Should I Become a Mathematician?

[mathwonk]

well i did think about it once and had a proposal for proving it false, but did not try doing the heavy lifting to see if it worked. I told it to some much smarter people more expert in the topic and had the pleasure at least of seeing them think about it seriously.

It is a very hard problem. it says that something very unusual only happens in a geometrically restricted situation. So most of the time it holds vacuously. And in all reasonable situations where the hypotheses hold, it has been shown the conclusion does as well.

So there are hundreds of papers out there saying "the hodge conjecture holds for cubic threefolds" or in some other case. But no one knows how to show it holds in general. One of my coworkers, Elham Izadi, has an inductive approach that may be useful.

Thanks for the suggestion I may be on it. It takes courage to work on something that hard.

 

[mathwonk]

here was my idea. the hodge conjecture is about recognizing the cohomology classes of algebraic subvarieties of a given algebraic variety.

I.e. every algebraic variety is a topological space, and if smooth, is a manifold. So it has a fairly computable cohomology group. recall that a homology or cohomology group is a group of equivalence classes of triangulable topological subspaces, where two classes are equivalent if their difference is the boundary of the class of a triangulable subspace, think submanifold, with boundary. All this is topology.

Now the analytic side of algebraic varieties allows one to represent all cohomology classes using differential forms, and by defining a metric, by differential forms which are harmonic, in the sense that the real parts of holomorphic functions are harmonic functions. This decomposition turns out to be independent of the choice of metric.

So then harmonic forms can be written as sums of terms involving dz's and dzbar's, and it turns out that the cohomology class of an algebraic subvariety always has the same number of dz as dzbar representatives, i.e. has "type (p,p)", for some p.

So as I understand it, which is minimally, the hodge conjecture asks if this is also a sufficient condition for algebraic representability of a cohomology class, i.e. that its harmonic representatives have class (p,p).

My idea was to look at the space, let's see now, its been so long ago, of hodge substructures of type (p,p), i.e. those which could be hodge structures of algebraic subvarieties. And in there to look at the subspace of actual geometric hodge structures, those which come from algebraic subvarieties.

So the Hodge conjecture is to see if those two are equal, or not. but if two subvarieties are equal, then their tangent cones are also equal, so my idea was to compute the tangent cones of these subvarieties at some interesting yet accessible element, and hopefully show they are different. That would disprove the Hodge conjecture. That would not win the prize money, but would settle it.

the reason this is an approachable tack is that tangent cones are far easier to compute than anything else.

 

[Vagrant]

I have just finished with high school and will be starting with engineering college in two months time.
I am a little weak in the following topics:
Functions,
Continuity and differentiability,
Permutations and combinations,
Equations and inequations.
I am looking for a book that will have more emphasis on theory and proofs, because I have a few books which contain problems for practice.
In school basically we were told how to deal with specific problems and given formulas.
Will "What is Mathematics?" by Courant and Robbins be a good choice?

 

[Werg22]

I have just finished with high school and will be starting with engineering college in two months time.
I am a little weak in the following topics:
Functions,
Continuity and differentiability,
Permutations and combinations,
Equations and inequations.
I am looking for a book that will have more emphasis on theory and proofs, because I have a few books which contain problems for practice.
In school basically we were told how to deal with specific problems and given formulas.
Will "What is Mathematics?" by Courant and Robbins be a good choice?

No, What is Mathematics can't be considered a rigourous textbook. Courant and Fritz John's books first out of three books is what you're looking for, though permutations and combinations will need you to be looking somewhere else. I must warn you though; Courant's text is no easy one, especially if it's your first dive into mathematical rigor.

 

[mathwonk]

i guess i agree that What is Math? is not a textbook, but it strikes me as mathematically and logically rigorous. I like it and think it has a lot to offer. It is not as dry as a regular textbook, and covers more topics. But the author Richard Courant, is a much better mathematician and scientist than most textbook authors.

Rigor is a relative concept. In the 1960's when set theoretic topology was growing in influence in textbooks, Fritz John rewrote Courant's book to make it more modern and "rigorous" by using more point set language, but to me the effect was more to make it less appealing.

Rephrasing the definition of continuity from epsilon /delta to the open set version in my opinion only makes it less intuitive and no more rigorous. But these are matters of taste. Surely there are discussions in What is M? that lack full details, but they are still valuable.

Here is a little example from What is M? The usual proof of uniqueness of prime factorization begins by developing the theory of the gcd and the lemma that a prime number cannot divide a product of two integers unless it divides one of the two factors.

Courant observes that the proof of prime factorization can be done without this lemma, if one observes that the lemma definitely holds for integers which do have prime factorization.

This way one is able to do the proof by induction, building up from cases where the lemma holds. It then follows as a corollary of uniqueness that the whole theory of gcd's goes through.

This argument as given by Courant, is not only completely rigorous, but contains insights one finds almost nowhere else. A typical textbook would merely present the usual theory of gcd's and then prime factorization, with or without perfect rigor.

E.g. the proof in Dummit and Foote, unlike Courant's, has a major logical gap, as I have observed elsewhere, although DF has the appearance of a rigorous text. But Courant is a master.

Although of course you are right that Courant is not written in the style of a usual textbook, still it is useful to read the masters no matter how they express themselves.

I myself have struggled for years with trying to write out this proof of unique prime factorization, troubled by the need to reorder the factors and give the induction in modern over precision, maybe using permutations notation.

Than I read Gauss, where it is done very clearly indeed, with only the amount of precision that illuminates the proof, and not so much irrelevant over- precision as serves only to obscure the argument.

So as long as you can provide any missing details, then Courant should be adequate, or even if not, it is a good introduction to topics one can read in more detail elsewhere later.

I agree though that as a student I sometimes found other treatments more understandable than Courant.

 

[fopc]

i also like hartshorne's recent book, geometry: euclid and beyond.

I can see why.
I read the first several pages of it that are available at Amazon.
Interesting, the Amazon list price is about $51.
I see copies out there (new or like new) for under $18.

I imagine you have major experience with his other monster textbook.

By the way, any experience with Goldblatt's book, "Topoi, the Categorical Analysis of Logic"?

 

[mathwonk]

no, i am not familiar with goldblatts book.

i think $50 is a good price and about right for Hartshorne. new copies of a book like that for $18 suggest something is amiss, i.e. that they are pirated, or "international editions: not intended for sale in the US.

So I avoid buying them.

 

[DeadWolfe]

What is the difference in those 'international editions". I know a guy who gets them and seems to have no problems with them. Are they illegal, or what's the deal with them?

 

[Chris Hillman]

Category Theory books?

By the way, any experience with Goldblatt's book, "Topoi, the Categorical Analysis of Logic"?

I found it valuable for some things, but the discussion of logic is IMO insufficiently clear. The best first book on category theory is Lawvere and Schanuel, Conceptual Mathematics. Good second books include

1. Saunders Mac Lane, Categories for the Working Mathematician,

2. Colin McLarty, Elementary Categories, Elementary Toposes,

3. Robert Geroch, Mathematical Physics (dont' be fooled by the title, it's really a reprise of standard undergraduate math major courses from the perspective of categories).

 

[fopc]

Thanks Chris. I know about the books you mentioned, except for Geroch.

Regarding Goldblatt, it's his focus on logic that got my interest.
But if the (logic) development is not sufficiently clear or weak, then I'm not too interested.

Incidently, I now see his book is available for viewing at:
http://historical.library.cornell.edu/math/

So I can check it out for myself.

 

[JeffN]

hello mathwonk,

in another thread, you listed four theorems in single variable calculus which you thought were the most important. The intermediate and extreme value theorems, as well as rolle's theorem and the mean value theorem. Can you explain why?

 

[mathwonk]

well i was trying to outline the content of a typical non theoretical calculus course for my class. the usual content covers 4 types of problems:

1) proving equations like x^3 = 2 have real solutions.
2) solving max/min problems.
3) graphing functions.
4) integrating to find areas and volumes.the first problem is solved by the intermediate value theorem, the second by the extreme value theorem, the third by the rolle theorem (which implies that a function can only change direction at a critical point, and can only change concavity at a second order critical point), and the 4th is covered by the corollary to the MVT which implies that a function is determined up to a constant by its derivative. (That implies that since the derivative of the area function is the height function, then you can find the area function by antidifferentiating the height function.)

actually, theoretically these theorems are not too different. the proof of the extreme value theorem is similar but a little more complicated than the proof of the intermediate value theorem, and the rolle thm is actually implied by the extreme value theorem, and MVT is a slight generalization of rolle, and is implied by rolle. thus really there are only two essentially different theorems there, the IVT and the EVT, but i called them 4 different thms because they have 4 different uses in the course, and they look different to the students.

 

[mathwonk]

here is a survey of the hodge conjecture by a friend of mine who does understand it:

http://math1.unice.fr/~beauvill/basically it asks for ( and proposes) a characterization of those homology classes on an algebraic variety, which arise as the fundamental classes of algebraic subvarieties.

 

[threetheoreom]

Hey mathwonk, your advice is well heeded .. i was wondering if you heard of the is book or ( or anyone for that matter ) Fundations and Fundemental Concepts of Mathematics?I am trying to wrk through it for the summer.

 

[mathwonk]

well there are extensive reviews at amazon, but the excerpts viewable there do not reveal much. this seems a book for the general public, apparently a good one.

 

[mathwonk]

there is a great algebraic geometry conference starting in paris on monday, unofficially in honor of my friend arnaud beauville's 60th b'day. i'll be there, let me know if you will, and maybe we can have lunch or something.

 

[mathwonk]

well i am in paris at the institut henri poincare, and it is wonderful to be here.

the city of paris alone is intellectually stimulating in a way one cannot at all believe coming from the US south. ~I passed a public bookstore today with the complete works of galois on display in the window, unheard of even in a typical university town in US.

And the leadoff talk in the conference was a wonderful account of recent work on determining when certain varieties constructed by group quotients, are rationally connected or not.

A variety is rationally connected if you can connect any pair of points by some rational curve, and this checkable property is conjectured to imply the variety is the image of a rational variety, which is unknown.

the distinction is between finding lots of maps from P^1 to the variety and finding one map from some large P^n onto it.

This is what I came for, the instant bringing up to date on interesting and current questions by masters, in a single hour.

there are also people sitting around discussing "political" matters like how to raise funds to support the education of mathematics students in the developing world, people having a wider impact than just by their own research program.

All this makes one ask what one could be doing to enlarge the reach of mathematics education, such as the keepers of the flame on this site are doing. bravo to them!

so it is both educational and inspiring to be here. best wishes to you all. hope to see some of you sometime at one of these meetings. If you are here and want to recognize me, I am the nerdy American touristy looking guy in a t shirt with the honoree's picture on the back and the conference poster on the front. come up and say hi.

 

[mathwonk]

Apparently MSRI in berkeley is having a big algebraic geometry session spring 2009, so think about coming there if you like the topic and want some immersion in it.

 

[mathwonk]

i just heard a nice talk by a friend, Fabrizio Catanese, vastly generalizing an old result of serre on the action of Galois groups on algebraic surfaces.

if you have an algebraic surface defined by equations with complex numbers, and you change those numbers to their complex conjugates, would you think it changes the surface much? actually it only changes the complex struture and not the differentiable structure so at least the fundamental group does not change.

but what if the equation is in terms of algebraic numbersa and you let the galois group of the algebraic numbers act on it? Catanese showed that for every non trivial element of that Galois group, except complex conjugation, there is a surface whose fundamental group is changed by the Galois action.

somehow that seems odd. for one thing there is an algebraic form of the fundamental group, which turns out to be the completion of the topological one, and these groups do not change under Galois action, so one gets a large collection of groups that are different but whose completions are isomorphic.

lovely talk, very concrete, with all the surfaces constructed explicitly as group quotients of products of explicit plane curves with very simple equations. it was very apporpriate at this conference dedicated to Beauville, since these surfaces generalize a construction of Beauville.

 

[mathmuncher]

Sounds like you're having a blast. Have a safe return trip.

Anyhow, I just came across a very light-hearted joke that some of you oughta like:

Why do so many math majors confuse Halloween and Christmas?
Because Oct 31 is Dec 25.

 

[threetheoreom]

Because Oct 31 is Dec 25.

Old One.. still interseting though

 

[DefaultName]

quick: integral of e^(x^2) dx

 

[ice109]

quick: integral of e^(x^2) dx

not integrable

 

[J77]

uhh... before those last two posts degenerate into a derail -- this isn't a thread for doing sums on!

 

[mathwonk]

just heard a talk on strange duality that has a moral for all young researchers: look for what is natural, as answer to a question.

the setup concerned the study of spaces of vector bundles on a given riemann surface. if we fix the rank r of the vector bundle, and the riemann surface, there is a "moduli" space M(r) parametrizing all these bundles (with some other fixed data).

This moduli space M(r) itself has a distinguished "divisor" the subvariety of codimension one of bundles having non trivial sections, and this in turn defines a line bundle L on M(r).

now we can ask about sections of this line bundle L, and of its powers L^k.

It was noticed early on, again i think by the honoree Beauville, that in some cases there is a duality between sections of L^k on M(r), and section of L^r on M(k), called strange duality. All this has some interest to physicists, since i guess vector bundles on Riemann surfaces probably have some connection to Witten's theory of quantum gravity, or string theory.Anyway, eventually this duality was proved in general for all Riemann surfaces, by linking it to a familiar duality. I.e. in projective n space there is a classical duality between points and hyperplanes, and in general between k planes and n-k planes.

e.g. a point of projective space has coordinates which may be viewed as coefficients of a linear form defining a hyperplane. this is the usual duality defiend by dot products in euclidean space. the more generals ewtup is called grassman duality. (the space of k planes in n space is called a Grassmann variety.)but the point is there is a way of interpreting the strange duality of those exotic sections of bundles as just the duality in a suitable grassmannian.

moral, if something new looks like something familiar, try to see why, i.e. try to relate them somehow. the idea is that there is often a reason an apparently new phenomenon resembles a classical one.

 

[mathwonk]

here is a tiny example:

compare

arctan'(x) = 1/(1+x^2)

tan' = sec^2 = 1 + tan^2,

f' = 1+ f^2 is soved by f = tan(x).

these are all basically the same statement.i.e. if you know that arctan'(x) = 1/(1+x^2), or that sec^2 = 1+tan^2,

then the diff eq f' = 1+f^2 should remind you of those facts, and that enables you to solve the de.

 

[Darkiekurdo]

Sounds like you're having a blast. Have a safe return trip.

Anyhow, I just came across a very light-hearted joke that some of you oughta like:

Why do so many math majors confuse Halloween and Christmas?
Because Oct 31 is Dec 25.

I do not understand your joke.

 

[mathwonk]

i didnt either until i decided the month names were suggestive of modular equivalences.

 

[Darkiekurdo]

I am obviously too stupid for this forum.

 

[Pseudo Statistic]

not integrable

Looks (Riemann) integrable to me; it doesn't have a closed form solution, though.

 

[mathwonk]

no you ain't.

 

[Darkiekurdo]

Haha, I get the joke.

 

[DeadWolfe]

It's really a computer science joke.

 

[mathwonk]

seriously now (belive that?),smart or dumb is our ancestors' doing, achievement is our own.

not to malign my ancestors, who were very bright, but i used to call myself the "rocky" of algebraic geometry, dumb, but persistent.

 

[InbredDummy]

i always hear algebraic geometry being used in terms of string theory and quantum field theory, can you talk a bit about this mathwonk?