The Should I Become a Mathematician? Thread

samspotting

What is graduate analysis like?

Yeah math 55 looks crazy, but I guess the bestof harvard math students can take it.

I read about it in the newspaper and there was this imo competitor that said "screw it math 55 is too hard" and dropped to the easier one.

axeae

Math 55 seems pretty over-hyped, my analysis course wasn't much different ( it was mostly questions from rudin for hw, around the same number of problems too) and I took it as a freshman---it's tough but not as tough as people make it out to be (http://www.thecrimson.com/article.aspx?ref=516216)

Vid

Have you looked st the homework problems on the course page I posted? Those are very difficult proofs to have to come up with not to mention the fact the class covers over 2 chapters of rudin a week.

http://www.math.harvard.edu/~ctm/hom...9/html/hw.html

samspotting

Wow that homework is insane for a freshman. I bet these students that make it can do graduate courses by their fourth semester.

axeae

Yeah I looked at the homeworks, they're rough but it's somewhat similar to what my class had. my class wasn't made for freshmen though, it was a class for juniors, but I took it as a freshman.

To answer samspotting's question, graduate real analysis typically covers some measure theory and lesbegue integration, and maybe some introductory functional analysis (or other topics) depending on the professor. I think math55 at havard covers some complex variables in their class too

samspotting

At my university my fourth year pure math courses are first year graduate courses, ive heard they are substantially more brutal than third year pure math courses.

I'm totally looking forward to them.

Does doing putnam style prep questions help you improve math skills for core topics like analysis and abstract algebra, or maybe combinatorics? seems like something fun to do over the work term this summer. If not ill probably concentrate on doing some texts like munkres topology or naive set theory.

axeae

I found first year grad courses easier than the third year pure math courses. the topics may be more "difficult" but at that point you should be better prepared and more experienced as a math student.

I've never done the putnam but learning more topology and set theory is usually never a bad idea if you're interested in analysis/algebra (for the second part of munkres you will need to learn a bit of abstract algebra though)

jgg

If I've missed a complete discussion on this, forgive me. I've only been able to find scattered opinions on the issue.

I recently got a copy of James Stewart's Calculus for a nearly-free price. Is this sufficient for self study, or should I fork over the $50 - $150 for an author more often recommended on this forum (I was thinking Apostol seemed pure and rigorous, which is what I wanted)? Also, someone already mentioned Strang's book (http://ocw.mit.edu/ans7870/resources...strangtext.htm). Is this better or worse than Stewart (or Apostol) for a newbie? I remember someone, I think it was one this forum, regarding Stewart as a superficial 'cookbook' of sorts.

(I'm a high school senior declared as a double major in computer engineering and math. I did the latter for fun, thus I want to actually learn Calculus (i.e., proofs) as a starting point for future study; I'm not looking to just be told how to do the power rule well enough to score high on an exam.)

axeae

while apostol is more rigorous, you might be better off with stewart's calculus for learning the material for the first time. I skimmed over the pdfs of strang's calculus---it seems similar to his linear algebra book, which was a good book but I personally didn't like his style.

if you can get stewart for nearly-free I'd probably do that. you could always check your local library for apostol, spivak, or courant---they all nearly the same level of rigor---for a more in depth treatment of calculus

qntty

I agree with axeae.
I'd say go with Stewart until you learn how to do the basic computations that you need to learn. Then you can pick up Apostol or Spivak. I'm also in high school and chose spivak after looking over both because, IMO, spivak is easier for a high schooler without sacrificing rigor.

jgg

I should have probably mentioned that I'm finishing an A.P. Calculus class.

qspeechc

Well, I think it depends, then. If you're hoping to become a mathematician, I would rather pick up Apostol, Spivak or Courant and plow through one of those. I did something similar. I was in high school and I didn't know whether to study calculus rigorously or out a cookbook like Stewart. I ended up doing the latter, and as a mathematics major, I think that was the wrong choice. Because in the first year at university we basically did all that crap again.

I you're not planning on being a mathematician, then Stewart is fine, I suppose. Stewart is written for biologists, economy students etc. after all.

Vid

If you want to major in math, master stewart and go to a college that offers a rigorous honors freshman sequence. You might have to shop around to find one, but if you can manage this it would be the best option by far. Or if the college you go to doesn't offer an intense honors calc, then master stewart extremely well and move on to analysis for your freshman year.

jgg

I'm thinking of doing Spivak's Calculus now. I reviewed a couple and I liked this one best. Thanks everyone.

Bourbaki1123

I have noticed a bit of a problem I have with the efficiency of my long term memory when it comes to mathematics. I have a very detailed memory for math when it comes to the short term and even midterm memory, but, and I suppose this is actually pretty common, I find that going back I will have forgotten a great deal of detail.

For instance, the material in my first semester algebra course from only a few months ago seems to have slipped out. I'm talking about some results dealing with normal subgroups that are fairly important, like that the normal subgroups of a group R containing some subgroup I are isomorphic to the normal subgroups of the quotient group R/I. I had to look this up when going over some commutative algebra to prove the analogous theorem for rings and ideals to myself.

Is there any strategy to use to keep all of the details in your head for the long haul? Is it the obvious answer of simply reviewing material? After enough algebra classes will it become second nature? Is this even really a problem or is it a common thing?

desti

I've noticed that the most important thing is to just review material, but often you forget to review which is the biggest problem. However, when I've forgot something, usually just seeing a definition or statement of a theorem regarding the subject is enough to reremember it.

I began using one of these "smart" flash card programs some time ago and since I began doing that, I have pretty much memorized everything from my courses without forgetting anything. The program I use is the following:

http://www.mnemosyne-proj.org/

I pretty much type in every new definition I see and all the important points and theorems together with the ideas of proving them. Then I spend about 15-30 minutes a day going through the cards. The program pushes stuff I know well far into the future, so stuff I really remember shows up in a card maybe every 4 months or so, so you don't waste time. Instead it usually shows me stuff I am almost beginning to forget.

Bourbaki1123

That looks interesting. I think I might try something like that. I'm very good at remembering things for a few weeks, I can just look at something and see the picture of it in my head, but that seems to only work for me in the relatively short term. I'm not sure why it is that these things don't seem to get stored into my long term memory, but its probably because I never bother to look at anything more than once or twice and then forget about it when the class has ended(unless I need to know it for my next class, and then I pick it back up).

As far as classes go its not so big a problem yet, but doing research or a competition, I could see it becoming a much bigger problem. I would definitely be at an advantage if I could keep sharp on everything in the long term.