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How much math does a math professor remember?

  1. Dec 19, 2006 #1
    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
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  3. Dec 19, 2006 #2

    matt grime

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    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.
     
  4. Dec 19, 2006 #3
    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).
     
  5. Dec 19, 2006 #4
    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
  6. Dec 19, 2006 #5

    chroot

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    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
     
  7. Dec 19, 2006 #6
    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
  8. Dec 20, 2006 #7
    Ok. A function is reimann integrable iff it's discontionuities have measure zero.
     
  9. Dec 20, 2006 #8

    matt grime

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    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.
     
  10. Dec 20, 2006 #9
    It's not at all obvious to me (though the reult in the case of finite discontinuities certainly is)
     
  11. Dec 20, 2006 #10

    Hurkyl

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    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
  12. Dec 20, 2006 #11
    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
  13. Dec 20, 2006 #12
    I've had professors who couldn't even tell you the jist of the IFT.
     
  14. Dec 21, 2006 #13
    you can state it, but its quite incorrect.
     
  15. Dec 21, 2006 #14
    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
  16. Dec 21, 2006 #15

    matt grime

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    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.
     
  17. Dec 21, 2006 #16
    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.
     
  18. Dec 21, 2006 #17
    Hence why I said "REIMANN" integrable.
     
  19. Dec 21, 2006 #18

    matt grime

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    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
  20. Dec 24, 2006 #19
    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.
     
  21. Dec 25, 2006 #20
    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
  22. Dec 25, 2006 #21

    mathwonk

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    there are two things that make it easier to remember something, 1) repetition, and 2) discovery.

    I.e. we can remember things better that we teach over and over. and we can remember things that were our own discoveries.

    i cannot remember say the proof of the radon nykodym theorem or the closed graph theorem, stuff that I memorrized as an senior in 1965 just to get an A.

    but stuff like the inverse function theorem in one variable si almost trivial.

    harder might be the infinite dimensional version.

    the better you understand something the easier it is to remember.


    as an example i will illustrate with the idea of the proof of the infinite dimensional inverse function theorem.


    the idea is to use a versiion of the geometric series. i.e. the easiest inversion process is the series 1/(1-r) = 1=r + r^2 + r^3+.....

    now that looks like just inversion of multiplication, but properly understood it becomes also inversion of composition.


    notice that convergence for this series holds if r is small.


    the idea is then to recall that a function f has the identity as derivative at 0, if there is a very small function h such that f = I - h. i.e. where h(x)/|x| goes to zero as x does.

    Then the inverse of f = I-h is given by something like I + h + h^2+....

    only this doesnot quite work. I am forgetting the exact formulation now, but it is something like this: you form a sequence: I, I+r,I+r(I+r), I+r(I+r(I+r)),.... and prove this conversge to the inverse.

    having gotten thsi far I would have to sit down for a few minutes or hours and get it straight.



    with really hard stuff (for me), like deformation theory, sheaves, cohomology, high dimensional abelian varieties, biratiuonal geometry, even my own research, I actually tend to forget it within about three months.


    I also forget stuff i never elarned that well, like little tricks for anti - differentiating ("integrating") weird trig functions.


    So most of us are the same as other people about forgetting, but we look at some of this stuff wayyyy more than an ordinary person.

    And there are exceptions. I have a friend who never seems to forget anything. He is extremely smart. But I suspect he also spends a lot more time than others do thinking about it and understanding it in the first place.
     
  23. Dec 25, 2006 #22

    mathwonk

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    i do aspire to being able to prove, in multiple ways, all theorems leading up to my area. But I will probably die first.

    I love understanding absolutely every detail, I like to be able to see where every proof relates to another one for the same result. But I do not really separate the areas as anything you understanbd can be used in another field, once you understand it.


    I like finding my own proofs for results as often as possible, since as I said above I remember those better. The key for me to rememeber a proof is to analyze it down to its smallest parts, find the central point and put it back togotehr. then just remember the basic central idea. like the geometric series idea for inverse function theorem above.


    here is another example: one of the most complicated basic proofs seems to be poincare duality ni algebraic topology. the argumkent goes on for pages in greenbergs book.


    but i was sitting around once with John Morgan and (Fields medalist) Simon Donaldson and John told Simon he ahd found a nice proof of Poincare duality, using Morse theory. Simon thought for about 2 seconds and said " oh yes, just turn it upside down".

    well if you know morse theory you know that it changes simplicial vertices and edges, or singular chains of various dimensions, into maxima, minima, and various kinds of saddle points. So turning the manifold upside down as Morgan and Donaldson realized, changes the maxima into minima, ..etc..., proves Poincares duality.


    I have never remembered the detaield proof in Greenbergs book, and have never forgotten the one I heard that day from Morgan and Donaldson.

    So if you are ever nearby and they say would you like to go to lunch with the speaker? say yes! thank you. you may hear something you will easily remember all your life.
     
    Last edited: Dec 25, 2006
  24. Dec 26, 2006 #23
    This is precisely the dilemma I'm in. There are too many proofs of theoremes leading up to my area of interest that I cannot provide, and I want to know these proofs by heart before I read on to new material. But for every proof I read from the lower level math, I lose the time I could have used to read new material from higher level math.

    What would you consider to be the best ratio for time spent on reviewing old material to the time spent learning new material? Currently the time ratio I'm using is about 1:1 (which includes the time I spend on doing problems in old topics), which many may consider wasting too much time on old material and I'll be much older than I should be by the time I reach the frontiers of my area of interest (though I will know the topics leading up to it much better).

    Thank you mathwonk for your very honest answer about how well you remember the proofs leading up to your area. For a second there, Matt Grime convinced me that a mathematician should be able to prove from scratch every theorem leading up to his area of research.
     
    Last edited: Dec 26, 2006
  25. Dec 26, 2006 #24

    mathwonk

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    well matt is younger than i am and sharper, and remembers more than i do.

    so as i get older i use more tricks for remembering.

    i can probabl;y prove a function is rimenna integrable if and only if the discontinuities have measure zero by the way, in a few minutes.


    the idea is as matt said, the same as the proof that a functio with a fin ijte set of discon tinuities is inbtegrable.


    i.e. just approximate your function by step functions very well except on a set which isnt very big. then although we cannot control the height on that set, it wont matter since the base is so small.

    lets try:
     
  26. Dec 26, 2006 #25

    matt grime

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    Really? Then I can only presume you didn't read what I wrote. The ability to recreate 'bookkeeping' type proofs is something I suspect a lot of research mathematicians can do with a little time. There are no deep ideas, nor tricks, to remember. Since one of the results in my pre-phd stuff was the Riemann mapping theorem, I certainly don't claim to be able to reproduce all of the material I've ever seen. Heck, I can't remember a lot of it. And I even said so in my first (or perhaps second) post in this thread.
     
    Last edited: Dec 26, 2006
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