Other Should I Become a Mathematician?

[mathwonk]

just do what you enjoy. self study is very useful, but you will find when you get a teacher that you will gain more insight on the same things you have self studied. if you like competitions then do them. I did. it made me feel i was talented and gave me motivation to do more math.

but take advantage of the free time you have now to read those great books. if you could read apostol and do the probolems with little difficulty, then you are very strong. If you are having fun, keep going and enjoy. And keep in touch.

I suggest next ted shifrin's linear algebra book, actually the one by shifrin and adams. but it costs money, if you want a free one, try mine, off my web page. if you are really strong and motivated to work hard on your own, you may be able to read my 15 page linear algebra book. but if it is too hard don't feel bad, so far no one has said they could read it. but maybe you would be first!

for analysis at a a high level. try to get hold of a copy of Dieudonne's foundations of modern analysis.

heres a nice looking copy for $20. buy it by all means! you will never see a better bargain in your life.

https://www.amazon.com/dp/B0006AWGOM/?tag=pfamazon01-20

 

[MrJB]

Hi. I'd like to be a mathematician, but I'm not sure how to get there.
I graduated in '06 with a BS in Physics, GPA ~2.93. Only realized after graduating that I loved math more than physics. Not sure how to convince a grad school admissions committee that I'm both willing to work hard, and able to do the math. I took some math classes in undergrad, though not enough to be a complete background prep for grad school.
So, I'm wondering what my best options are?
Any advice would be appreciated.

 

[strings235]

great. could you give me the link to your book? and also, would you recommend michael artin's book on linear and abstract algebra?

 

[quasar987]

What is so great about Dieudonne's book?

And what does it cover?

 

[Emmanuel114]

http://www.archive.org/search.php?query=dieudonne

Online version?

 

[quasar987]

Great. The DJVU file doesn't work for me but at least I got to see the table of content.

 

[mathwonk]

please forgive brief answers as work has begun again. feel free to ask for more details. my links are on my public profile, or search math dept uga,

Dieudonne was a brilliant member of the famous bourbaki math group and wrote one of the first books in the early 1960's which went well beyond the traditional books on advanced calculus to rpesent the modern point of view on analysis.

he also wrote up the works of grothendieck, arguably the most famous mathematician of the last 50 years.

as to how to become a mathematician, I myself after a checkered career, just read the books and taught the courses and went and sat for the prelim exams at the university of washington, apparently outperformed the students there, and got an offer from them.

 

[ekrim]

hey mathematicians, i was wondering if its too late for me to live the math dream. I'm a junior math/mech eng major, and I've taken cal1,2,3, linear alg, diff eq, and complex analysis. I will be taking partial diff eq this fall. However, I have no training in proofs and had to use my own crackpot proofs on complex analysis tests. I also never paid too much attention and just studied the night before to con an A out of the class. Naturally, I've forgotten everything even though i have a 4.0. What should I do to make myself grad school worthy despite my ineptitude?

also, I say "math" rather than "maths." Could this be holding me back?

thanks

 

[leakin99]

Hey guys, first time posting in this thread. I am an undergrad math student who is mostly interested in applied math/medical physics. Anyway…I found this great website with a free ebook called “Intro to Methods of Applied Mathematics”. When I opened it, my eyes just lit up. It has everything! Well, at least for the undergraduate level. It’s 2000+ pages and about 9mb so it takes a bit to load up. I have not gone through the whole thing, but from what I’ve seen so far, it seems example based. Just thought I share this since I found it so amazing lol. Sorry if it has already been posted…

http://www.cacr.caltech.edu/~sean/applied_math.pdf

 

[mathwonk]

mike artin's book algebra is probably the best out there for those who are ready for it. i would not have written my book if i had been better acquainted with it, i would have just used his book in my class.

 

[mathwonk]

maybe artin's algebra would bring you up to speed in proofs.

 

[Darkiekurdo]

I just picked up Spivak's 'Calculus', third edition. :!) I'm so happy.

 

[mathwonk]

i learned one variable calc from spivak's calculus, and i learned several variable calc from his calculus on manifolds.

 

[symbolipoint]

i learned one variable calc from spivak's calculus, and i learned several variable calc from his calculus on manifolds.

What kind of book is that "manifolds" book of Spivak? Does it contain a section on several-variables Calculus, or IS IT a book on several-variables Calculus? Further, who applies the concept of manifolds (I really have no idea about manifolds)?

 

[quasar987]

mike artins book algebra is probably the best out there for those who are ready for it. i would not have written my book if i had been better acquainted with it, i would have just used his book in my class.

How is Dummit & Foote?

 

[MrJB]

as to how to become a mathematician, I myself after a checkered career, just read the books and taught the courses and went and sat for the rpelim exams at the university of washingtona nd outperformed the studebnts there, and got an offer from them.

That's rather cool. I guess it comes down to how well you know the math.

 

[ircdan]

mike artins book algebra is probably the best out there for those who are ready for it. i would not have written my book if i had been better acquainted with it, i would have just used his book in my class.

What do you like about Artin? I will be using it for a course and so I'm just curious.

Thanks.

 

[Darkiekurdo]

Wow, Spivak is really rigorous!

 

[mathwonk]

manifolds are like n- space except they can be smoothly curved. this means you lose the precise value of a derivative, but you can still tell if a derivative is zero or not, i.e. you can tell if two curves are tangent even in a curved space.

thus calc on manifolds is just a generalization of several variables calc. so spivak begins with a chapter on sev vbls diff calc, then one on sev vbls integ calc, then goes into the algebraic machinery for discussing differentials and integrals abstractly, (chains and differential forms), then defines manifolds and generalizes the sev vbls calc to the setting of manifolds.

everyone who studies spaces that can curve uses manifolds, like space scientists, relativity theorists...artin is a world famous algebraist and algebraic geometer, so his grasp of the subject goes deeper than that of people like me. This means the material is so clear to him he can make it seem easy to the reader as well. as a master he is not dependent on copying the same proofs from other books, he often makes his own simpler ones.

he also wrote this book for sophomores at MIT, so although the entry level is high in terms of ability it is low in terms of sophistication. This is the best possible situation, when a deep master of a subject undertakes to explain it to beginners. the result is often that the rest of us can understand it too.

this is a very successful book. the only thing it lacks for some of us is that it does not treat every graduate topic, being an undergraduate book, so he does not give us the benefit of his explanation of tensors.

 

[mathwonk]

dummitt and foote is a fine book, written in more detail than artin's book, for average students, and is very popular. i prefer artin myself, although DF have succeeded in stating some facts very very clearly. e.g. i like the section on products and semi direct products of groups.

they say very clearly how to recognize a group G which is a product of two subgroups H,K, namely the subgroups, should satisfy HK = G, HmeetK = {e}, and both H,K are normal.

analogously, if the same holds except only K is normal then G is still a semidirect product of H,K.

this is very clear and very useful.

there is also much more coverage of more topics in DF than in Artin. But in my opinion, the discussion in DF just does not have the ring of the master, as Artin's does. Still they are obviously expert algebraists, and researchers, and they know what they are doing, and even their problem sets have a huge amount of information. I havea copy of DF, I just do not read it much. (Later I read it more and earned a lot.)

I myself disagree with their way of presenting some material as I have said elsewhere here. E.g. they give proof of the decomp. thm for fg modules over pids that to me is useless in understanding or using the theorem. then later when they need to use the theorem themselves, they refer you to a different proof in an exercise. i do not like this way of doing things.

Why waste your time reading a proof that will not be used, and omit the approach that wil be used? it's just not my cup of tea. but it is a good book with a lot of clearly explained and userful material. in particular it is easy to read.

 

[Darkiekurdo]

I just finished the first chapter of Spivak's 'Calculus'. And although very rigorous, I like his style. It is unlike everything I have read before. In all other books i have read they took the properties of numbers for granted. Spivak does niet. He shows, for example, why 'a times 0' equals 0.

 

[mathwonk]

yes, he will also make the key "least upper bound" axiom for reals very very clear.

but be sure to do the exercises, or his smooth explanations may fool you into thinking you understand more deeply than you do.

 

[Darkiekurdo]

yes, he will also make the key "least upper bound" axiom for reals very very clear.

What version did you read? And what chapter is that?

but be sure to do the exercises, or his smooth explanations may fool you into thinking you understand more deeply than you do.

I will do that. :)

 

[mathwonk]

well somewhere there is a chapter called "three hard theorems". the lub axiom is used there. i read the first edition of course in 1969 or so.

 

[Darkiekurdo]

Wow.

 

[mathwonk]

chapter 7 is three hard theorems, and chapter 8 is least upper bounds, where he actually proves those 3 theorems.

 

[Darkiekurdo]

I'm afraid I can't finish the whole book. It took me a lot of thinking to complete the first chapter. :(

 

[uiulic]

What a thread is this! I don't even have patience to read all through.I have been wanting to ask a question, and please allow it to be asked here.

For me, saying I want to be a mathematician is like saying I want to be King.

Looking back at how much time we (including you mathematicians) spent in mathematics,I then realized how difficult mathematics really is. At primary school, I were forced to recite 1*1=1 to 9*9=81 and from that time the language of maths was embedded in our brains and later on we are using what we remember in the way/style it is. I just feel lucky in passing those exams related to mathematics (or involved), which is a big release.

But now, there is such a situation for me (maybe also for many others). Whenever I read a bit more advanced mathematics books (say topology or advanced algebra, I never really started it), I find I am so disappointed with myself. In order to grasp an easy concept, I need recite it and then do a lot of exercises, which is what I did in primary school. I just fear this kind of experience, and I can't afford it because of my age.

How did you (those mathematicians and those who want to be) overcome such "fear"? Or share your experiences please if possible?

 

[mathwonk]

well it is just as hard for us to learn new things. the will to try to overcome our fear and struggle to learn is something we always have to work at.

it helps to realize it is natural, for everyone to feel uncomfortable in new situations, and everyone has to work to learn, and also remember the pleasure of succeeding at learning, and remember the added benefits of doing so.

just try one new thing today, and i will try one too! for me it will be something scary like trying to put my homework system online for the first time in my life. or even learn some deformation theory, which for me is challenging but more fun.

 

[mathwonk]

my webbooks are on my page whose address is visible in my public profile. just click on my name and the public profile link will appear.
pivoxa15, i now recommnend Hartshorne's book geometry: euclid and beyond as the best book on euclidean geometry, in connection with euclid's own work, available free online.

i am now using Hartshorne together with a book called geometry for the classroom by Clemens and Clemens. these are both books written by world renowned algebraic geometers for classes of young students at the undergrad or even middle school level, in Clemens' case. i love these books.

there is nothing like an elementary explanation from a high powered scientist.

 

[mathwonk]

any reaction to my new 100 page algebra book?

 

[quasar987]

I didn't know you had a new book, I'll go check it out immediately.

In the meantime, I was in the middle of applying to a university and it asks me if I want to apply for a

MA with thesis
MA w/o thesis
Ms.c with thesis
Ms.c w/o thesis

Would you be able to explain the difference btw the four, and why would one want to chose one instead of the other? Thx!

 

[mathwonk]

not fully, the best people to ask are the grad dvisors at the uni. but with thesis is for people who want to spend some time learning one topic rather well, and then practice writing by writing it up carefully. this is a good experience and will result in really knowing that one topic.

without thesis is for people who prefer to just take courses. this results in broader but less deep knowledge of more topics, and may take less time.

the 100 page algebra book has been up there for several weeks now, so may not be new to you, but is pretty new.

 

[mathwonk]

darkiekurdo, here's an idea, you do not need to read a math book linearly from front to back, in fact it is usualy a bad idea pedagogically.

so just turn right now to chapter 7 and read about the three big theorems.

of course you will realize you do need some stuff from earlier chapters, (namely the definition of continuity), but that will motivate you to go back and get it.

 

[Darkiekurdo]

I'll do that right now. Can I ask some questions while reading that chapter? Or you could refer me to parts where he explains that what I don't understand?

 

[mathwonk]

of course, good for you!.

you might think of it as a basketball game, if you get stopped driving down the side, you flip it out to the middle and go that way, if that doesn't work, you cross court. you cannot afford to let one blocked lane stop you from getting to the goal.

 

[ice109]

mathwonk is like the one man math subforum at pf

 

[Darkiekurdo]

I think I'll start at the previous chapter about continuity.

 

[Darkiekurdo]

I think my understanding of limits is not sufficient. So I'll go to the previous chapter about limits.

 

[Darkiekurdo]

I don't understand page 91; we have a function

x\ \sin\ \frac{1}{x}

and we want x to be within

\frac{1}{10}

of x, so,

-\frac{1}{10} < x\ \sin\ \frac{1}{x} < \frac{1}{10}

I understand it until that, but then he writes:

"|x\ \sin\ \frac{1}{x}| < \frac{1}{10}"

Now my question is probably very stupid, but why does he use the absolute value?

 

[mathwonk]

ok. but you have an intuitive idea of continuity, and the three big theorems just say:

1) if f is continuous on an interval, then the values of f form an interval also.

2) if f is continuous on a closed bounded interval, then the values also form a closed bounded interval.

well ok i got them to two theorems.

so then the question becomes, how do i define continuity precisely so as to make these intuitive theorems actually true?

the answer is to say that small changes in the inputs produce only small changes in the outputs. but this has to be made very precise, with letters for the degree of change in input (delta) and output (epsilon).

 

[Darkiekurdo]

But why do they use the absolute value?

 

[quasar987]

Try to answer that question for yourself by proving that |x|<a iff -a<x<a.

So writing |x|<a is just another more compact way of expressing the fact that x is btw -a and +a.

 

[Darkiekurdo]

Aha. Thank you! I understand.

 

[mathwonk]

if x is between -a and a, then since sin is always smaller than 1 in absolute value, then x sin(anything) is also between -a and a.

 

[Darkiekurdo]

Is it strange that after I read a chapter of Spivak's book I have no idea how to do the problems? And after I read the chapter again I still can't do the problems.

 

[symbolipoint]

Is it strange that after I read a chapter of Spivak's book I have no idea how to do the problems? And after I read the chapter again I still can't do the problems.

Which of Spivak's books? Some or many of us have not seen any of his books (although others of us certainly have - not me, though); we may like to know which is the one you are finding trouble understanding, in case we might like to obtain a copy to use for study. Is it his Calculus book which you find difficult?

All enthusiasts of Spivak's books, please give your discussions about this.

Also, something interesting - do a web search and you can find a wikipedia article on Michael Spivak.

 

[quasar987]

It is a bit strange, but not totally surprising since it's your first time dealing with higher math.

It could be that you are reading the book like a novel. You have to read it as slow as it takes for the material to sink in.

It could be that you are just not used to the level of the problems; as soon as a problem begins with "show that" or involves something more original than a direct application of one theorem in the book, you are lost. I think the cure to this for me was that there were hints at the end of the book. So after thinking hard for a while about each problem and getting nowhere, I would look at the hint and try again. The key is really to genuinely try hard to solve the problem on your own though. Because I believe the best time to learn is when you are convinced that you have explored all the possible ways to approach a problem and they all failed. In this situation, I found that after I look at the solution, I sticks.

As you progress, you will find that there is in fact a finite number of methods/tricks to solving problems that appear again and again. I found it helpful to make a list out of these tricks and systematically try them out on every problem.

Here's my list.

-multiply by the conjugate
-add 0
-multiply by 1
-exploit the properties of exp and log
-can I use an identity? (Gauss' sum, Bernoulli inequality, etc)
-factorisation
-triangle inequality
-put fractions on the same denominator
-decompose in a sum of partial fractions
-suppose W.L.O.G. (without loss of generality)
-change of variable
-proof by contradiction
-prove something that is equivalent but easier
-decompose the problem in a sum of smaller problems
-complete the square
-by induction

These will begin to make sense to you when you encounter them again and again.

 

[d_leet]

Which of Spivak's books? Some or many of us have not seen any of his books (although others of us certainly have - not me, though); we may like to know which is the one you are finding trouble understanding, in case we might like to obtain a copy to use for study. Is it his Calculus book which you find difficult?

All enthusiasts of Spivak's books, please give your discussions about this.

Also, something interesting - do a web search and you can find a wikipedia article on Michael Spivak.

I believe he was using Spivak's Calculus book. About the problems, they are not all easy and computational problems, many require proofs using theorems and techniques learned in the chapter. Some problems are harder than others, but I believe this is the case with practically any textbook. Most of Spivak's problems require more thought than those in books like Stewart's where the problems are almost entirely computations.

I do not find Spivak's problems overly difficult, but that is not to say that I understand how to do all of them (right now I am having difficulty with a problem in the continuity chapter). I would say not to take one person's issues the problems to be necessarily true for everyone, and that you should judge for yourself whether or not you like his book, just be warned that you should not expect Spivak's Calculus book to be like other such books, it is more rigorous and if you have not had experience with mathematical proofs before than you may have some difficulties doing the problems.

Forgive me if that made no sense at all, I'm sure I repeated myself about five times or so, hopefully that helped you out some.

 

[Darkiekurdo]

It is a bit strange, but not totally surprising since it's your first time dealing with higher math.

It could be that you are reading the book like a novel. You have to read it as slow as it takes for the material to sink in.

It could be that you are just not used to the level of the problems; as soon as a problem begins with "show that" or involves something more original than a direct application of one theorem in the book, you are lost. I think the cure to this for me was that there were hints at the end of the book. So after thinking hard for a while about each problem and getting nowhere, I would look at the hint and try again. The key is really to genuinely try hard to solve the problem on your own though. Because I believe the best time to learn is when you are convinced that you have explored all the possible ways to approach a problem and they all failed. In this situation, I found that after I look at the solution, I sticks.

As you progress, you will find that there is in fact a finite number of methods/tricks to solving problems that appear again and again. I found it helpful to make a list out of these tricks and systematically try them out on every problem.

Here's my list.

-multiply by the conjugate
-add 0
-multiply by 1
-exploit the properties of exp and log
-can I use an identity? (Gauss' sum, Bernoulli inequality, etc)
-factorisation
-triangle inequality
-put fractions on the same denominator
-decompose in a sum of partial fractions
-suppose W.L.O.G. (without loss of generality)
-change of variable
-proof by contradiction
-prove something that is equivalent but easier
-decompose the problem in a sum of smaller problems
-complete the square
-by induction

These will begin to make sense to you when you encounter them again and again.

Yes, this is the first time I am learning from a rigorous book. All the others just took the properties for granted. I like it, but it is hard.

And indeed, I have difficulty with problems where I have to prove something or show that something works like that. But I am able to do the problems after I get a hint. I just don't know how to start.

I will try to see if I can use your tips on problems.

Thank for you for all your responses!