Should I Become a Mathematician?

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  • #3,026
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A sketch of how it works in this case is that certain logics have algebraic counterparts (usually based on Boolean algebras or lattices), and there are various ways, largely based on Stone and Priestley dualities, for interpreting these algebraic structures as topological spaces (possibly equipped with some extra structure, like the ordering in Priestley duality), and the maps between them as continuous functions (usually with additional properties) with domain and range switched. I know that these dualities have been used to prove completeness results for various logics, though I can't give details off the top of my head.
 
  • #3,027
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Many thanks for the input, mathwonk.
Also, I noticed the many books listed in the beginning of the thread. I found previews of some of them on the internet, and they seemed a little too complex for my high-school level skills. Can you recommend any mathematics-related books that would be interesting for someone who is passionate about math without much knowledge of university math? What I'm looking for is not a textbook, but something that can be read (analyzed and worked-on) recreationally as well. I just want to develop skills beyond the curriculum, and know things that don't require a mere substitution into, or use of, a formula (so essentially, skills which would benefit me in a contest-type problem). So I'm not looking for a textbook, and nor am i looking for a novel, but a mix of the two. My skills extend to enriched (AP) grade 12 calculus (canadian curriculum) So any recommendations would be greatly appreciated, and thanks once again for your help on the previous question I posted.
 
  • #3,028
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Try What is Mathematics? by Courant and Robbins.
 
  • #3,029
mathwonk
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exactly my choice.
 
  • #3,030
Cod
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Try What is Mathematics? by Courant and Robbins.
A great book. I've read through it and Stewart's Concept of Mathematics numerous times. I highly recommend both for individuals jumping into higher maths.
 
  • #3,031
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Wow, "What is Mathematics?" by Courant and Robbins it is. Thank you all very much for the suggestion. I have also been recommended "Lessons in Geometry" by Jacques Hadamard (ISBN 0821843672). In the book description, it states "The original audience was pre-college teachers, but it is useful as well to gifted high school students and college students, in particular, to mathematics majors interested in geometry from a more advanced standpoint." so I think that this book would be suitable for me.
Can any of you confirm the good things I heard about this book? :)
Thanks in advance.
 
  • #3,032
thrill3rnit3
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Any knowledgeable folks in here have any idea about the algebra group at UCLA? I'm looking at it as one of my potential graduate schools and am wondering if their algebra research is thriving..
 
  • #3,033
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I am not at UCLA but I went to the prospective grad students open house a few weeks ago and talked to a few of the algebraists there. I asked Haesemeyer about algebraic K-theory and he said that UCLA might be the best place to do K-theory, since besides the K-theorists at UCLA there are strong K-theorists nearby at USC as well. So there is a lot of interaction between the two departments. I also talked to an algebra grad student who said that there were plenty of people interested in algebra and algebraic geometry (including more than one person working in motives). In any case it seemed like a great place to do algebra (which I began considering much more strongly after talking to two extremely enthusiastic algebra professors, even though I have always been more interested in geometric/topological things).
 
  • #3,034
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bublik13, while I am not familiar with the book you mentioned, it looks good. Hadamard was a great mathematician (he was the first* to find a proof of the prime number theorem) so I would expect anything by him to be valuable.

*de la Vallée Poussin discovered a proof independently at the same time.
 
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  • #3,035
morphism
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Any knowledgeable folks in here have any idea about the algebra group at UCLA? I'm looking at it as one of my potential graduate schools and am wondering if their algebra research is thriving..
What kind of algebra? If it's with a number theoretic bent (algebraic number theory, galois representations, etc.), then UCLA would be great for that.
 
  • #3,037
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I can safely that towards the end of your undergraduate or atleast during your Ph.D there should be some topics that excites you (in the sense that you feel passionate towards learning, thinking and asking questions about that topic). It need not necessarily be your Ph.D topic as not all people have the chance to work exactly on their topic of interest (but something related). Basically at your Ph.D stage you should atleast heave dreams about studying a certain topic when you become a faculty :p
 
  • #3,038
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One thing I do to research the best regarded people in math is to look at the invited speakers to the ICM. The one in Hyderabad in 2010, featured Paul Balmer, algebraist from UCLA.

http://www.icm2010.in/scientific-program/invited-speakers

This was one of the professors I spoke to. He works on tensor triangulated categories which, if I understood correctly, allows you to prove things about algebraic geometry, motives, noncommutative geometry, symplectic geometry, and more, all at once. Crazy powerful stuff. This is his survey on the topic: http://www.math.ucla.edu/~balmer/research/Pubfile/TTG.pdf
 
  • #3,039
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Hi everyone.

Sorry to cut in on your discussion like this and change the topic.

I graduated a few years ago with my bachelors in maths, and have been working since, and recently I have been reviewing the maths I did at university. I have worked through Herstein's algebra book, and I wanted to know if I should work through Artin, since everyone talks so highly of it. My aim is eventually to read grad-level books (my interest isn't in algebra, but everyone needs to do graduate algebra, right?).

But here's the thing. I really don't have money to spare, and even used copies of Artin are expensive (for me at least). Instead of getting another book on undergrad algebra, which I already know, I'd rather spend the money on a book on another topic, maybe even Lang's algebra book.

So, do you think Artin is really worth getting, or should I get some other book?
 
  • #3,040
mathwonk
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well Artin's book is better than Herstein's in my view, but if you are poor, why not take a look at my free notes for math 843-4-5 on my page

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

I am not in Artin's league, but my book has helped some pretty good people.
 
  • #3,041
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well Artin's book is better than Herstein's in my view, but if you are poor, why not take a look at my free notes for math 843-4-5 on my page

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

I am not in Artin's league, but my book has helped some pretty good people.

Well, there's no need to put it quite like that :blushing:

What I meant was, if you think it's really worth it, then I guess I'll save up for Artin, and I'll just have to postpone on getting some other book.

In the mean time I'll take a look at your notes, thanks.
 
  • #3,042
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I;m taking a calculus II course. Partial fractions seem very unmotivated and ugly to me. But I'm sure there has to be some beauty behind it. Can anyone link me to the underlying theory of it all?
 
  • #3,043
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I agree that partial fractions are ugly in the sense that they can be a pain. But, I don't get the unmotivated part. Aren't you decomposing a complicated quotient into the sum of several easier quotients that you can integrate? That is the motivation.

As for underlying theory, I really think it is just algebraic manipulations, like partial fractions or something.
 
  • #3,044
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I agree that partial fractions are ugly in the sense that they can be a pain. But, I don't get the unmotivated part. Aren't you decomposing a complicated quotient into the sum of several easier quotients that you can integrate? That is the motivation.

As for underlying theory, I really think it is just algebraic manipulations, like partial fractions or something.

Well unmotivated because they seem to just come out of nowhere. The book I'm using says do this and this and you will get this. But I don't blame it, deriving it seems tricky-- you need a bunch of clever manipulations that aren't so straightforward.
 
  • #3,045
658
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I;m taking a calculus II course. Partial fractions seem very unmotivated and ugly to me. But I'm sure there has to be some beauty behind it. Can anyone link me to the underlying theory of it all?

I agree with Robert1986 that the motivation is simply just decomposing it into a useful form. If you want a more general form of it check out wikipedia:

http://en.wikipedia.org/wiki/Partial_fraction
 
  • #3,046
Cod
324
4


Well, there's no need to put it quite like that :blushing:

What I meant was, if you think it's really worth it, then I guess I'll save up for Artin, and I'll just have to postpone on getting some other book.

In the mean time I'll take a look at your notes, thanks.
Keep an eye on the price of international editions. They're a lot cheaper and usually contain the same material as their US counterpart.

http://www.abebooks.com/servlet/SearchResults?isbn=9780132413770&sts=t&x=54&y=13
 
  • #3,047
840
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Keep an eye on the price of international editions. They're a lot cheaper and usually contain the same material as their US counterpart.

http://www.abebooks.com/servlet/SearchResults?isbn=9780132413770&sts=t&x=54&y=13

Yes, the problem is I'm not in the USA, so the price may be good, but the shipping is dreadful, the first few are all over $40 or $30 for shipping! And then there's import tax, duties, etc., which adds another 40% or thereabouts, so a $60 2nd-hand book (including shipping, I think) comes out at $84 etc.

Also, I've tried a few times to buy from bookseller in India, but they won't ship to where I am.

But thanks for the tip, I'll definitely keep my eye out for a good deal.

I'm in no rush anyway, there are many, many books I'd like to read, and maybe one day I'll get round to Artin (hopefully not too long from now).
 
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  • #3,048
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As for underlying theory, I really think it is just algebraic manipulations, like partial fractions or something.

Even I, Mr. Conceptual himself, would agree with that. However, it makes much more sense with complex numbers than real numbers. Maybe people avoid complex numbers because calc students aren't 100% comfortable with it.

Partial fractions just aren't that great of a thing.

I didn't really understand the algebraic tricks when I first saw it, so that was annoying. But after I figured out how to derive it myself, it was somewhat less annoying. It's kind of analogous to multiplying both sides of an equation by something. It just isn't anything to write home about. But it's also not something to get upset about, either.

I think the motivation in calculus is also to write things in a form where they can be integrated.
 
  • #3,049
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Does university reputation matter? I have an offer from a top 20 world university, and top 10 UK universities to study mathematics, but also one which is closer to home, but has less reputation? I'd prefer to go to the one with the lower reputation, as i'd like to stay at home, but I'm not sure if I should just suck it up and go to the one who should give me more career prospects. After graduating I plan on going onto actuarial, or investment banking jobs, or perhaps graduate work, if i'm good enough.
 
  • #3,050
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With a very few exceptions at the very top or bottom, I would say university reputation does NOT matter going into actuarial work. Actuarial work is not like law where only going to the top few schools makes it worth the price.
 

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