Should I Become a Mathematician?

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  • #2,026
mathwonk
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great suggestion. is this about the fundamental group? i think this was the first book i read as senior that really made me understand algebraic topology for the first time! if so, it is really clear and thorough for beginners just trying to grasp the concept of homotopy.
 
  • #2,027


What's the difference between an undergrad journal and the typical kind? I was under the impression that the usual journals also published undergrad research.

What are living costs like in the USA? I live in aussie and 14K doesn't really sound like it's enough to live like a pauper but that's compared to our currency and living costs. Do students get much more from teaching?
 
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Undergrad journals publish expository articles on a topic rather than just new research.
 
  • #2,029
mathwonk
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14K is not very much. But in Athens, Georgia life is cheaper than in many places.

Our problem is our average stipends are low, but our good stipends are high.

So I would suggest applying for our best stipends, and deciding what to do if you only get the average one.
 
  • #2,030


I have just been made aware that many universities require one to be able to read maths texts in German/Russian/French to do a PhD. I don't know either. *panics*
 
  • #2,031


Mathwonk: do you know what the current state of research into Topology is? I mean, is there still a lot of interest in the topic?
 
  • #2,032
mathwonk
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well with perelman's fairly recent solution of the poincare conjecture, yes, i would say topology is one of the hottest subjects.
 
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Mathwonk,

You seem to give quite a bit of praise to Michael Artin's book on algebra. What do you think of his father Emil's book on the subject?
 
  • #2,034
mathwonk
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them only books i know of by the father are "galois theory" notes from notre dame lectures, and "geometric algebra". these books are great classics, but they are not as easy to read as mike's book. mike write his book for sophomore students whereas emil seemed to write his books for eternity. i.e. whoever can read them is welcome, and not one word is wasted.

i myself never could really learn from e. artin's galois theory book as it was too condensed for me. he also has some algebraic geometry notes from nyu but those also leave much to be desired from my viewpoint for learning ease. But it is almost sacrilegious to criticize anything written by e. artin, who is regarded with great awe by many people.

but i regard mike's books a much more user friendly.

but as i meant to imply, i am not aware of any books by e. artin strictly on abstract algebra. of course the great book by van der waerden is based on lectures of e. artin and e. noether. Is that what you mean by e. artin's book? I like it quite well and learned a lot from it as a student.

If that is representative of e. artin's lecture style then he was a very fine teacher. Indeed I have read in his own works that he always tried to write more than usual on the board when lecturing so that the student who was not following could recover the lecture from his notes. this struck me as admirable and i long followed this practice in my own lecturing.
 
  • #2,035


well with perelman's fairly recent solution of the poincare conjecture, yes, i would say topology is one of the hottest subjects.

That's good to hear. I started reading Cairne's "Introductory Topology" and so far I've found it pretty fascinating. I can't wait to be able to take a class on it.
 
  • #2,036
mathwonk
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topology is the most fundamental branch of geometry. as such i believe it will always be one of the most fundamentally important topics.

the ideas developed in topology of ways to understand different types of connectivity, are absolutely crucial in all areas of mathematics.

the tool of cohomology, which is present in algebra, geometry, and analysis, received its greatest development within topology. Sometimes I think the greatest ideas in mathematics grew there.

that is probably unfair to analysis, but anyway.
 
  • #2,037


Your post went a bit over my head. :)

I really liked Abstract Algebra when I took it. It looks like group theory plays a roll in Topology, from skimming some things. Am I right in assuming this?
 
  • #2,038
mathwonk
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i am just saying that the ideas that were developed in the 30's, 40's and 50's within topology, like bundles, characteristic classes, and sheaves, and cohomology, grew outward and illuminated complex analysis and algebraic geometry in the 60's and 70's and are universally used now.

you are currently at the beginning, studying point set topology, but later when you study algebraic topology this will be meaningful.
 
  • #2,039
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Mathwonk,

In the first page of the thread you said that a high school student should explore probability, linear algebra, calculus after having a thorough grasp of geometry and algebra. What constitutes knowing Euclidian geometry and algebra well?
 
  • #2,040
mathwonk
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i would say mastering harold jacobs' books on those topics are a minimum for a high schooler. if more ambitious you might search out smsg books from the 60's. say arent there numerous such recommendations in that thread? have you only read page 1?
 
  • #2,041


Do they really expect PhD students to learn 2 foreign languages in 3 years?
 
  • #2,042


Do they really expect PhD students to learn 2 foreign languages in 3 years?

I don't see why this requirement would be intimidating. Two semesters in college is enough to teach the average student the basics of a language; with the generally higher capabilities of PhD students, I would imagine this time could be shortened. From there, it's just practice.
 
  • #2,043
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From what I've heard, the language exam is usually just to translate a mathematical paper from the language into English. I can't imagine that it's too difficult.
 
  • #2,044


Would Spivak's Calculus on Manifolds be a good reference text for a undergraduate course on multivariable analysis?
 
  • #2,045
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Calculus on manifolds book is primarily useful for the exercises, which are quite good. The writing and explanation is too terse in my opinion, but some people swear by it.
 
  • #2,046
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I just took a course using the book and found it to be really good. Munkres Analysis on Manifolds is kind of like an expanded version of CoM and is really good as well.
 
  • #2,047


I am trying to prepare a good foundation for math. I am learning from a few sources but I will be proficient these areas from classes and books:

Real Analysis (Learned from pugh and baby rudin, and class)
Linear Algebra (Learned from Friedberg, Insel, Spence, and class)
Set Theory (Learned From Naive Set Theory)
Combinatorics (Learned from Class)

What is a good way to learn geometry? I never paid much attention to any of my high school math classes and never really got much out of it, besides the basic identities. It seems like it could be very interesting.

I was looking at Beyond Euclid's Elements, and was surprised to find Mathwonk as one of the featured reviews on amazon. Maybe he can offer some advice and input.

Is there anythink else that math majors should know before moving on? One very interesting book that caught my eye is Inequalities by hardy, littlewood, and polya. It looked intense though, is that book my level?
 
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  • #2,048
mathwonk
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i liked calculus of several variables by wendell fleming.

as i said in my review, hartshorne's book is an excellent guide to euclid.
 
  • #2,049
quasar987
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IMO, "Inequalities" is a reference book, as opposed to a book you read from back to back... Say you're stuck on a problem and realize that if you had some kind of inequality then it would work... you go look in "Inequalities".
 
  • #2,050
mathwonk
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do you think these proof questions are too hard?

I.A i) Recently, my only guests for Thanksgivings have been turkeys.
ii) No mathematicians fail to solve crossword puzzles faithfully.
iii) The only faithful crossword puzzle solvers I know are my recent Thanksgiving guests.
Conclusion (using all the hypotheses):

IB. i) The Americans who exploited the Hawaiian natives ended up doing quite well.
ii) Some American missionaries who came to Hawaii originally to do good, started pineapple plantations.
iii) The pineapple planters in Hawaii exploited the natives’ land and labor extensively.
Conclusion(using all hypotheses):

IC. i) I consider money not spent enjoyably, to be wasted.
ii) I have had little joy out of anything lately other than comic books.
iii) An intelligent person does not waste money.
Conclusion(using all hypotheses):

ID. i) Dr. Smith has discovered the most wonderful beach.
ii) Some things are really fine, but nothing is as fine as the sand at the beach.
iii) If a person discovers something really fine, he should bury his head in it.
Conclusion(using all hypotheses):
 

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