Other Should I Become a Mathematician?

[uman]

I think you've got things backwards when you say we can discover new pure mathematics through string theory.

 

[Gib Z]

I think you've got things backwards when you say we can discover new pure mathematics through string theory.

I was thinking exactly the same thing lol.

 

[mathwonk]

odd as it may seem, he may be right. check out the story of counting rational curves on the quintic threefold, by candelas, et al...

 

[mathis314]

i still want to be a mathematician

 

[Gib Z]

odd as it may seem, he may be right. check out the story of counting rational curves on the quintic threefold, by candelas, et al...

Our object wasnt in that mathematics was derived from string theory, but that the mathematics is "pure" (not applied), because it inherently can not be; it is applied in string theory.

 

[mathis314]

Our object wasnt in that mathematics was derived from string theory, but that the mathematics is "pure" (not applied), because it inherently can not be; it is applied in string theory.

Its funny, however, that sometimes epiphanies in science beg for higher mathematics.

 

[muppet]

Ed Witten became the first theoretical physicist to win the Fields Medal because he worked out new maths needed to tackle string theory
That being said, I'd say that string theory is really applied rather than pure math?

 

[Gib Z]

Its funny, however, that sometimes epiphanies in science beg for higher mathematics.

Not at all. In fact, its been like that throughout the history of science and mathematics.

 

[mathwonk]

to me, counting the rational curves on a quintic threefold is about as pure as math gets.

of course you can define pure to mean it did not come from physics, but then your argument is tautological.

 

[uman]

Well what is pure mathematics then?

 

[mathwonk]

good question, but i have no interest in answering it. its what i do, not what i philosophize about.

i guess my point is that the question answered in this case is one of pure mathematical interest, with no physical application. The answer was however obtained via insights stimulated by very theoretical physics, i.e. quantum gravity and string theory.

you will also find more than a few people who do not believe string theory is physics, but is merely pure mathematics, since it has apparently no predictive power in physics. Not only have no predictions been shown correct, but critics say it has not even produced any checkable predictions at all.

It does however have predictive power in pure mathematics, as exemplified above.

 

[Darkiekurdo]

Is it weird that when I see an expression I need to simplify I don't automatically think 'Right, obviously I need to apply so-and-so rule'? Or is that something that comes with lots of practice?

 

[RonL]

I'm not sure if Lord Kelvin has been mentioned, (1364 post) a lot to go thru, what impressed on me the need for math, was a quote by kelvin, and doing a search, produced so many variations, that i'll just use the words that are in my mind.

" Any idea, that cannot be given a numerical solution, is not worth the paper it is written on"

 

[pivoxa15]

Grothendieck was a very complex person, and you might enjoy reading one of the articles about his life in the Notices of the AMS. Even though he quit young, he accomplished far more than most people in a much longer period. From his own remarks, he may have overdone the hard work, and needed a rest.

Just read it. Very interesting. Some of you were discussing what is pure maths. I think the mathematics done by Grothendieck is an extremely good example of pure maths. I love it when he says he doesn't like to use tricks to solve problems but using many small steps that are completely natural to crack open a problem as if by no force at all. There seems to be many similarities between him and the philosopher Ludwig Wittgenstein. For one, they both try to get to the absolute foundations of their disciplines.

 

[Gib Z]

With my due respects to Lord Kelvin, he obviously was wrong on that point. The Fundamental theorem of algebra has no numerical quantities. Is it not worth the paper it is written on?

 

[RonL]

Not sure what i have learned today, but one thing is to be more careful where i put a comment. -:)
As for the Fundamental theorem of algebra, after a quick read my brain started hurting.
It seems more of an assertion, than an idea.
I have always thought of Kelvin's statement, more in the action, reaction, physical, and mechanical world.
So much to learn.

 

[quasar987]

mathwonk,

How important is the topic of research for my master's degree? By "important", I mean, how easily will I be able to switch subject for my doc? How relevant is it to a professor that a student seeking to do a doc under him has no research experience in his field or no experience at all despite possibly one or two relevant course followed at the undergrad or grad level?

 

[mathwonk]

i did not do a research based masters degree myself. usually in math people go straight into the phd program. a masters thesis is often an expository paper on a topic one has learned, involving no original research.

this may be changing now, as even undergraduates are often involved in some research, frequently involving computers, where one does not need a great deal of technical expertise.

 

[proton]

i have a question about applied math grad school admissions:

what do the top grad schools look for in applicants? are gpa, letters of rec and gre scores the main factors? do research, REUs, etc matter?

 

[mathwonk]

i do not know about applied math in particular, but i know about all grad programs in general. what is sought is sheer mathematical strength and tenacity, creativity and potential to do good research.

it is assumed that the best candidates will also have high scores of most every sort, but the really distinguished candidates will have impressed someone personally, who will say so in a letter.

 

[proton]

oh yeah, i had 1 more question about grad schools:
do they only really care about your major gpa? like if you majored in physics and math and apply to math grad school program, do they only care about your math gpa? not your physics gpa?

 

[mathwonk]

i think i answered this above.

 

[eastside00_99]

i do not know about applied math in particular, but i know about all grad programs in general. what is sought is sheer mathematical strength and tenacity, creativity and potential to do good research.

it is assumed that the best candidates will also have high scores of most every sort, but the really distinguished candidates will have impressed someone personally, who will say so in a letter.

oh my god! I waiting to hear from the schools I applied too; this sounds really intimidating.

 

[mathwonk]

well that's what we want. but there are not a lot of those, so we'll take what we can get.

 

[torquerotates]

Hey, guys. I'm new to this forum. I have been interested in math for a while now, but have now decided to major in. I face a major concern. I'm currently a community college student intending to transfer to a top school. I have great grades and straight A's in math courses up to differential equations and am currently expecting an A in linear algebra. But I'm afraid that all this will not prepare me for upper div work at the 4yr. We've covered all the standards that the AP covers, i.e computing derivatives and integrals etc. But we lack rigor completely. I fact I haven't even seen proofs until linear algebra. I'm transferring next fall. Is there still time to catch up? If so what should I do?

 

[PowerIso]

I wouldn't worry to much. You might want to work on some of your proof writing, since the courses after linear algebra are pretty heavy into it. Especially if your university requires math majors to take a calculus proof based class. If you want to get a good idea on general proof ideas and terminology you can read:

"A transition to Advanced Mathematics" by Chartrand. It isn't a long book, but it gives you a good overview on the type of proofs out there and how to attack certain problems.

 

[aXiom_dt]

How does co-op work for a (undergraduate) pure math program? Is there a way to get some kind of research related job?

 

[eastside00_99]

well that's what we want. but there are not a lot of those, so we'll take what we can get.

Its amazing! I got an acceptance letter today from U of ill in urbana. I didn't see that coming; this was the highest ranked school I applied to.

 

[morphism]

Congratulations!

 

[Evariste]

Is it always best to read one subject at a time?

 

[mathwonk]

congratulations!

Sheldon Katz is a good friend of mine there, and I have met William Haboush, also very nice.

 

[Gib Z]

Is it always best to read one subject at a time?

Not always. Sometimes two different subjects can aid each others learning. For example, many people recommend taking multivariable calculus and linear algebra together.

Its amazing! I got an acceptance letter today from U of ill in urbana. I didn't see that coming; this was the highest ranked school I applied to.

Congrats =]

 

[Helical]

mathwonk, my main interest is physics and that is what I've planned on majoring in. But I've found that I'm also quite interested in math and am thinking about doing a double major. Is this feasible in four years as well as having a life other than my studies? I think I'm more intelligent than average but I'm not a genius so maybe staying for 5 years is a better option. I'll be waiting to make this decision until my first year to make sure I enjoy calculus. Also I am a senior in high school and am currently only taking pre-calculus.

 

[mathwonk]

helical, i don't really know the answer to this, but i do think it is worth trying.

you can give it your best informed shot, discussing it with college advisors at your chosen school, then see how it goes.

the only way to find out is to try, intelligently, i.e. by first getting advice and planning as you are doing.

if you don't try, you'll never know.

 

[Darkiekurdo]

How can I get better at solving Olympiad-type problems?

 

[mathwonk]

practice. i presume there exist books of problems.i have never done this in college, but in high school we practiced for contests by working lots of them and did very well.

 

[G01]

How can I get better at solving Olympiad-type problems?

There is nothing better than practice. So as many problems as you can. As you keep doing them you'll develop personal patterns and algorithms to solve problems. You won't be tied down to formula sheets as much, etc. Practice, practice, practice!

 

[jhicks]

How can I get better at solving Olympiad-type problems?

My library of math books is pretty limited due to a few years of uh... languishing, but one book I enjoyed to the fullest is called "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics". All of the problems are very accessible to someone who has done up through precalculus, but they range from easy to extremely difficult and cover just about any type of elementary problem you could think of, from divisibility to word problems to limits. I worked on the 320 problems for maybe a year total out of the 5 years I've had the book, and I have solved maybe 100 of them.

Edit: It's 12 bucks on Amazon. https://www.amazon.com/dp/0486277097/?tag=pfamazon01-20

 

[PhysicalAnomaly]

You've provided plenty of good book titles. However, you haven't said what has what prerequisite. Could someone please list good popular books for undergrads (like Spivak, Apostol and Kreyzig) in sequential order? I would like to cover linear and abstract algebra, topology and real analysis on my own (the uni course is a bit slow; my foundation is uk A levels and the australian system is a bit sluggish) but don't know which order of books to use. Also I am taking a double degree with chem eng as one side and would like to know some maths that'll be useful for engineering (I'm guessing something like fourier, greens, pde's). Any good books for this purpose? Will Kreyzig's book cover enough for me to skip things like Spivak and Apostol? What do you think of Strang's linear algebra?

I'm thinking it would be quite safe to follow Cambridge's maths syllabus (it's on their maths department's site). Would it be overkill? What they cover in 2 years is probably what mine covers in 3!

 

[eastside00_99]

Could someone please list good popular books for undergrads (like Spivak, Apostol and Kreyzig) in sequential order? I would like to cover linear and abstract algebra, topology and real analysis on my own

You could probably learn any of those topics right now. I don't see one as a prereq for another depending on what you mean by topology. If you mean point-set topology, real analysis will give you intuition but is not really necessarily if you have a propensity for abstract thinking. If you mean algebraic topology then you will need abstract algebra, point-set topology, and possibly linear algebra for intuition in homology theory. I don't know of any "standard" textbooks for these things save Munkres "topology" for point-set topology. Of course, you have Lang's "algebra" and other similar books for all of those subjects but they are not necessarily the best books to read to first learn the subjects but rather after a second go around at the subjects. The most economical list that I can come up with is the following:

linear & abstract algebra ------ Artin's "Algebra"
point-set topology ------ Munkres' "topology"
Real Analysis ------ Marsden's "Elementary Classical Analysis"

That list is pretty damn difficult enough. In the end, you will have to decide what books are most accessible to you and which challenges you enough.

 

[PhysicalAnomaly]

Thanks. I seem to be in luck. My university's library has Artin and Munkres.

More questions:

I've gathered from trawling the forum that Spivak would be easier than Apostol. That true?
Kreyzig or Stroud? (And would I need strang after that?)
If Stroud, what's the difference between engineering maths, further engineering maths and advanced engineering maths?
I'm partway through strang's linear algebra. Which book do you recommend after this for linear algebra?

What is point-set topology and would it be required to something else?

Is Marsden the standard complete text for real analysis? Will it be redundant if I read Spivak's calculus? Does Spivak's book on manifolds follow from his Calculus or is there overlap?

Thanks for your guidance.

PS If I study the books that go in depth for maths, does that mean I can do without engineering maths books?

 

[Gib Z]

PS If I study the books that go in depth for maths, does that mean I can do without engineering maths books?

Not always true. Just because you know the theory does not necessarily indicate you can also apply it effectively. Chances are your math book won't go into much detail about applications to engineering, so for that course you may need a good separate book that is filled with those types of problems.

 

[quasar987]

Is Marsden the standard complete text for real analysis? Will it be redundant if I read Spivak's calculus? Does Spivak's book on manifolds follow from his Calculus or is there overlap?

I wouldn't call Marsden standard, but it is in my opinion, the best analysis book for beginners. However, it is deficients in some ways (rudimentary treatment of power series) and overdoes it in others (it develops the Riemann theory of multiple integrals! this is useless)

Spivak's calculus is not a real analysis text. It's a calculus text, whatever that means. In either case, 'Calculus on Manifold' does not follow and even if it did, it is a bad introduction to calculus in higher dimensions because it is extremely dense with no examples.

 

[eastside00_99]

I've gathered from trawling the forum that Spivak would be easier than Apostol. That true?
I thought it was the other way around; I have read neither.

Kreyzig or Stroud? (And would I need strang after that?)
Kreyzing for what? "introductory functional analysis"?

I'm partway through strang's linear algebra. Which book do you recommend after this for linear algebra?

Any linear algebra book that discusses canonical forms.

What is point-set topology and would it be required to something else?

Point-set topology is a generalization of the concepts of space of R^n that are not associated with distance. It would be required for a myriad of things: algebraic topology, manifold theory, several complex variables, algebraic geometry, etc, etc



Anyway, take it slow. I mean unless you are just some freak of nature or study 18 hours a day, you are not going to be able to master these four subjects in a month. It wouldn't really be possible within a semester while you are also taking other classes. Next semester I would just recommend you taking a more advanced math course and working hard in it.

 

[jhicks]

Kreyzig or Stroud? (And would I need strang after that?

My personal opinion of Kreyzig was that it was an information overload (like a lot of engineering classes) with minimal theory discussion. To me it's a reference book only, but it does cover absolutely everything you're likely to see in engineering.

 

[PhysicalAnomaly]

I'm not that crazy. I intend to take a year or two to finish. But I have limited time so I have to make sure that I'm using the best books since I won't have time to go back and use an alternative. Australian universities don't usually have very advanced maths classes. Not until 3rd year anyway. Cambridge students cover almost our entire 3 year syllabus in 2 years and in more depth too!

 

[mathwonk]

you should read both spivak and apostol to see which you like better. there are no applications and no physics in spivak at all, and this is a limitation for many people.

one nice thing about apostol, is that although differentiation makes evaluating integrals easier, integral calculus is much older than differential calculus and does not depend on it at all, up to a certain point. it seems it was essentially invented by eudoxus and especially archimedes.

this is obscured by most books which do differential calc first. apostol is unique in thoroughly explaining integral calc first, which is historically sound.

 

[Hydrargyrum]

I am interested in starting this discussion in imitation of Zappers fine forum on becoming a physicist, although i have no such clean cut advice to offer on becoming a mathematician. All I can say is I am one.

My path here was that I love the topic, and never found another as compelling or fascinating. There are basically 3 branches of math, or maybe 4, algebra, topology, and analysis, or also maybe geometry and complex analysis.

There are several excellent books available in these areas: Courant, Apostol, Spivak, Kitchen, Rudin, and Dieudonne' for calculus/analysis; Shifrin, Hoffman/Kunze, Artin, Dummit/Foote, Jacobson, Zariski/Samuel for algebra/commutative algebra/linear algebra; and perhaps Kelley, Munkres, Wallace, Vick, Milnor, Bott/Tu, Guillemin/Pollack, Spanier on topology; Lang, Ahlfors, Hille, Cartan, Conway for complex analysis; and Joe Harris, Shafarevich, and Hirzebruch, for [algebraic] geometry and complex manifolds.

Also anything by V.I. Arnol'd.

But just reading these books will not make you a mathematician, [and I have not read them all].

The key thing to me is to want to understand and to do mathematics. When you have this goal, you should try to begin to solve as many problems as possible in all your books and courses, but also to find and make up new problems yourself. Then try to understand how proofs are made, what ideas are used over and over, and try to see how these ideas can be used further to solve new problems that you find yourself.

Math is about problems, problem finding and problem solving. Theory making is motivated by the desire to solve problems, and the two go hand in hand.

The best training is to read the greatest mathematicians you can read. Gauss is not hard to read, so far as I have gotten, and Euclid too is enlightening. Serre is very clear, Milnor too, and Bott is enjoyable. learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

If these and other top mathematicians are unreadable now, then go about reading standard books until you have learned enough to go back and try again to see what the originators were saying. At that point their insights will clarify what you have learned and simplify it to an amazing degree.


Your reactions? more later. By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me. Please correct me on this point, since nothing this general is ever true.

Remark: Arnol'd, who is a MUCH better mathematcian than me, says math is "a branch of physics, that branch where experiments are cheap." At this late date in my career I am trying to learn from him, and have begun pursuing this hint. I have greatly enjoyed teaching differential equations this year in particular, and have found that the silly structure theorems I learned in linear algebra, have as their real use an application to solving linear systems of ode's.

I intend to revise my linear algebra notes now to point this out.

Where can I find writing from Gauss, Newton, Euler, etc?

 

[quasar987]

Here's a book by Euler (translated of course): http://www.amazon.com/dp/0387985344/?tag=pfamazon01-20

Newton's PRINCIPIA MATHEMATICA is also easily found in english, but not easily read.

For people of the era ofGauss, Riemann, Weierstrass, Cauchy, etc. and onward, if most universities are like mine, the library should have books going by names like "Collected works of [...]"

 

[Hydrargyrum]

i was thinking about books that i could download . . . for free