Why Require Graduate Core Classes in Math?

[phreak]

This is a bit of an odd question, but it's been bothering me for a while.

The graduate sequences at top universities (Algebra, Geometry/Topology, Analysis) have always been rather esoteric in nature. Most students entering top universities have a rather strong background in these topics and need no further treatment to continue with their own research. Moreover, if by some chance, the content of some core course were to pop up in their research, it would probably be minimal enough that one could learn it very quickly.

What then is the point of requiring core classes in Algebra, Geometry/Topology, and Analysis if the students are probably never going to use a significant portion of the curriculum outside of their own area again in their research? Is the reason political? Something to keep students busy during their first year? A test to see if students can actually learn material outside of their specialty so that they can "earn" a PhD? Or does it actually serve a purpose?

(The title of the thread results from the specific case that I'm a mathematical physicist. Though the material in analysis, topology and geometry are quite useful to me, the chance that I'll actually use, say, principle ideal domains or Noetherian modules ever in my work is rather unlikely. This is very frustrating, as I think classes' primary purpose should be to serve as a supplement to research.)

 

[Unknot]

This will be a poor attempt to answer the question:

I really think you should know them, because just knowing them ables you to reach many different areas of mathematics.

Well, for my area in those three sequences you described, I needed to go further for every one of them. Having those basic things down just gives your research program more potential. (This is very speculative. Not too sure about this)

And I suppose to be a mathematicians you need to know the basics, and those are the basics. I agree with the choices for quals in most universities. It's reasonable. (OK, maybe not Harvard)

Also I should point out that there are virtually no schools that MAKE you take these courses if you know the material already. You can just take the quals right away.

 

[jambaugh]

Here are some points;

1.) Group theory was once though a play-toy of mathematicians. Now it is a central theme of theoretical physics. Who is to say what mathematical tools will be needed in future research. Often problems though expressible in one mathematical domain are made clearer and even "non-problems" when translated into an alternative mathematical framework. SR was never so clear until I saw it in terms of a deformed Lie group structure.

2.) What I tell my mathematics students is that though they may never use a particular topic we are studying they will certainly use the cognitive ability they develop by learning the topic. I use the analogy of a football player doing sit-ups. He doesn't do sit-ups as part of the game he plays but he uses the muscles thereby exercised.

3.) Whereas some mathematical subjects may not by their very nature apply to your field of study they may provide examples within a larger classification (say category theory) which in turn gives insight into this classification and which meta-subject can give insight into enumeration of possible examples of mathematical constructs which do directly apply to your area of study. I am thinking in particular of category theory and understanding of it via algebraic topology. The language of exact sequences is a very nice way to express quotient structures which helped me immensely in enumerating Lie group deformations. I'd have never been able to deal with the problem had I not, out of curiosity taken a course in algebraic topology.

 

[confinement]

Of all the first year core disciplines I feel that algebra is the most important, this class most of all consists of non-trivial theorems. Algebra is essential for studying algebraic topology, lie groups and lie algebras, etc. What kind of mathematical physics do yo intend to do without algebra? Perhaps you would prefer a course of graduate study in physics.

 

[phreak]

Of all the first year core disciplines I feel that algebra is the most important, this class most of all consists of non-trivial theorems. Algebra is essential for studying algebraic topology, lie groups and lie algebras, etc. What kind of mathematical physics do yo intend to do without algebra? Perhaps you would prefer a course of graduate study in physics.

Actually, from the graduate programs I've seen, most classify algebraic topology as a different subject matter than algebra or incorporate it into the geometry/topology curriculum. At my school, for instance, algebraic topology is strictly classified under topology.

When I say algebra, I mean things like principal ideal domains, artinian modules, galois theory, etc., which, as far as I know, are almost completely distinct from mathematical physics. As for non-triviality of theorems, I'd say that at most top programs, the theorems are almost strictly non-trivial. Would you say that the Radon-Nikodym Theorem is trivial? Lebesgue Decomposition? Whitney Embedding?

I personally (will) specialize in Constructive Quantum Field Theory. There exist algebraic techniques, sure enough, but these techniques compose such a small part of the typical algebra course that I have branded algebra courses as practically useless at the personal level. Basic group theory, basic ring theory, modules, direct sums, and tensor products are all I've used, and I'm sure a short one-week course could cover all of this.

Regardless, the spirit of my question was more "why should it be mandatory to explore other fields?" rather than a personal question. Realistically, one will not encounter too much of the content of at least one of the core courses in his research, so why not let the student have his own choice in what courses he should take? From what my research tells me, the only top universities that take this approach are MIT and Cornell.

1.) Group theory was once though a play-toy of mathematicians. Now it is a central theme of theoretical physics. Who is to say what mathematical tools will be needed in future research. Often problems though expressible in one mathematical domain are made clearer and even "non-problems" when translated into an alternative mathematical framework. SR was never so clear until I saw it in terms of a deformed Lie group structure.

2.) What I tell my mathematics students is that though they may never use a particular topic we are studying they will certainly use the cognitive ability they develop by learning the topic. I use the analogy of a football player doing sit-ups. He doesn't do sit-ups as part of the game he plays but he uses the muscles thereby exercised.

3.) Whereas some mathematical subjects may not by their very nature apply to your field of study they may provide examples within a larger classification (say category theory) which in turn gives insight into this classification and which meta-subject can give insight into enumeration of possible examples of mathematical constructs which do directly apply to your area of study. I am thinking in particular of category theory and understanding of it via algebraic topology. The language of exact sequences is a very nice way to express quotient structures which helped me immensely in enumerating Lie group deformations. I'd have never been able to deal with the problem had I not, out of curiosity taken a course in algebraic topology.

I think these are fair points. However, algebra (in the sense which I have described it above) is often so irrelevant to what I'm doing that I don't think it serves as useful mental exercise nor does it give me a way to express mathematical results in a visual and/or aesthetically pleasing manner. The fact is that algebra is, in many ways, too abstract to be present in nature and will almost surely never make an appearance in my field since my field is rather dependent on the physical. I do agree that looking at things a different way is often beneficial to the student, but if that's the case, then why not require, say, graph theory or combinatorics instead of algebra? (Not that I think either one of these deserves a place in standard mathematics curricula.)