Should I Become a Mathematician?

[G037H3]

my apologies for the thread killer.

I simply think no one had a response. :P

 

[mathwonk]

Not unless I get another job, or volunteer. Our retirement system motivates people to retire at a certain age. Many people would prefer a graduated withdrawal from work, with part time duties as they age, but this is not available at some schools. At mine, if you work less in your last years, you retire on less.

One option is to polish my several nearly finished books, on algebra, algebraic geometry, Riemann surfaces, complex analysis, calculus, linear algebra, differential topology, ... most of which are just lying on my computer in an outdated font and a word processor that isn't even readable by current versions of the same program (guess what famous software company produced this marvel of usefulness), and publish them for profit instead of giving them away. I am told however that publishers pay authors so little that it is hardly worth it. This may be why Mike Spivak publishes his own works.

 

[lisab]

Not unless I get another job, or volunteer. Our retirement system motivates people to retire at a certain age, roughly 67, and I have done so. Ideally many people like me would prefer a graduated withdrawal from work, with part time duties as they age, but this is not available at some schools. At mine, if you work less in your last years, you retire on less.

One option is to polish my several nearly finished books, on algebra, algebraic geometry, Riemann surfaces, complex analysis, calculus, linear algebra, differential topology, ... most of which are just lying on my computer in an outdated font and a word processor that isn't even readable by current versions of the same program (guess what famous software company produced this marvel of usefulness), and publish them for profit instead of giving them away. I am told however that publishers pay authors so little that it is hardly worth it. This may be why Mike Spivak publishes his own works.

Congrats on your retirement, mathwonk !

 

[G037H3]

hey, can you give me your opinion on the books,

Naive Set Theory, by Halmos

Polynomials, by Barbeau

?

 

[mathwonk]

i think i read halmos in high school and really liked it. he also wrote a really good book on finite dimensional vector spaces. i don't know the other one.

amazon reviewers make it sound quite good though.

 

[G037H3]

I intend on doing Naive Set Theory after I finish Elements of Algebra, but before Principles of Mathematics. Since Principles of Mathematics uses set theory, I figure that going through Naive Set Theory will help me understand how to prove things more easily, etc. and work through it more quickly. As for Polynomials, I may do that before calculus, idk o.o the problems look pretty hard :D

 

[mathwonk]

naive set theory as i recall is probably more sophisticated than principles of math, but may be easier to read since it has less content.

 

[G037H3]

It didn't look too difficult. I have the impression that Principles of Mathematics covers a lot of other things though.

 

[mathwonk]

yes i agree, sophisticated is not the same as difficult.

 

[ritzmax72]

big big posts. I'll read later(may b)
I stay in india. Doing engineering at NIT RKL. (clg ain't good, students are good).
I am also very much interested in maths, and want to become mathematician.
I'll be contributing to projects u guys make. And would promote my projects too. Projects here i meant is new innovative thing we'll be doing in maths. And maths get beautified if applied in physics and general people sees it happen.
With regards. And we'll do it.

 

[sentient 6]

just as a quick question... would I benifit more from doing my undergrad in the UK as opposed to Canada or the US.. that is, does the UK cover more material or anything like that on average? Where would encourage more creative thinking? or does it just not really matter...

 

[G037H3]

just as a quick question... would I benifit more from doing my undergrad in the UK as opposed to Canada or the US.. that is, does the UK cover more material or anything like that on average? Where would encourage more creative thinking? or does it just not really matter...

here: http://www.arwu.org/SubjectMathematics2010.jsp

 

[hunt_mat]

The UK maths degree consists of maths only, so if it's just maths you want then that would be a good option. As for creative thinking, this is just the sort of courses you like. There are a number of very good universities apart from oxford and cambridge that will give you a very good grounding in maths.

They are: Edinburgh, St Andrews, Glasgow, Heriot-Watt, Warwick, Newcastle, Manchester, Bristol, Imperial, King's College London and Exeter are a few very good universities.

You would expect that a good degree will cover the following subject areas: Algebra, analysis, geometry, probability, statistics and applied maths. If the university department can only cover a small proportion of these then it most likely not a very good university.

The root of the question, it calculus is calculus wherever you go really, a group is still a group in china or paris. The only difference is the amount of study you put in or if there are a group of you who want to beyond the syllabus.

 

[Ryker]

just as a quick question... would I benifit more from doing my undergrad in the UK as opposed to Canada or the US.. that is, does the UK cover more material or anything like that on average? Where would encourage more creative thinking? or does it just not really matter...

I'd say you learn more Maths per se, yes, but if that's a good thing or not is left for you to decide. I used to think it is, but you don't see a lack of great mathematicians (or other experts, for that matter) coming from US/Canadian schools, so I guess it's not all in the amount of material you cover.

 

[sentient 6]

thanks a lot for the answer to my question guys.. it really helped clear my head..!

 

[G037H3]

lol where is mathwonk, i start posting on the forum and he disappears :O

i hope he's having fun :3

 

[backthen]

Want to ID 40-year-old book

I graduated from college in 1966. Between then and ~1970, I saw on open shelf at Big Public Library a book that seemed to be principally on methods of integration, authored by someone from one of the Minnesota liberal arts colleges. I thought I would pick it up on a later visit but I never saw it again.

If someone can reference this book for me, I may be able to obtain it through interlibrary loan and, upon verification, locate and purchase it using a metasearch engine. Thanks.

 

[mathwonk]

I'm here. I didn't know the answer to the question about universities in UK, not having been to any of them. I am also retired and hence more busy than before with everyday stuff. I did have fun at my retirement dinner the other night. I assumed it would be a roast of sorts, but everyone was very kind.

I would guess that a student learns most at a school that pushes her/him, or by working with other students that do so. But some students can self motivate quite a lot. And I have learned that even calculus is not the same everywhere. E.g. compare the treatment in Hass, Weir, Thomas, to that in Spivak or Courant.

Indeed that was the first insight I had at xmas of my freshman year when I contrasted what I was getting in calculus at harvard with what my friend was getting at georgia tech.

It also matters whether you have the love of the subject. That's what keeps you going or coming back when things get tough. I also think it helps to be taught by an expert, since real understanding seems to come from personal contact with someone who embodies it, not just by reading a book of facts. The expert also has to want to make it clear to you, not just push his own agenda.

E.g even after reading Halmos, Munroe, Riesz - Nagy, etc..., listening to lectures by Loomis, and teaching real analysis myself from Lang, I never quite understood why you could approach Lebesgue integration so many different ways until I sat in on the first day introductory lecture by an analyst here at UGA who explained it clearly and answered my questions helpfully.

 

[sentient 6]

I'm here. I didn't know the answer to the question about universities in UK, not having been to any of them. I am also retired and hence more busy than before with everyday stuff. I did have fun at my retirement dinner the other night. I assumed it would be a roast of sorts, but everyone was very kind.

I would guess that a student learns most at a school that pushes her/him, or by working with other students that do so. But some students can self motivate quite a lot. And I have learned that even calculus is not the same everywhere. E.g. compare the treatment in Hass, Weir, Thomas, to that in Spivak or Courant.

Indeed that was the first insight I had at xmas of my freshman year in 1960 when I contrasted what I was getting in calculus at harvard from john tate with what my friend was getting at georgia tech.

It also matters whether you have the love of the subject. That's what keeps you going or coming back when things get tough. I also think it helps to be taught by an expert, since real understanding seems to come from personal contact with someone who embodies it, not just by reading a book of facts. The expert also has to want to make it clear to you, not just push his own agenda.

E.g even after reading Halmos, Munroe, Riesz - Nagy, etc..., listening to lectures by Loomis, and teaching real analysis myself from Lang, I never quite understood why you could approach Lebesgue integration so many different ways until I sat in on the first day intro lecture by an analyst friend of mine who explained it clearly and answered my questions helpfully.

hmm well I definitely do love the subject... I can't live without it. I'd also like to think I am fairly self motivated since i have been teaching myself for about 2 years or so... but only recently have I been trying to give myself a more rigorous treatment of it all with apostol's mathematical analysis (1st edition) and some random linear and abstract algebra book I found. Unfortunately I can't.. or don't... want to go to school in my country, especially not for math, because the program seems so limited... there isn't even a course on geometry. We don't do SATs here within schools... so I am looking at either the UK or Canada... this bothered me at first as I might be blocking out options in the US, but since hunt_mat made the point that calculus is calculus no matter where I do it once you work for it.. it cleared my head. But you too have a good point.. someone who really explains things well and takes the time out to explain it really does help... which now brings back some confusion... I have been thinking maybe to apply to university of toronto and just go there.. do you think that they have a good department? There seems to be a wide range of courses to choose from... I hate making life decisions hahaa..

 

[G037H3]

I mainly don't want to have the math-physics hole in my knowledge like Goethe did. To be a respectable polymath, mathematics is a fundamental subject.

 

[Math Ghost]

Hi I'm new to this forum but I have some questions. How integral to the graduate school process are the various gpa's? I have taken many classes (in fact all that are at my school short of a few grad classes) and I have been contemplating going to grad school somewhere but I have a feeling that my breadth of knowledge wouldn't be as much of a bonus as much much as my poor grades would be a detriment. I have taken (* are in progress): real analysis, complex analysis, topology, algebraic topology*, pde, ode, adv. ode, linear algebra, adv. calc, modern algebra I, galois theory,modern algebra II*, mathematical statistics, number theory, analytic number theory, mathematical modeling, stochastic processes(grad level), dynamical systems* (grad level). I have good work ethic and dedication but my grades suffer from unit overloads in addition to working full time while taking a full load. another question would be is a letter of reference from my workplace advisable? any other insights would be helpful as well.
Thanks

 

[Incognition]

I wouldn't try to explain the string theory in any other way. http://bayarearoster.com/js/includes/34/b/happy.gif

 

[mathwonk]

sentient 6. I could answer your question about whether toronto has a good math dept easily "yes" without even looking, but after looking at their faculty list and seeing bierstone and arthur there e.g., I say "YES!".

many people here ask questions like: what department will give me the best leg up politically, or mathematically, in my career? But these questions are sort of pointless for most of us. all it takes is one good advisor and a few competent mathematicians to get started on a career. I.e. ANY competent mathematician can teach you the basics off his specialty, and then you only need one good advisor to help you do some research. Then you are on your own to a large extent. The main decision to make is to get down to work, every day for a long time.

math ghost: your course background is much greater than the average incoming student at UGA. The point is whether you understand any of the stuff you took and can use it. E.g. you took algebraic topology. Can you decide (and prove) whether the identity map of a circle is null homotopic? What about a continuous map from a circle to a 2 sphere? you took complex analysis, can you use it to prove the fundamental theorem of algebra? do you know whether there exists a holomorphic isomorphism from the open unit disc to the open upper right quadrant of the plane? why or why not? Is there a way to extend the exponential function to the whole complex plane? What about the log function? Why or why not? If there is a way, is there more than one way? Why or why not?

Its not what you took, or what grade you got, it's what you know and can do. If you know something, you can convince someone of that and get into school at the appropriate place.

 

[Math Ghost]

Thanks! But in hindsight if its what I understand and can use then my grades would be better more then likely. I have shown that the circle is not contractible and after some thot I could probably answer most of these questions assuming I was taught the definitions of certain words. Thanks for the response!

 

[sentient 6]

thanks a lot man, and yeah I see what you're saying.. to become a competent mathematician.. the real work is on me, not them, their job is just to show me the ropes of the math world... well I guess.. it's time to start working

 

[Annonymous111]

sentient 6. I could answer your question about whether toronto has a good math dept easily "yes" without even looking, but after looking at their faculty list and seeing bierstone and arthur there e.g., I say "YES!".

many people here ask questions like: what department will give me the best leg up politically, or mathematically, in my career? But these questions are sort of pointless for most of us. all it takes is one good advisor and a few competent mathematicians to get started on a career. I.e. ANY competent mathematician can teach you the basics off his specialty, and then you only need one good advisor to help you do some research. Then you are on your own to a large extent. The main decision to make is to get down to work, every day for a long time.

math ghost: your course background is much greater than the average incoming student at UGA. The point is whether you understand any of the stuff you took and can use it. E.g. you took algebraic topology. Can you decide (and prove) whether the identity map of a circle is null homotopic? What about a continuous map from a circle to a 2 sphere? you took complex analysis, can you use it to prove the fundamental theorem of algebra? do you know whether there exists a holomorphic isomorphism from the open unit disc to the open upper right quadrant of the plane? why or why not? Is there a way to extend the exponential function to the whole complex plane? What about the log function? Why or why not? If there is a way, is there more than one way? Why or why not?

Its not what you took, or what grade you got, it's what you know and can do. If you know something, you can convince someone of that and get into school at the appropriate place.

math wonk. TO be fair ... I could answer those questions before I even opened books on the subjects. The proof of the fundamental theorem of algebra is so popular that you need only know the statement of Liouville's theorem to know it. You don't need to derive the statement. Similarly, whether the identity map on the circle is null-homotopic?? I mean come on! I did algebraic topology for 1 week and I could answer that. Surely one has to have a greater mastery of the subject if one took a course in it?

 

[Annonymous111]

Thanks! But in hindsight if its what I understand and can use then my grades would be better more then likely. I have shown that the circle is not contractible and after some thot I could probably answer most of these questions assuming I was taught the definitions of certain words. Thanks for the response!

You should know the meanings of all those words if you've really taken the courses you've claimed you've taken. Retract your claim that you've taken courses in those subjects, or substantiate it. I find a clear contradiction in your posts.

 

[Annonymous111]

sentient 6. I could answer your question about whether toronto has a good math dept easily "yes" without even looking, but after looking at their faculty list and seeing bierstone and arthur there e.g., I say "YES!".

many people here ask questions like: what department will give me the best leg up politically, or mathematically, in my career? But these questions are sort of pointless for most of us. all it takes is one good advisor and a few competent mathematicians to get started on a career. I.e. ANY competent mathematician can teach you the basics off his specialty, and then you only need one good advisor to help you do some research. Then you are on your own to a large extent. The main decision to make is to get down to work, every day for a long time.

math ghost: your course background is much greater than the average incoming student at UGA. The point is whether you understand any of the stuff you took and can use it. E.g. you took algebraic topology. Can you decide (and prove) whether the identity map of a circle is null homotopic? What about a continuous map from a circle to a 2 sphere? you took complex analysis, can you use it to prove the fundamental theorem of algebra? do you know whether there exists a holomorphic isomorphism from the open unit disc to the open upper right quadrant of the plane? why or why not? Is there a way to extend the exponential function to the whole complex plane? What about the log function? Why or why not? If there is a way, is there more than one way? Why or why not?

Its not what you took, or what grade you got, it's what you know and can do. If you know something, you can convince someone of that and get into school at the appropriate place.

in fact, i suspect that if you've read chapter 10 of rudin's R&C (which is approx 20 pg.), you could answer those questions

 

[Annonymous111]

Thanks! But in hindsight if its what I understand and can use then my grades would be better more then likely. I have shown that the circle is not contractible and after some thot I could probably answer most of these questions assuming I was taught the definitions of certain words. Thanks for the response!

You should know that the circle is not contractible. That's the first thing they teach you in an algebraic topology course. You should be able to answer all those questions. Which college are you in for undergrad.?

 

[mathwonk]

a111. would you mind giving more details of your answers to those questions? i.e. how does liouville imply fta? and then how do you prove liouville? the point is to understand the reasons for these phenomena, not just to be able to quote a theorem which is so strong that the corollary is rendered almost trivial from using it. do you know why the fta is almost an immediate consequence of the open mapping theorem? similarly it is not at all trivial to prove that the identity map of the circle is not homotopic to a point. what is your argument? and again, if you derive it from assuming some powerful machinery of algebraic topology, why is that machinery valid? do you know why both fta and non triviality of the identity map on the circle both follow from green's theorem? I am not trying to challenge you as I believe you can answer these questions, just to push you to think, and get beyond standard answers. In my opinion using liouville to prove fta is a little unnatural and unmotivated, ( and unnecessary). Of course you have to use something, since the result is non trivial.

 

[mathwonk]

Try these questions from elementary complex analysis:

1. Evaluate the path integrals of these differentials around C, and explain your method:

where C = {z: |z| = 2}.

i) (6 z^5 -5z^4 +1)dz/(z^6 -z^5 + z + 1)

ii) (6 z^6 -5z^5 +z)dz/(z^6 -z^5 + z + 1)

iii) dz/(z^6 -z^5 + z + 1).



2. Let Aut(D) be the group of holomorhic automorphisms of the unit disc D.
i) Prove that those elements of Aut(D) consisting of linear fractional transformations preserving D, are “transitive” on D, i.e. they take any point of D to any other point of D.

ii) Let D = {z: |z| ≤ 1}, and prove every holomorphic automorphism of D fixing 0, is a rotation.

iii) Prove that Aut(D) consists entirely of linear fractional transformations.


3. Assume f = u(x,y) + i v(x,y), is a function on the complex plane with u,v, real valued functions with two continuous derivatives, and R is a rectangle in the complex plane.
Assume also for all z = x+iy, that has a finite limit as h-->0,
and basic results of real differential calculus, and prove that:

i) ∂u/∂x = ∂v/∂y and ∂v/∂x = - ∂u/∂y.

ii) ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0.

iii) the integral of f(z)dz = 0 taken around ∂R, the boundary of a rectangle in the complex plane.

4. Use apropriate theorems of complex analysis to give a proof of the fundamental theorem of algebra, i.e. if f(z) is a non constant polynomial, then f has a complex root.

5. Assume f is a non constant holomorphic function on some neighborhood of the closed unit disc, such that |f(z)| is constant on the unit circle. Prove that f has at least one zero inside the unit disc.

6. i) Prove every meromorphic function on the Riemann sphere is necessarily rational.
ii) Prove a meromorphic differential on the Riemann sphere has two more poles than zeroes, each being counted with multiplicities.


7. If a function f is analytic on a neighborhood of the unit disc, is it possible for its values at the points 1, ½, 1/3, ¼,...to equal:
i) 0,1,0,1,0,1,...?
ii) 0, ½, 0, 1/3, 0, ¼, 0, ...?
iii) 1,1/4,1/4, 1/6, 1/6, 1/6, 1/8, 1/8, 1/8, 1/8,...?
iv) ½, 2/3, ¾, 4/5, 5/6,...?

Why or why not?

 

[mathwonk]

If you are having trouble with these problems, let me admit that the other professors could not do them either so they made me make them easier. problem 1i should be ok, and then for 1ii use the proof from the result that does 1i. then for 1iii think about what happens at infinity on the riemann sphere.

 

[wisvuze]

I just wanted to say that I saw this thread a long while back and picked up ,
PRINCIPLES OF MATHEMATICS - SECOND EDITION (Allendoefer ) off your recommendation of it; it's a great book, I really knew nothing before reading that book.This was years ago, but I just felt like saying thanks :)
By the way, how is the career market generally for applied mathematics in areas such as cryptography or data mining? Would you say that the jobs are particularly scarce? Competitive?

 

[mathwonk]

as a retired guy, i know almost nothing of the job market, except that i myself don't know where to find one. those areas sound promising to me though. has anyone got any information to offer? has anyone found a job lately?

 

[Cod]

I'm finally getting into upper level mathematics and had a quick question. How should I treat definitions? Should I memorize them before learning to apply? I ask because it seems like once I learn to apply a definition to solve a examples, but I usually end up twisted on problems unlike the examples. Is there a good way to rectify this issue I have? Do I just need to work as many types of problems with a definition as possible? Only problem with this method is that a text typically only gives a few different problems, nowhere close to all the types you may experience.

Any guidance is greatly appreciated.