Other Should I Become a Mathematician?

[johnnyp]

I love this thread - very insightful. And I definitely like it unstuck; stickies are labeled in such a way that it's bound to get unnoticed some time or the other--just keep it the way it is!

Mathwonk -- I have a question, if you don't mind. I'm currently taking real analysis. I'm understanding everything so far but the homeworks have always been [for the most part] difficult, and that's not what my conception of math was -- I was never stuck in Calculus and Differential Equations, unlike now. Is this a danger sign? Should I not pursue math as a major? It's just that in analysis, I seem to need significantly more time to solve problems (mostly proofs) than I would have in Calc and ODE's.

 

[ultranet]

this is well guided posts. I am goin to check out some books...
thanks

 

[JasonRox]

I love this thread - very insightful. And I definitely like it unstuck; stickies are labeled in such a way that it's bound to get unnoticed some time or the other--just keep it the way it is!

Mathwonk -- I have a question, if you don't mind. I'm currently taking real analysis. I'm understanding everything so far but the homeworks have always been [for the most part] difficult, and that's not what my conception of math was -- I was never stuck in Calculus and Differential Equations, unlike now. Is this a danger sign? Should I not pursue math as a major? It's just that in analysis, I seem to need significantly more time to solve problems (mostly proofs) than I would have in Calc and ODE's.

If this is your first time doing proofs, I wouldn't worry about it. That seems to be normal for first timers.

Just be sure to strictly justify each step during a proof. Read lots of proofs too. And justify each proof you read. Don't just read along.

I come by proofs I don't like myself sometimes. Feeling as though there could be another way, then I try it out myself. Sometimes I get a new and sometimes I don't. If I don't, I then learn to just enjoy that proof a little more.

I wouldn't worry about it for now. Just keep practicing. If you're determined, good things are bound to happen.

 

[mathwonk]

good point johnnyp. this way (unstuck) when it dies it will fade away gracefully.
no, analysis is just harder than those other subjects, we all think so.
and it does not prevent one from being a mathematician to find analysis hard.
there are three kinds of thinking in math, algebra, analysis, and geometry.
i.e. finitistic, infinite (limiting), and visual.

few people are good at all of them. i am very visual-geometric. i majored in algebraic geometry because it was halfway in between algebra (hard for me) and geometry (easier for me.

no slight intended, but topology to me seemed "too easy". i found the challenge of seeing the geometry behind the algebra stimulating.

analysis on the other hand was painfully hard. I did ok in complex analysis of several variables while i tried that topic, but my head hurt when I was thinking about it.

It felt pleasant the whole time I pondered geometry or topology. i wanted to enjoy myself, not suffer. You cannot get a PhD taking several years, if you are suffering the whole time. It is hard enough in the best of circumstances.

that said, one should not avoid the subject one finds hard, as it too will be useful learn as much of it as you can, and try to change your attitude to it. work with someone who likes it and try to see why they think it is beautiful.

 

[mathwonk]

i like arnol'd's definition of math: "that branch of physics where experiments are cheap."

 

[courtrigrad]

mathwonk, does one pick up proof writing techniques when they learn real analysis? I know some institutions offer classes that teach students how to write proofs. Would it be better to learn the technique by yourself?

 

[mathwonk]

learn it as soon as possible, from any source that helps. learn it in as many ways as one can. better not to wait until reals as then it is very hard and coupled with very hard topics too.

i started learning it in high school, from the book principles of mathematics, by allendoerfer and oakley. i also took euclidean geometry, whose absence is one of the main reasons proofs are no longer understood by today's students.

i.e. removing geometry proofs and inserting AP calculus from high school I think is a prime culprit for our current demise as a math nation.

 

[courtrigrad]

ok, yeah I will use the textbook by Solow then. https://www.amazon.com/How-Read-Pro...bbs_sr_1/ 102-6215276-8882554?ie=UTF8&s=booksAlso what programming languages do you think one should learn? Should he learn Java? Because I think knowing a programming language will be extremely helpful (or am I wrong)?

 

[Ki Man]

http://www.math.uga.edu/~roy/

nice picture :]
were you talking about the links to all of the algebra notes and everything?

 

[mathwonk]

i forget what i recommended. the linear algebra notes link is a very condensed review of linear algebra for a strong student who either wants to work out all the theory for himself, and is already good at proofs, or has learned it before and wants to review for a PhD prelim.

the 843-4-5 notes are for a detailed first year grad algebra course, for any grad student or upper level good undergrad student, or bright motivated high school student who knows whaT a matrix is. actually even that is reviewed in the 845 notes.

the rrt notes are for advanced students who know some complex analysis.

the research papers are for people interested in prym varieties and other abelian varieties, and the riemann singularity theorem.

 

[mathwonk]

oh i remember. i was referring you to some web based notes by other people on linear algebra. ill look them up. or just search on google for linear algebra notes, books.

 

[mathwonk]

oops, here's another elem one. http://www.numbertheory.org/book/

 

[mathwonk]

http://www.etsu.edu/math/gardner/2010/notes.htm

 

[mathwonk]

heres a more advanced one. but ilike the one by Ruslan Sharipov, who posts here, his second level linear algebra notes are truly excellent.

 

[Ki Man]

wow thanks guys

 

[dontbesilly]

learn it as soon as possibvle, from any source that helps. learn it in asmany ways as one can. better not to wait until reals as then it si very hard and coupled with very hard topics too.


i started elarning it ni high school, from the book principels of mathematics, by allendoerfer and oakley. i also took euclidean geomnetry, whose absence is one of the main reasons proofs are no longer understood by todays students.

i.e. removing geometry proofs and inserting AP calculua is a prime culprit for our current demise as a math nation.

I've heard this said before...that proofs have been removed from geometry classes. That was not true for me. I took it back in 2003/2004 and we proved everything we did, all the time. Most of our work infact involved proofs, or constructions if I recall correctly. Of course, I live in a pretty good school district, and was on the advanced track. Maybe it's different elsewhere, or for those who take it later than I did.

But yeah, understanding those proofs was pretty key to my mathematical developement.

 

[mathwonk]

glad to hear it. of course now that we are grown up and geting our own info and motivating ourselves, we can fill any gaps that were left by our schooling. so thanks for the input and the questions.

my personal goal for the next few weeks is to learn either Grothendieck's version of Galois theory (etale maps and etale cohomology), or learn about Galois theory of ring extensions.

 

[Ki Man]

That was not true for me. I took it back in 2003/2004 and we proved everything we did, all the time. Most of our work infact involved proofs, or constructions if I recall correctly.

I believe he was referring to how they take the Geometry-style proof and fill it in with Calcula, not them taking out the proofs completely.

Geometry proofs and Calculus proofs are different, right?

Hmm... Last week I got my hands on a Calculus textbook in pretty good condition for $1 at a library booksale along with ancient sheet music of 'The Messiah' Copyrighted 1918. its amazing what you can find if you look in certain places :P.

Calculus: early transcendentals (3rd edition)
James Stewart.

I can't understand any of it right now, but maybe later i'll undertsand it along with E=hv :]

 

[johnnyp]

Thanks Jason and Mathwonk. My real analysis class is using a text called Principles of Mathematical Analysis, but I'm not liking it very much. The professor's classnotes, on the other hand, are amazingly clear and motivated, and the proofs are a bit longer than Principle's, but more instructive and less terse.

Oh and real analysis here is the transition to proofs class - is that a bad thing?

 

[quasar987]

Sharipov's book:

http://uk.arxiv.org/abs/math.HO/0405323/

Sharipov's bookS:

http://www.geocities.com/r-sharipov/e4-b.htm

 

[mathwonk]

who is your professor johnnyp/ and where are you studying? i like to know about good profs and good schools.

 

[mathwonk]

uh oh. be careful what you wish for. we are stuck!

 

[mathwonk]

a nice thing in sharipov's is inclusion of a first chapter with basic set theoretic terminology about maps, images inverse images, and so on. this is crucial in all advanced course work.

then his treatment of jordan forms begins at the essential point, namely explanation of structure of nilpotent operators.

i.e. jordan form tells you how to understand an operator based on its minimal polynomial. since the simplest polynomial is X^n, this is the basic case.

a polynomial T that satisfies X^n, i.e. such that T^n = 0, is called nilpotent, because some power of it is zero, or nil. [zero-power = nil-potent.] you get it.

 

[mathwonk]

by the way, if my recent plaint caused us to be stuck, i apologize to those of you who did not want that.

it makes it easier to find, as a sticky, but harder to gauge the level of current interest. so either way is ok with me.

whoever put it up here presumably did so either to prevent our feeling ignored, or to make it more useful.

either way it was thoughtful, and i thank you.

 

[mathwonk]

to get into a phd program, you have to pass 2 hurdles: 1) admission to the university grad school and 2) admission to the departmental program.

math departments are not so bureaucratic as university grad schools are. the latter will require degrees and certifications for admissions.

moreover, those requirements are there for a reason, since people without them are almost always lacking some quality that would help insure success.

however if you are that rare bird, a truly exceptional mathematics talent, who knows all they need to, and can do the work, then you might get in without a degree.

let me say this is highly unlikely, and unrecommended. why would anyone want to avoid the college experience, which many people recall as the best time of their lives?

and why would anyone think it more likely to succeed in grad school without being instructed for four years by experts?

i recommend you give yourself every opportunity. take all the usual courses, convince people you have the ability to do a graduate degree.

besides, one thing you would be missing without this experience is the ability to convince someone to write a recommendation letter.

here is one possible scenario: show up at a grad school, on your own nickel, and sign up for a course that advanced undergrads are taking, people who are thought of as grad school material, and outperform them.

or show up at a grad school and take and pass their phd prelims.

 

[radou]

I'm currently on the 3rd year of civil engineering and am interested in math. Well, 'to be interested in math' is a slippery construction, since interest alone doesn't imply anything. Anyway, we had four math courses on the first two years which contained a standard calculus, basic linear algebra and numerical methods overview, as a probability and statistics course. But, since a faculty of engineering in general isn't a place where you'll learn math on a bit higher and more precise level, I decided to take additional courses on the faculty of mathematics, since I don't believe (in my case only, though) in the possibility of a good-quality-self-study. I'm currently attending a linear algebra course, which I find highly interesting and enlighting. If I'll have the time in the nearer future, I plan to attend more math courses and build a small 'additional database' in my student record. So, actually, there are some 'ways around'. But, they're still just 'ways around'.

 

[JasonRox]

to get into a phd program, you have to pass 2 hurdles: 1) admission to the university grad school and 2) admission to the departmental program.

math departments are not at so bureaucratic, but university grad schools are. they will require various degrees and certifications for admissions.

moreover, i have learned by experience that those requirements are there for a reason, i.e. people without them are almost always lacking some quality that would help insure success. [i once went to bat for someone without paper quals, who afterwards was indeed not well qualified for our program.]

however if you are that rare bird, a truly exceptional mathematics talent, who knows all they need to, and can do the work, then you might get in without a degree.

You can get into graduate school without a degree?

 

[mathwonk]

the research experience is quite different from the passive learning experience, and after passing through it i had new respect for my colleagues who had done it before me.

moreover, it is not entirely about talent, but persistence and stamina play an equal role. by skipping the undergrad degree one misses the chance to develop this stamina.

just not giving up, is as essential as being smart, as many average intellects have obtained phds (tataa!), but no one who gives up ever does, no matter how brilliant.

 

[mathwonk]

jasonrox: i tried to make it clear that a math dept may be interested in a very talented person, degree or not, but a grad school will not want to accept that person, and with good reason. you have seized on one phrase in my long statement and taken it out of context. read it all. i am not advising or encouraging anyone to seek entrance to grad school without a degree.

no it is unlikely you can get in and unwise to try.

i'll give you one successful example, Barry Mazur apparently has no undergrad degree. he's the guy Andrew Wiles sent his manuscript on Fermat's last theorem to check it. and presumably he was unsure about it, when it was indeed wrong.

but most of the rest of us are not like Barry. and besides Barry had all the academic requirements and time spent in school, he just declined to attend required ROTC. the school (MIT) afterwards seems to have eliminated the requirement, possibly as a result of the subsequent embarrasment at having denied Barry Mazur a degree.

 

[balletomane]

How can I figure out if I am cut out to be a math major? I really love my math classes, but there is always a sense of not being good enough at it. (I seem to be very dense compared to my classmates.)
Er, I'm not doing a good job of articulating what I mean. I guess, put another way, what qualities should a person pursuing a career or degree in mathematics possess? (I do realize that mathematicians/math majors are very diverse, but are there common qualities?)