Hello All,
This post is actually a question. I realized throughout my course at JHU that I had failed to learn a good chunk of advanced undergraduate concepts (for example, Jordan Blocks and Diagonalization in Linear Algebra, and I have completely forgotten most common examples/methods in Calculus from Series in Calculus II into the entirety of Calculus III with the exception of basic operations of vectors (adding, cross product, dot product,etc.) and I never even learned Stokes and Green's Theorems. Now, here's my question. I just finished going through (reading and doing all exercises) in a Linear Algebra textbook. So, I've filled in those gaps as far as Jordan Blocks, Diagonalization, Projections, Orthogonality, etc. But, I don't know what the next step is. I'm tempted to go back to Calculus and really nail down all of that, but fear it will take too long, and it should be stuff that I can pick up as I need it by reading a few sections in a standard calculus textbook. It's only really methods and basic tools that I've lost, but upon reading it I should be able to get it back fairly easily. I taught Differential Equations last summer, so I'm fairly comfortable with that material, and the next logical thing would be to move towards an Abstract Algebra textbook (at least for me). I know that I did not pass the analysis qualifying exam, but I'm comfortable with undergraduate analysis (a lot due to the fact that I did most of an undergraduate textbook last summer, while studying for my analysis exam, thinking that I had missed something in undergrad, but I didn't). Now, if that is the next step, I'm not sure which book I should use. I have 3 or 4 at my disposal:
Lay's A First Course in Abstract Algebra
Hungerford's Abstract Algebra: An Introduction
Nicholson's Introduction to Abstract Algebra
Lanski's Concepts in Abstract Algebra
It is my knowledge that Lay's is the most elementary of the 4, and it is the one I used as an undergraduate. At Hopkins, we used Lang's Algebra, so you can see that there was quite a gap. I have heard that Hungerford is good, and JHU is using Nicholson for their undergraduate course right now. Wesleyan will use Dummit and Foote, which I also semi have access to, but not fully, so I would like to avoid using that one at the current moment.
My question ... Help! Which book do I use!? And is this even the right direction to go? Maybe I should go back to Calculus? Or, since I may be studying moving to study Graph Theory at Wesleyan, do I do my Graph Theory text?
Thanks all!
11 comments:
Lang's Algebra is a beast to learn from. As an undergraduate, I had Hungerford in UMD's MATH600. D&F is also good, but you'll do well to see a different viewpoint on your own.
As for "is this the right direction"? Yes. You need a solid foundation in algebra, even if you're working in something like graph theory. Analysis isn't quite so important, but you should still get comfortable with basic measure theory concepts. Your advisor will be there to point to what you should learn in graph theory.
This is good information! Thank you!
Dummit and Foote is much more accessible than Lang; I' m not familiar with the other texts on your list. I've heard good things about Michael Artin's Algebra text, but I've never read it; it might be worth checking out. http://www.amazon.com/Algebra-Michael-Artin/dp/0130047635/ref=tmm_hrd_title_0
man, you suck.
reading your post reveals that your level is far below PhD.
You should spend a couple of years in hard exercises, before you even think of applying for a PhD.
I'm teaching abstract algebra right now out of Fraleigh. I think it's a really good book. It's certainly not the most advanced by any means, but it's very good for beginners. The standard for a long time in undergrad classes was Herstein. That's solid but a little dense. I second whoever recommended Artin. It contains a lot of material, including what I call intro graduate material, but I think the presentation is very accessible.
And PS, I'm astounded that even on this sort of blog we have cowards who hide behind anonymized logins and spend their time being jerks. Please don't let that person be a voice for the mathematical community.
Work through Apostol's two volume set on Calculus. You'll get a nice handle on undergrad single variable calculus, multivariable calculus, basic set theory, linear algebra, ODEs, probability theory, basic real analysis. It's an incredible rigorous set.
Second the comment on anonymous jerks. I am nearly done with my PhD (applied math, though), and have enjoyed your postings and never once questioned the level of your understanding.
Apostol is good. I am a Rudin junkie, but his works are analysis not algebra, although I tend to consume algebra and analysis simultaneously. For Linear Algebra in a readable form it's hard to top Hoffman and Kunze.
Good luck -- keep us up to date and pay no attention to anonymous losers.
While you are likely well on your way, there still might be some space for the following:
http://www.extension.harvard.edu/openlearning/math222/
Lots of good information in these lectures.
HI
I HAVE PASSED OUT MSC.MATHS. IN 1998. I 'M MARIED NOW.I WANT O DO PHD.BUT I'M NOT IN TOUCH.MAY I DO IT OR NOT.WILL U SUGGEST ME?
I myself am on a similar path, although I believe we have started our journeys in different ways and I am 1 year behind you, as I am applying this fall for next year.
Just wanted to comment that I really look forward to hearing about your upcoming adventures, but my main point is with regards to your self admitted forgotten information: That stuff is the irrelevant part!
Advanced knowledge of mathematics and ability to conduct quality research is what is most important. I think of it as being similar to spelling: any dumbass with a dictionary and spellcheck can spell (I'm assuming Anonymous above falls in this category). But it takes a special quality to write a successful novel.
I wish you luck and encourage you to focus on the algebra. More than a little of what is most important in math is creativity. Algebra will show you how to tackle many problems from different angles and be creative with your proofs.
People do forget things. I will be starting my fourth year as a math Ph.D. student, and I know there are some things I have forgotten that I once know intimately.
I prefer Hungerford's text over Dummit and Foote's. Hungerford's tendency is abstraction; though it may have fewer examples, I believe it covers more content.
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