How Much More Knowledge Does a Mathematician Have Than a Math Graduate?

[mathwonk]

by the way, i am having a birthday party and you are all invited!

http://www.math.uga.edu/~valery/conf07/conf07.html

 

[andytoh]

I would like to be the first in this forum to wish you happy birthday and at least 40 more to come. I think its rare for professors to post regularly like you, Matt, and Chris, especially when you could otherwise be using your valuable time doing research or posting in some forum reserved for other mathematicians whom you could benefit from more than from us amateurs.

 

[Crosson]

I also wish to everyone that they follow there dreams!

In fact I think it is the publish-or-perish establishment to blame for the ills I spoke of, and that anyone can turn an interest in math into quality research, if they were not pressured by such a system.

 

[Gib Z]

Haha Nice mathwonk! (or should I say, Roy Smith). Guess what, check out my profile! It my birthday tomoro as well, I am turning 15!

My jesus, you've got some smart fiends, I've heard of some of those people...don't know much on algebraic geometry though lol.

And one i thing i keep forgetting to do is apologizing to matt grime for my estimation of your age at 40. Take it as a complement to your knowledgeableness (I didn't know there was such a word, but my dictionary says so..). Not even 30, you've definitely got a bright future.

 

[andytoh]

I've forgotten a bit, As you can see in my thread in the Calculus/ Analysis section. Can't even integrate sqrt tan x anymore...but when other people do it i still understand what their doing, so yea..

Learning Calculus at such a young age is great, but make sure you are learning the foundations of calculus as well, what the derivative and anti-derivative really means, what limits and continuity are really about. Don't just learn how to calculate derivatives and anti-derivatives.

 

[mathwonk]

happy birthday Gib Z! I first read Lincoln Barnett's "The universe and Doctor Einstein" at about 15, but knew nothing at all of calculus, so you are way ahead of me.

 

[Gib Z]

Well then its nice to know I have a small chance of one day becoming as knowledgeable as yourself :D

Ahh feels a little lonely to be on a physics forum on the morning of my birthday, no one at home...but o well, andytoh, i know all of those fundamentals, however I am not 100% sure I have the formal definition of a limit in my head...and also, somethings that worries me, I understand, but don't like, using the formal limit approach to differentials, I prefer viewing them as infintesimal quantites, helps me understand, rather than just calculate. I find it more intuitive, and makes the chain rule a breeze :D.

 

[mathwonk]

see if you like this approach; there is no need for limits to do derivatives for polynomials, as descartes realized.

the limit point of view says the tangent line is the line which is the limit of secant lines, but algebraically this means it is a limit of lines that meet the curve twice. this should be visible algebraically by saying the intersection point obtained by setting the curve and the tangent line equal to each other should have a double root.

so solve for the slope m that makes the line y = m(x-a)+a^2, (which passes through the point (a,a^2) with slope m), meet the curve y = x^2, twice at the point (a,a^2).

I.e. set y = m(x-a)+a^2 = x^2, and note that x=a is a root. Then solve for the unique m that makes x=a, a double root. see if you get m = 2a.

 

[mathwonk]

dont feel too lonely Gib Z, my mom is 98 today, and is in an assisted living home, and i cannot be there, because I have a doctors appointment myself today.

It is true often that the teen age years are a lonely time, because we have not yet found our community. Look forwaRD TO COLLEGE, and choose it well.

But haVE FUN IN HIGH SCHOOL TOO, YOU MAY NEVER SEE MOST OF THosE PEOPle AGAIN, AS THEY WILL GO DIfFERENT WAys.

 

[Gib Z]

Thanks for the advice, I am going to take College to be the same as University, Australia doesn't really have colleges...

The descartes approach is quite good, though what I meant was I prefer to think of differentails as infinitesimals in my head, and when doing calculations. ie all my textbooks say that it isn't rigourous to treat a derivative as the ratio of 2 infintesimals, and can't use them like normal fractions, but it does it anyway..

 

[Hurkyl]

Well then its nice to know I have a small chance of one day becoming as knowledgeable as yourself :D

Ahh feels a little lonely to be on a physics forum on the morning of my birthday, no one at home...but o well, andytoh, i know all of those fundamentals, however I am not 100% sure I have the formal definition of a limit in my head...and also, somethings that worries me, I understand, but don't like, using the formal limit approach to differentials, I prefer viewing them as infintesimal quantites, helps me understand, rather than just calculate. I find it more intuitive, and makes the chain rule a breeze :D.

You haven't learned differentials yet. (And the chain rule is even more obvious in differential geometry!)

Just remember, thinking in terms of infinitessimals is a crutch -- it might help now to imagine derivatives in terms of naïve infinitessimals, but in the long run you are going to want to turn things around: to use the concepts of calculus to define your intuitive notion of infinitessimal.

Now, a definition is needed for mathematical study, but that doesn't mean the definition is the most important thing. The limit definition of a derivative is just one of its many properties -- we just happened to pick that one as the starting point.

(And that, I think is because it's more "concrete" -- it would probably be difficult for most beginning students to understand what's going on if calculus texts started with an abstract characterization of the derivative)


By the way, have you seen nonstandard analysis? There is a calculus text that teaches with the hyperreals.

 

[Gib Z]

Thanks, I've seen nonstandard analysis in wikipedia, and it acquaints to my simplistic definition needs :S From what I saw in the wikipedia article for differentials, I've learned them...but o well...I can't really envision a form of geometry making the chain rule obvious, buy you obviously know more than me. Btw, Happy 28th yesterday :)

 

[mathwonk]

chain rule: (linear + higher)o(linear + higher)

= (linear)o(linear) +( linear)o(higher)+ (higher)o(inear) +(higher)o(higher)

= linear o linear + higher.

 

[Gib Z]

I think du's canceling out in fractions is simpler, if you don't mind me :) no disrespect guys.

 

[mathwonk]

well that is the genius of leibniz!

 

[interested_learner]

If you are going to be any good at all, you have to keep learning all your life. I am 30 years out of college and I still read constantly. I learned NOTHING that I do at work in college, nothing and I went to about 8 years of college. It was all picked up afterwords. Math is a vast field. You will never catch up. That is the fun!

My only answer to you is to keep running. You are off to a good start. Keep it up.

 

[Tom1992]

i don't agree with the definition of a senior unit of knowledge. i doubt that a senior student knows all that, and definitely doesn't remember all that even if he took all those courses.

i am hoping to get my phd by the time I'm 18, but in order to get there in time, i will have to avoid some of those courses. for example, i have no intention of doing my phd in anything related to pde's, so I'm not going to waste my time taking it. i already feel that i wasted my time reading a textbook in number theory because I'm not going to go into that field either (i never use congruences and mods in any of the subjects that i am studying now). you should only know deeply what you want to specialize in, and just have a mild familiarity of subjects that are very distinct from your specialty.

i found that when you move from one level to the next, you rarely use topics from the previous levels again except very common foundations like derivatives, jacobians, vector spaces, open sets,... for example, after learning implicit differentiation in calculus 1, i never needed to use it again. then after learning curl, div, and grad, stokes' thm, gauss' div thm in calc 2, i never used them again. after studying rings, i never saw rings again. after studying curves and surfaces in R^3, the diff geo moved onto R^n, and i never saw them again nor did i ever used the gaussian curvature, mean curvature, mainardi-codazzi equations, gauss-bonnet thm ever again... my point is that to know everything is a waste of knowledge space. you should only know very well what you want to specialize in and just have a basic familiarity with the other topics.

 

[mathwonk]

well yes and no. i also have worked in a somewhat narrow specialty most of my career and never used certain things, that is true.

but occasionally i have been confronted with problems i could not solve, partly because i ahd not bothered to learn something which i THOUGHT was unrelated, but turned out to be useful after all.

One wants to learn, not as many topics, but as many ideas, as possible. Ideas are useful, but only if we have them in mind.

also sometimes you get bored being specialized and want to have some fun teaching number theory even while doing a career in abelian varieties.

at some point you may enjoy talking collaboratively as well with people in adjoining fields, and it helps to be able to communicate with them, knwing their language. Some one who not only does his own research but also answers questions for others is considered more valuable to a department.

So do specialize deeply, but try to keep up a certain familiarity with things that are generally agreed to be important. what is your special field of interest? maybe we can make some suggestions as to how some of the other fields touch it usefully.

good luck with your PhD.

 

[mathwonk]

heres a little example of stuff that came up and surprized me by its usefulness. I knew little about gauss and stokes, etc, but was interested inm topology and our department was having a seminar to learn de rham cohomology, sheaf version. then i had to teach several variable calculus.

somehow i began to wonder how one would prove thata circle really does wrap around the oprigin at the same time, and i noticed that stokes theorem i.e. greens thm was the right tool.

i.e. the fact that dtheta has integral 2pi around the circle implies the circle does not bound any disc that misses the origin, by greens thm.

i was so excited, i relaized this was actually the key idea behind de rham cohomology, but the people lecturing on the sheaf theory version of DR apparently did not know this.

i went on to include a proof that a sphere has no never zero vector fields by gauss thm in my calc class. later i saw an article on this topic in the American math Monthly, but not as elementary as the version I had discovered myself.

In my thesis i was studying the degree of a mapping of moduli spaces, and the usual technique for that is to use "regular values", but i did not have any at my disposal. It turned out the inverse and implicit function theorems could be applied to the normal bundle of a fiber to substitute for them.

This method was very effective, and had not been used before. It only came to mind because years before i had thought long and hard about those theorems from advanced calculus. I was using them in algebraic geometry but the ideas were the same, once understood deeply.

a beautiful technique in studying theta divisors of abelian varieties is to use the gauss map, as introduced by andreotti, or rather as re - interpreted by griffiths, from andreotti's proof. this is a classical idea used by gauss to measure curvature of surfaces in 3 space, but adapted by andreotti to study the geometry of jacobian varieties of algebraic curves.

fundamentally, the gauss map is an invariant of an embedded hypersurface (or more general surface). once understood, it becomes of interest to calculate it also for a divisor embedded in a complex torus, because like affine space, a complex torus has a trivial tangent bundle!

once you get past the nuts and bolts details of the gauss map and curvature in 3 space, and realize the gauss map is an invariant of all hypersurfaces in manifolds with trivial tangent bundle, you can use it more widely.

even in manifolds with non trivial tangent bundle, the "gauss map" taking a map on points, to its derivative, a map on tangent spaces, is of interest in measuring properties of maps, as developed again by griffiths and carlson as the method of infinitesimal variation of hodge structures, in many settings.

the beautiful and powerful techniques in analytic number theory show that zeroes and poles of complex holomorphic map[pings are intimatel connected to number theoretic proe\perties. see the proof by dircihlet of the reslt on primes in arithmetic progression, which also uses crucially group theory.

the idea behind groups is just that of symmetry, which is why it is useful in many places especially physics.

etc etc...any ideas you understand can be useful. so specialize, but try to understand as deeply as possible those ideas you encounter.

 

[Tom1992]

i also have worked in a somewhat narrow specialty most of my career ...also sometimes you get bored being specialized .

so what does a specialist mathematician do when after several years he gets bored in his specialized area? can an abelian varietist suddenly become a number theorist like you suggested?

 

[Gib Z]

Anyone can suddenly change their area of expertise, but it will take time for them to work up to a certain point. A mathematician can change their area of focus to physics, but it will take some time before the mathematician is able to know as much as a fully fledged physicist?

As to you thinking you are wasting you time with some of the fields, I could not disagree more. It seems you care more about your PhD than the beauty of the mathematics. It does not matter if you ever need it again. Having the knowledge, and even better, an elegant proof is all i desire.

Study mathematics for the beauty of it. It is an art. It does not matter if it has no practicality, no physical usefulness, no human interpretation. If you find a field of mathematics that may be useful, learn it. But take the time to appreciate it.

 

[matt grime]

so what does a specialist mathematician do when after several years he gets bored in his specialized area? can an abelian varietist suddenly become a number theorist like you suggested?

Mathwonk didn't suggest changing speciality. He suggested learning about it and teaching it. Abelian varieties are algebraic geometry, a large field, which has links to algebraic number theory and arithmetic geometry. Fermat's last theorem is a proof that very much requires knowledge from such apparently unrelated-to-the-outsider areas.

If you want a reason to learn more than just one narrow area, then I suggest you read up on one of the Fields Medal winners, Terry Tao.

 

[Tom1992]

Study mathematics for the beauty of it. ...take the time to appreciate it.

perhaps you are right. i still need time to figure out which branch of math i really like and so should do some exploring with various courses to find out is right for me. and as mathwonk and mattgrime suggested, any math course we take may surprisingly turn out useful in whatever area we specialize in later on.

 

[mathwonk]

tom, i do get bored sometimes with my speciality. it does seem possible however to switch to number theory, from abelian varieties, since those subjects are closely related.

also algebraic geometry is so broad, that moving to many other fields, like diff geom, diff top, several complex variables, commutative algebra, number theory, maybe algebraic topology, or even mathematical physics such as string theory or quantum field theory, is quite feasible.

i have friends who have done such a transition. I myself have been an invited speaker at the institute for theoretical physics in trieste, while still a specialist in abelian varieties.

in fact as an algebraic geometer, i have learned and used almost all pure math fields.

it took me a while to choose a specialty as well, as i started in algebra, then algebraic topology, then several complex variables, then algebraic geometry.

i have also taught measure theory and functional analysis, and number theory, but never numerical analysis, lie groups, representation theory, or pde.

but i have used the heat equation in my research. so compared to some, certainly not all number theorists, my training is pretty broad.

 

[mathwonk]

i feel however a certain insecurity at changing specialities, since i am a recognized specialist in my area, and if i change, i start over as a newbie.

but hey, you have to go with what interests you before it is too late, right?

 

[mathwonk]

an algebraic function is also analytic, hence differentiable, hence continuous. thus algebraic geometry is a subspecialty of analytic geometry, analysis, differential topology, and topology. thus one can move backwards into any of those other fields, at least in principle.

 

[andytoh]

i certainly don't know all that stuff, especially not the pde...

i have used the heat equation in my research.

How did you incorportate the heat equation in your research when you don't feel up to par in pde's? Did you crash-read a pde's textbook out of interest and then the heat equation sparkled a light in your algebraic geometry lore? And to do research related to the heat equation, even if it isn't the main focus of your article, don't you have to have great in depth knowledge of the techniques of pdes in its solution methods? I don't get it.

 

[mathwonk]

i examined the heat equation and thought about it and its implications. you don't need a general education in an area to use specific topics from that area.

in particular i read carefully the paper of andreotti and mayer, on period relations for algebraic curves.

after many years, i have an ability to use things in my research that i barely know.

 

[mathwonk]

here is a little example: it is difficult to compute the circumference of a circle, but it is easy to prove that the area of a circle is (1/2) the product of the circumference times the radius.

i am like the guy who cannot compute the circumference, but who can prove that the area is the product of the radius by half the circumference.

 

[andytoh]

i do get bored sometimes with my speciality. it does seem possible however to switch to number theory, from abelian varieties, since those subjects are closely related.

also algebraic geometry is so broad, that moving to many other fields, like diff geom, diff top, several complex variables, commutative algebra, number theory, maybe algebraic topology, or even mathematical physics such as string theory or quantum field theory, is quite feasible.

i have friends who have done such a transition.

How much time off does a specialist usually have to take to read up on the prerequisites in another (related) area to start doing research in this new area? (now here is where being a learning machine really helps)