematics

I would suggest that before you choose and declare your major, to explore the different course offerings at your University and talk to the faculty on your interests. This may give you a better understanding on the career choice for you

Good luck and best regards
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