How do mathematicians find things to investigate?

[wooby]

I am curious about this. Not every master's and Ph.d. thesis can be ground breaking work that adds significant findings to mathematics. How does someone find the mundane things to investigate and publish? The low hanging fruit if you will.

 

[mathman]

Theses topics are usually proposed with a thesis advisor, who presumably is pretty knowledgeable about what is going on at the frontiers of mathematics.

 

[mathwonk]

i think one looks at new situations and sees analogies with other older ones, and then asks oneself if certain properties of the old case apply again in some form here as well.

or one understands a proof quite well and asks oneself if it can be made simpler. or one looks for a theorem whose proof actually proves more than the theorem states.a trivial example in abstract algebra is this exercise i gave my class: we knew eisensteins criterion for irreducibilityof a polynomial, namely if the coefficients are integers, and p is a prime that divides all but the leading term, and p^2 fails to divide the constant term, then the polynomial is irreducible.

i noticed this criterion looked asymmetrical whereas polynomials are very symmetrical. so i asked them to prove the mirror version, that a polynomial is irreducible if p divides all except the constant term, and p^2 does not divide the lead term.

but no one did. one woman tried though.

 

[mathwonk]

in my own recent work, i noticed that although in many ways "Prym" varieties were analogous to jacobian varieties, still many of the properties of jacobins were unknown for pryms, like multiplicity of the points of the theta divisor, the riemann singularities theorem, so i tried to establish this for pryms. the first and lowest hanging case was noticing that as above, in a certain rather general case, the old proof actually still worked.

i.e. the kempf proof of rst for jacobians stated that if a certain map were birational and had smooth fibers, and the derivative was injecftive on the normal bundle, then the tangent cone to an image points is the image of the normal bundle.

My coworker and i noticed that birationality was not needed for this, just smoothness of the fibers and the rank condition on the derivative. this gave a new result, although perhaps known to some experts. to boost the significance of this easy result, we gave also some applications.

later we proved a more precise result on exactly when this happens by different techniques. this attracted attention from other workers who obtained stronger results and more important applications. i would have liked to do those too, but at least we got it started.

 

[mathwonk]

how many times will this browser throw away my posts before i elarn to always save them before trying to post? i wrote a rather lengthy answer to this but have no will to repeat it now.

basically it said the better you understand a proof the more easily you will see places to use it again, and cited a case where a published paper proved A, then cited a reference for the proof of B, when in fact A implies B, but the authors semed not to be aware of that.

 

[mathwonk]

i am reminded of a quote in hadamrds book on the psychology of invention in the mathematicL SCIENCES WHEREIN A PROFESOR ASKEDE A STUDENT, how long have you been taking our courses? and in all that time have you not noticed anything that warranted further investigation?

(when i read this, i had not.)

 

[mathwonk]

one brilliant profesor of mine suggested taking a good paper and reading only the statements of the theorems in that papaer and trying to prove them oneself. he claimed that whenever he did this his own proofs of the resukts always contained some new resuklts not stated in the paper.

he was also the man who taught me to look at a proof to see if it in fact proved more than stated. (maurice auslander)

 

[mathwonk]

to get started on even hard problems restrict the problem to small cases. the great david mumford used often to look carefully at the case of hyperelliptic curves to investigate a significant problem which in principle was of interest for all curves.or one could look only at genus 2 or 3 and see what comes out. once you do all you can in genus 2, you have a list of problems to try to generalize to higher genus. notice although there is a definition (by kontsevich) of moduli spaces M(g,n) of stable curves with marked points, much of the research out there is on the case g = 0.

 

[mathwonk]

it os hard to find the low hanging fruit since the experts often do those cases first. people who are fast learners jump on new ideas and work them out in easy cases right away. e.g. when quantum cohomology was discovered people began calculating it for the simpeklst accesible cases. thos of us who were late to elarn about it were too late to get in on the easy calculations even though we could have done them too. sometimes a thesis dvisor is well placed to inform a student of new results before they are widely disseminated but less so now in the internet age. not being so quick myself, i like the approach of understanding things deeply so as to see how to use them again where others have not noticed the similarity. sometimes my students do not have the patience i ask to learn things deeply however, preferring to only hear the statement of a result, to studying the proof carefully. later however some of these same people have come back and asked your question: how do i do research?

 

[0rthodontist]

i noticed this criterion looked asymmetrical whereas polynomials are very symmetrical. so i asked them to prove the mirror version, that a polynomial is irreducible if p divides all except the constant term, and p^2 does not divide the lead term.

If P = QR, P, Q, R are polynomials, then Reverse P = Reverse Q * Reverse R, where the reverse of a polynomial the polynomial formed by reversing the order of the coefficients. So P is reducible iff Reverse P is reducible, so eisenstein's criterion applies.

 

[mathwonk]

well that was basically what i wanted them to notice, that the same proof applied if it were reversed. you have stated, if not proved, this in as succint a way as possible. but my student never grasped what you did so clearly, namely to prove the reverse statement, reverse the proof.

i am sure you can prove your statement, but one quick way seems to be to set X = 1/Y.

i like your argument better, but the reduction mod p proof also works exactly the same for both cases of eisenstein.

 

[mathwonk]

as i said, one looks for analogies. in my algebraic geometry course once we proved, from the book, that the intersection of any two projective varieties of codimensions r,s respectively in P^n, has codimension at most r+s.

the proof used the fact that the homogeneous coordinate ring of P^n is a ufd.

so later when we learned that the local ring of a variety X at a smooth point is also a ufd, i challenged them to prove the same local result, i.e. the dimension at p, of the intersection of two subvarieties of X passing through a smooth point p and having codimensions [wrt X] respectively r and s at p, is at most r+s. and to find a counterexample when p is not a smooth point of X.

but apparently no one had learned the proof of the earlier result well enough to do it, or even to notice it was feasible. The class was apparently used to solving easy problems that follow just from the statements of the big results, and having those problems given to them, and I had not succeeded in helping them break out of this pattern.

 

[mathwonk]

rather than looking for low hanging fruit, i suppose one just looks for whatever one can notice, and then tries to prove it, afterwards it may or may not turn out to seem low hanging.

i.e. find your own problems, instead of looking in books for problems that others have stated or could not prove. although that may stimulate your thinking.

i prefer trying to understand what has been done as well as possible and why it was done,a nd how it hoppens to explain some phenomenon. then that understanding will begin to recur to you as you look at other situations.

but if you do not put some real information in your brain thoroughly in the first place, in my experience it is unlikely to suggest itelf to you later as a solution method.

there are counterexamples to this, since when working with strong coworkers one can sometimes depend on their understanding and just wing it, counting on them to keep errors out, but when working alone i have only been able to use what i really knew very thoroughly, what i had in my pores so to speak.

 

[mathwonk]

if you are a student, this year in one of your courses try making up problems and questions that go beyond what is presented, and keep a notebook of these questions.

if your course is elementary calculus then probably all your questions will be eithere asy or the subject of later courses, but that will make those courses more interesting.

e.g., we all know that a continuous function is riemann integrable, and that the indefinite riemann integral( as a the function of the upper endpoint) is differentiable everywhere with the original functiion as derivative.

but what about a discontinuous riemann integrable function? does it ahve an antiderivative? i.e. is there a fundamental theorem of calculus for all riemnn integrable functions?

what would we mean by an antiderivative of a discontinuous but integrable function? how would we recdognize it?

the main point turns out to be the analog of the mean value theorem, or rather its corollary, that two functions with the same derivative differ by a constant. what should this say when the two functions are only differentiable almost everywhere?

said backwards, how do you recognize the integral G of a riemann integrable function f, where G is the integeral of f as a function of the upper limit? what properties does this function G have? yes it is diffrentiable where f is continuous, but what else?

you have to say more than that to pin it down.

did it ever occur to you that there was a gap in the presentatiuon of integration theory? that integration was defined for lots of discontinuus functions but the FTC was only stated for either continuousn ones, or onesw that have an antiderivative everywhere?

what abut integrable ones that do not have an antideriavtive everywhere? can they still be evaluated by the FTC somehow? you will not even notice this gap unless you understand the proof of the usual FTC very well.

 

[mathwonk]

also if you can make contact with people who are the leaders of a subject they will be most ikely to be aware of the open questions that have not been open too long, or are easier to tackle. go to their talks, take their courses, read their papers and preprints.

i have been too lazy to do this many times in my career and missed fine chances. excellent people have handed me their early works and i have been too intimidated or overwhelmed to make the effort needed to penetrate them at the time. years later people have poured all over these works and i have noted ruefully that i had them in hand for decades sometimes without studying them

if someone outstanding has clearly done some new imprtant work, one way to find low hanging fruit is to try be one of the first people to learn the new methods and ideas. of course this is hard to do. often we do not even hear about these results until others closer have already extended them and thanked the author for "showing us his unpublished work".

but when we do get the chance we can work at it. at least we will be ahead of most people.