How much math does a math professor remember?

1. Dec 19, 2006

andytoh

I've always wondered about this question. I've taken university math courses and gotten A+'s. But then years later, if I never used topics in that course again, I realize how much I have forgotten.

A math professor who does research in, say, number theory would essentially never use, say, the Gauss-Bonnet Theorem that he had learned many years ago in Differential Geometry. Would the number theorist be able to pick a textbook problem in the Gauss-Bonnet chapter and solve it from the top of his head? Are math professors so mentally powerful that the phrase "if you don't use it, you lose it" does not apply to them? Do they remember every math topic they have learned as much as they did just before walking into their final exam many years ago?

For example, how many math professors reading this post can prove the Inverse Function Theorem of second year calculus from scratch?

Last edited: Dec 19, 2006
2. Dec 19, 2006

matt grime

Erm, pick a hard one, not an obvious one. And, no, that isn't being arrogant. I can't remember plenty of proofs, but that is an obvious one to reconstruct.

3. Dec 19, 2006

pivoxa15

I'd say maths professor are as forgetful as the rest of the population.

I've witnessed an associate professor say he could not remember the basic trig identities (when it was needed in a complex analysis subject) because he hasn't taugh a first year maths subject for a decade or so.

Also I have asked a professor (who had taught the subject for 5 years in a row) a problem he had set but he only gave a sketch of the answer and said he couldn't remember the details (I don't think he was purposely trying to hide the answer but geniunely forgot).

4. Dec 19, 2006

andytoh

Of course a mathematician cannot repeat all the proofs he reads in research papers, but can they repeat the proofs of all the theorems and remember vividly all the topics that they had learned in, say, the first 3 years of university undergraduate math? (which I would consider the fundamentals for all math aspirees)

Last edited: Dec 19, 2006
5. Dec 19, 2006

chroot

Staff Emeritus
I don't really understand why you'd focus on the memorization skills of a teacher. After all, understanding is much deeper subject than mere memorization, and math is a huge field.

- Warren

6. Dec 19, 2006

andytoh

I agree that problem solving skills and ideas are more important than plain knowledge. I was just curious about how much they really know and remember? I personally get frustrated when I feel I mastered a subject, and then about 1 year down the road after not using that topic again, I cannot solve from a problem in that topic without looking up my notes and textbook (so now here is where problem solving does come into the discussion).

In fact, problem solving skills is what I am truly concerned about, but knowledge and memory is tied in to that. For example, just the other day I was trying to prove that the real projective space is a smooth manifold but had problem proving the diffeomorphisms of the coordinate transformations because I had forgotten the details of quotient topology that I had learned in my third year topology course years ago.

Last edited: Dec 19, 2006
7. Dec 20, 2006

DeadWolfe

Ok. A function is reimann integrable iff it's discontionuities have measure zero.

8. Dec 20, 2006

matt grime

That seems like a straight forward result. It's one of those that as soon as you start thinking about the definitions it is obvious. Like the Inv F. Th..

This is as opposed to say the classification of compact surfaces upto homeomorphism.

9. Dec 20, 2006

DeadWolfe

It's not at all obvious to me (though the reult in the case of finite discontinuities certainly is)

10. Dec 20, 2006

Hurkyl

Staff Emeritus
The proof idea is the same, isn't it? You split the domain into one part where the integrand is continuous, and one part of arbitrarily small size that contains all of the discontinuities.

(p.s. you forgot some conditions of the theorem)

Last edited: Dec 20, 2006
11. Dec 20, 2006

andytoh

Good stuff, guys! This is what I meant when I asked how much do mathematicians remember (I never said memorize). No one here has intentionally memorized the proofs of these theorems that we all studied in undergraduate university, but you have shown me that you "remember" the proofs in the sense that you can think it over and then repeat the proof by figuring it out on your own. I doubt anyone here has memorized the exact formula for the curvature tensor, but perhaps you can derive it from first principles. This is what I mean by remember.

So can I state that all mathematicians can prove every theorem that they had learned from university if all they were given was paper, a pencil, and sufficient time to think about it? Or is it possible that a mathematician would say (about a theorem from undergraduate math) "Yes, I know the theorem, but for the life of me I cannot repeat the proof" or even worse "What does the theorem state again?"

Last edited: Dec 20, 2006
12. Dec 20, 2006

DeadWolfe

I've had professors who couldn't even tell you the jist of the IFT.

13. Dec 21, 2006

Doodle Bob

you can state it, but its quite incorrect.

14. Dec 21, 2006

andytoh

I'm not talking about proofs like Fermat's Last Theorem. I'm talking about the proofs of the theorems that we have all learned from undergraduate math courses, all of which appear in any standard undergraduate math textbook. And the mathematician has all the time he needs to think about it, but without recourse to any aids.

If a mathematician can come up with their own new theorems, surely they can reprove any theorem at the level they would consider elementary, right? Similarly, they can solve any problem from a textbook from the top of their head without any aids too? For example, give a complex analysist a homework question in differential topology and he can solve it correctly and unaided if you just give him the time he needs.

Last edited: Dec 21, 2006
15. Dec 21, 2006

matt grime

It's continuous on all but a set of measure zero, and hence integrable. It's straighforward from the definition of the lebesgue integral, isn't it? I've never actually seen a proof of this, nor even the statement as given.

This is one of those proofs that requires no leap of imagination, and can be derived straight away.

16. Dec 21, 2006

gunnihinn

Yes, both of those are very possible. You must remember that undergraduate math is about showing you as big a spectrum of mathematics as possible, but your professors have spent years specializing themselves in one field.

For example, this situation would very likely come up if you went to your abstact algebra professor and asked him about, say, Dirichlet's problem in complex analysis. Or, vice versa, if you went to one of your analysis guys and asked about some of Sylow's theorems. They might remember what those theorems state, and even under what conditions, but it's not reasonable to expect them to be able to prove them from scratch in front of you, they're just human like you.

17. Dec 21, 2006

DeadWolfe

Hence why I said "REIMANN" integrable.

18. Dec 21, 2006

matt grime

But it is a short step from one to the other for 'nice' things like this. After all we're not trying to integrate the Dirichlet function (if I mean Dirichlet - 1 at irrational, 0 at rational). The key point is the 'set of measure zero' thing. Once you understand that and put in the correct criteria (like, bounded, I imagine, since the function 1/x for x=/=0 and 0 at x=0 is not integrable on the interval [-1,1])

Last edited: Dec 21, 2006
19. Dec 24, 2006

andytoh

Ok, is it fair to say that a mathematician should be able to prove all theorems in every topic (at the undergraduate level) that leads up to his field of expertise?

If so, then I intend to study all the proofs of every theorem in every topic that leads to my area of interest.

20. Dec 25, 2006

littleHilbert

How can you be sure that those theorems you leave out as somewhat irrelevant will not provide your area of interest with unexpected insights and applications in the future?

Nowadays number theory(!) finds unexpected applications in physics. Now, number theory is traditionally recognized as fairly irrelevant for physics. Modern theories are beginning to prove this assumption wrong. I think there is also luck and intuition in play when it comes to choosing the right tools for discovery and inventions at the frontier of maths.

Last edited: Dec 25, 2006
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