Other Should I Become a Mathematician?

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Becoming a mathematician requires a deep passion for the subject and a commitment to problem-solving. Key areas of focus include algebra, topology, analysis, and geometry, with recommended readings from notable mathematicians to enhance understanding. Engaging with challenging problems and understanding proofs are essential for developing mathematical skills. A degree in pure mathematics is advised over a math/economics major for those pursuing applied mathematics, as the rigor of pure math prepares one for real-world applications. The journey involves continuous learning and adapting, with an emphasis on practical problem-solving skills.
  • #2,101


Real analysis is kinda the staple upper division math class, its absolutely necessary for grad school even if you focus in another area like set theory or algebra. Most of the programs I've seen expect you to have had analysis and topology/axiomatic geometry
 
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  • #2,102


How much do mathematicians get paid and in what way. Is it a fairly politics-free occuption
 
  • #2,103


Les2.0 said:
Is it a fairly politics-free occuption
Every place where money is involved there's politics.
 
  • #2,104


Hey mathwonk,

I just had a question for you... Three pronged actually: First I am in medical school right now and have found that the more time I spend in the biological science the more I miss math! Anyway it has been quite a few years since I took Calculus in college and was trying to get back into it. I'm thinking of going through Apostol's slowly and really making sure I remember everything, what do you think?

Secondly, what would you suggest in terms of where to go after I get through Calculus? Linear Algebra? Does Apostol's Vol. II cover all the linear Algebra I would need? Just general advice would be great.

Finally, I was wondering if you know anyone personally who has come back to a math/physics career later in life and made significant contributions. For financial reasons at this point I need to see medicine through to the finish. (Loans!) I'm planning on going for a specialty with the most physics/math in it, but was just curious if I decided to go back to to get my PhD later in life if I would be laughed out of most departments.

Thanks a lot!
 
  • #2,105


Following Dr. Smith's advice to "read the masters, rather than their students", when I got to the sections on Lebesgue integration in my analysis book, before looking at them I decided to take a look at H.L. Lebesgue's original writings on the subject. Although I can read French, English would be more comfortable. Does anyone know if his "Intégrale, longueur, aire" has been translated?
 
  • #2,106


I just had my supervisor advise me to use Atiyah Macdonald's Introduction To Commutative Algebra for a first course (with a bit of Artin on the side). Can anyone tell me at what level the book is actually meant for? I had a look on Amazon and the first few pages seem like it's accessible (ie, had no problem understanding it). But I'm a bit intimidated by the fact that some of the chapters are 5 pages long. o.O
 
  • #2,107


For a first course on what? On algebra or on commutative algebra? And if the latter, have you studied rings and modules before?
 
  • #2,108


A first course in algebra with some prior knowledge of groups and applied linear algebra. Obviously, I'll do some preliminary work on rings and modules with Artin. But I just wanted to know what kind of student the book is actually written for so that I can prepare myself.
 
  • #2,109


Well if it's for a first course on algebra, I doubt the material in Atiyah & Macdonald will have any relevance to what will be studied in the course! A first course on algebra usually talks about group while Atiyah & Macdonald is about commutative rings.
 
  • #2,110


I should have made it clear - it's not a first course... it's more like a first proper encounter. I'm actually doing a supervised reading course. Hence having a supervisor.
 
  • #2,111


I am not sure why you are not looking at something like Hungerford, Lang, Jacobson, or something like that, but if your advisor thinks Atiyah-Macdonald is appropriate then it probably is. Although it's really weird from your description because that's such an awkward choice for a first PROPER encounter with algebra. More like a course in commutative algebra that is. Actually, it WILL be a course in commutative algebra. But if you find it accessible - well let's see I have a copy here.

Hm I guess it's possible to learn from it things about rings and modules, but...

Maybe your advisor has a secret plan? :rolleyes: Atiyah-Macdonald followed by Hartshorne is always mind-blowing, after all.
 
  • #2,112


o.O

Is that a good thing? What is this Hartshorne you speak of?

His specialty is in algebraic geometry if that helps...
 
  • #2,113


There we go. Commutative Algebra - Algebraic Geometry (Hartshorne) is the classic way to get into the subject.

He wants to teach you his specialty!
 
  • #2,114


How come you have a reading course for a first course in algebra?

wouldn't you normally have just as part of coursework?
 
  • #2,115


I don't see anything wrong with starting with commutative algebra from the get go. I think it could make sense to talk about fields first. I mean geez these are the things that say an a lot of undergraduates work with if they study engineering and physics. I could see a course where you start like this fields -> matrices, vector spaces -> rings -> algebras and modules and then bam you are at the Cayley Hamilton theorem which is in Atiyah & Macdonald.
 
  • #2,116


I'm special. =P

Well, my uni doesn't have a very strong pure maths programme per se since very few people are actually interested in pure maths so it's all pretty easy. However, they do allow and encourage the capable and ambitious ones to do advanced studies.
 
  • #2,117


That reminds me of my sorry state at uni.
 
  • #2,118


This is much like my school. There are only a handful of dedicated math students and even fewer students who want to go on to be mathematicians (probably less than 5? At least 2-3) so standard classes in upper level math beyond algebra and analysis are virtually non-existent(we have number theory, Discrete math, a graph theory course and PDE as well as a number of other classes but these don't run every year and sometimes not at all). However, since there are only a few strong math students, we have full reign over the professors who usually are happy to do an independent study/directed readings course.

I'm looking at Atiyah-Mcdonald and it looks like if follows directly from the Dummit and Foote material in chapter 9; it reviews ideals, maximal ideals, prime ideals nilpotent stuff, algebraic closure. I'm not sure how penetrable the material is without any background, however. It seems kind of odd to go right into a book which seems to presuppose a good deal of knowledge, however it will surely be manageable with the help of a professor.

What do you guys think of Eisenbud's book on commutative algebra?
 
  • #2,119


Matsumura I like better. But you need to look at both, Eisenbud's book shows that he is very interested in computation. Matsumura is more abstract.
 
  • #2,120


Well, I'm preparing myself with a few chapters from Artin. I'm hoping that'll make things easier to swallow.

Oh, and I'm confused... is homology a part of algebraic topology? Or is it it's own area? Would Munkres be a good place to learn algebraic topology or would a more specialised book be better?
 
  • #2,121


Homological algebra and homology (from algebraic topology) are similar.

For algebraic topology, I used Spanier but perhaps nowadays Bredon is much more reasonable choice. I never liked Munkres that much, too expensive.
 
  • #2,122


What about Massey?

Hmmm... I think I'll do algebraic topology in the second semester... best not to take on too much at one go.
 
  • #2,123


I have never read that book, so I am not sure. But Hatcher seems popular, just to give you another choice.
 
  • #2,124


what happened to mathwonk?
 
  • #2,125


I just realized today that I spent my whole undergraduate years thinking I was going into the topological field.

I'm doing my Master's in Number Theory.
 
  • #2,126


Don't you find number theory problems are either quite simple, or devilishly difficult? I wouldn't know, I haven't done too much number theory, but that's how it seems.

How did you go from topology to number theory. Did the subject matter of topology lead you to number theory?
I'm still an undergrad, and I thought I was going into algebra, until I took a fun analysis course (my first analysis course was not a fun experience...), and a horrid algebra course. These things depend so much on the text used for me.
 
  • #2,127


Problems in Number Theory can go from easy to mediocre to hard. Just like anything else.

I went from interests in topology to number theory simply because I didn't think I could do it. Then luckily for me the number theorists at my school asked me to work with him on my Master's. So, I took the offer.

Um... now I just work hard everyday. I'm working harder than I ever had before. With regards to course work.
 
  • #2,128


Mathwonk, I already asked this in another section but I'm interested in your answer particuliarly. Have you ever looked at Apostol's Mathematical Analysis? If so, have you looked at both the 1st and 2nd edition? Do you think the 1st is inferior to the 2nd and not worth getting?
 
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  • #2,129


Mathwonk hasn't been here in more than a month.
 
  • #2,130


Werg22 said:
Mathwonk, I already asked this in another section but I'm interested in your answer particuliarly. Have you ever looked at Apostol's Mathematical Analysis? If so, have you looked at both the 1st and 2nd edition? Do you think the 1st is inferior to the 2nd and not worth getting?

2nd edition. It will correct errors from the 1st edition.

Why even ask such a question? Just buy it if you want to learn Analysis.

Second, Apostol isn't the only one either. If you are nervous about the quality, buy something else.

The best books are those that explore the subject and provide the perfect questions. I used to think Spivak was good, but now that I think about it, I don't think it is. I think it is good only if supplied by another textbook to give that nice, even easy exploration or by a really good professor that puts the time into his lectures.

I've never seen an Apostol textbook except his Number Theory book (same guy?), and looks like any other to me.

Note: I judged it, not by its cover, but its table of contents and preface. (Essentially a summary of what to expect.)
 
  • #2,131


JasonRox said:
I just realized today that I spent my whole undergraduate years thinking I was going into the topological field.

I'm doing my Master's in Number Theory.

What area in number theory?
 
  • #2,132


How difficult would you(anyone who has completed some substantial part of the book) consider the problems in Atiyah MacDonald's Commutative Algebra? How long should it take to do say, one chapters worth of exercises?
 
  • #2,133


JasonRoxx, If I remember correctly you went to University somewhere in Ontario? I got accepted into the maths program at Waterloo and U of T - St.George campus, and I was wondering which one would you recommend for studying pure mathematics? Ignoring all other factors like location, money, etc.
 
  • #2,134


JG89 said:
JasonRoxx, If I remember correctly you went to University somewhere in Ontario? I got accepted into the maths program at Waterloo and U of T - St.George campus, and I was wondering which one would you recommend for studying pure mathematics? Ignoring all other factors like location, money, etc.

Hey, I'm actually studying at Brock University.

I'm very happy with my choice. My supersivor made sure I had a good funding package so I don't have to work. I'm so thankful for that.

I have free time to learn what I need to learn. He knows so many people in his field that I don't need to worry about where I want to go for my PhD if I do a good Master's of course. The schools I want to go to are not top notch though. Sure Icould have gone to Waterloo and stuff, but I think I am way better here.
 
  • #2,135


Though I may be beating a dead horse here, which school is better for math, Waterloo or U of T?
 
  • #2,136


JG89 said:
Though I may be beating a dead horse here, which school is better for math, Waterloo or U of T?

Undergraduate or graduate?

Either way, I see no difference.
 
  • #2,137


Bourbaki1123 said:
How difficult would you(anyone who has completed some substantial part of the book) consider the problems in Atiyah MacDonald's Commutative Algebra? How long should it take to do say, one chapters worth of exercises?

It depends on your preparation. It took me about 3 months to do every exercise in the book just after having finished my 3 year bachelors. After all, the exercises are the beef of the book. The text itself is pretty trivial.

There are a few exercises which take more time to do, some required knowledge of Tor and Ext which I had to look up form Weibel...
 
  • #2,138


Can anyone recommend a book on topology that would be suitable for (casual?) self-study? Most of the topology books I have glanced at in the library are either too basic or too technical.

My background: Freshman, know a trivial amount of group theory, read all of Spivak and most of Baby Rudin, somewhat comfortable with linear algebra.
 
  • #2,139


Wretchosoft said:
Can anyone recommend a book on topology that would be suitable for (casual?) self-study? Most of the topology books I have glanced at in the library are either too basic or too technical.

My background: Freshman, know a trivial amount of group theory, read all of Spivak and most of Baby Rudin, somewhat comfortable with linear algebra.

Maybe Alexandroff's elementary concepts of topology. It's a dover book that goes for under 10$ written by one of the founders of topology.
 
  • #2,140


Wretchosoft said:
Can anyone recommend a book on topology that would be suitable for (casual?) self-study? Most of the topology books I have glanced at in the library are either too basic or too technical.

My background: Freshman, know a trivial amount of group theory, read all of Spivak and most of Baby Rudin, somewhat comfortable with linear algebra.

You're going to want to start with a basic topology book. No need to know any analysis or linear algebra.

Theral Mooral - Elementary Topology
James R. Munkres - Topology

Both of those are suitable for you. The first is cheaper, but won't cover as much. Although, I find the first to get the reader more involved.
 
  • #2,141


Wretchosoft said:
Can anyone recommend a book on topology that would be suitable for (casual?) self-study? Most of the topology books I have glanced at in the library are either too basic or too technical.

My background: Freshman, know a trivial amount of group theory, read all of Spivak and most of Baby Rudin, somewhat comfortable with linear algebra.

Munkres makes me sick. If you know that much math, there is nothing wrong with getting something like Bredon. Much more concise with general topology, and it goes into some smooth manifolds and algebraic topology.
 
  • #2,142


JasonRox said:
You're going to want to start with a basic topology book. No need to know any analysis or linear algebra.

Theral Mooral - Elementary Topology

I believe you mean Theral Moore.

He's a professor at my school who recently retired. While I never had him, I've only heard great stories about him. He was blind but still taught calculus courses for many years, and had all of the problems in the book memorized so that he could answer students' questions.
 
  • #2,143


Thanks for the replies.

Another question: What would you recommend studying first, real analysis or topology? I am currently working through a complex analysis book (Cartan) and will start on an algebra book after that, which I hope to be done with by, say, mid-summer, with school.
 
  • #2,144


I am a High school junior who plans to do partial differential equations via EPGY at Stanford. However, due to the economic situation I can't afford to take the prerequisites at EPGY and so I have to self study everything. I have finished single variable and most of multivariable calculus, however, I can't find any good resources for studying differential equations.

Are there any good online resources except for MIT OCW to learn differential and what are some good textbooks?
 
  • #2,145


tyrant91101 said:
I am a High school junior who plans to do partial differential equations via EPGY at Stanford. However, due to the economic situation I can't afford to take the prerequisites at EPGY and so I have to self study everything. I have finished single variable and most of multivariable calculus, however, I can't find any good resources for studying differential equations.

Are there any good online resources except for MIT OCW to learn differential and what are some good textbooks?

Do you know much linear algebra? For ODE I really like Arnold's book, but I have trouble remembering what the prerequisites are. The one by Devaney is a good one too.

I don't know any undergrad PDE textbooks. Maybe Strauss.
 
  • #2,146


Unknot said:
Do you know much linear algebra? For ODE I really like Arnold's book, but I have trouble remembering what the prerequisites are. The one by Devaney is a good one too.

I don't know any undergrad PDE textbooks. Maybe Strauss.

I don't know much about it but I have a good textbook for reference if needed
 
  • #2,147


tyrant91101 said:
I don't know much about it but I have a good textbook for reference if needed

You should really learn linear algebra first. You have plenty of time to learn math, no need to learn your interests now.
 
  • #2,148


tyrant91101 said:
I don't know much about it but I have a good textbook for reference if needed

I don't think I would recommend Arnold's ODE text for you, for your first time encountering differential equations. You'd probably want a book along the lines of Boyce and Diprima (but cheaper probably)
 
  • #2,149


axeae said:
I don't think I would recommend Arnold's ODE text for you, for your first time encountering differential equations. You'd probably want a book along the lines of Boyce and Diprima (but cheaper probably)

Yes, you are right. Now Tenenbaum comes to mind. Widely used text and also cheap.
 
  • #2,150


Well, real analysis is a gentle introduction to point set topology but learning topology first will probably make real analysis pretty trivial (at least for the early parts). My friend did topology first and he thinks of real analysis as just a special case.
 

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