Suggestions for Graduate level Math books

[MelchyBeau]

Hey,

I'm working on my Masters in math, and would like to get a few books to work through during the summer. They have no graduate classes at my school during the summer. So just a little background information is required I guess.

I have a BS in Physics, not math. At this point the only class I have a 'deficiency' in is Abstract Algebra. I never took it as an Undergrad, and will be taking it in the Fall.

At this point I'm leaning towards focusing on Combinatorics, though I am unsure what specifically in combinatorics I want to do. In addition to the abstract class I'll be taking algorithmic graph theory and Random Probability.

So I am open for suggestions on any books I should go through. Right now I have Rudin's Principle of Mathematical Analysis which I'm partially through already, I plan on finishing that up as well.

So any suggestions would be helpful.

Melch

[BruceZ]

[ QUOTE ]
In addition to the abstract class I'll be taking algorithmic graph theory and Random Probability.

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As opposed to deterministic probability? /images/graemlins/smirk.gif

Feller is a classic probability text used as a first probability course by math majors. The second volume contains advanced graduate level material. DeGroot is also excellent and perhaps an easier read, but it is half probability and half statistics. Papoulis is a classic graduate level text on probability and random processes for engineering and physics students. It is packed with excellent examples and applications from engineering.

[lightw1thoutheat]

dummit and foote is a fantastic (although somewhat encyclopedic) algebra text. i highly recommend it.

also, if that seems a bit intimidating, which i still find it, hungerford has a book that is a bit easier to read. if you search for abstract algebra at amazon, these will pop up as some of the first.

[jason_t]

[ QUOTE ]
dummit and foote is a fantastic (although somewhat encyclopedic) algebra text. i highly recommend it.

also, if that seems a bit intimidating, which i still find it, hungerford has a book that is a bit easier to read. if you search for abstract algebra at amazon, these will pop up as some of the first.

[/ QUOTE ]

Hungerford is the most dry, boring book that I have ever read.

[armPitt]

Agreed. Do not go with the Hungerford text.

[sweetjazz]

Joe Gallian has an undergrad algebra book called Contemporary Abstract Algebra. I don't like the book as I think it is too easy and not meaty enough, but it might be a good book for self-study. (Abstract algebra is not a particularly easy subject for self-study because a lot of the intuition can be lost and you get left with trying to figure out dry symbol manipulation.) Ted Shiffrin has an undergrad book Abstract Algebra: A Geometric Approach, which I think is awesome. Some of the problems in that book can be quite difficult. There's a lot packed into the book, even though it is not nearly as encyclopedic as a graduate text.

As far as combinatorics goes, I think Richard Stanley has a two volume collection called Enumerative Combinatorics. I am not positive about this, but this would be a great place to get a start.

[cts]

In my undergrad algebra course we used Abstract Algebra, Fraleigh (http://www.amazon.com/gp/product/0201763907/002-3086149-9516025?v=glance&n=283155) which was pretty standard and you could definitely self-study from it. The more popular books seem to be Artin's and Dummit/Foote's.

[rufus]

Munkres topology book is excellent.
Spivak and Apostol on calculus are standards.

[srjunkacct]

Rudin's analysis text is a classic, but make sure you go through the "proofs" carefully, filling in all the missing details.

As for abstract algebra, Dummit and Foote is well-written and more comprehensive, but Artin is more fun, IMHO (lots of playing around with interesting group actions and such in the early part of the book). I agree that Hungerford is boring.

And you can't go wrong with any combinatorics text written by Richard Stanley. IIRC, the exercises are rated on a scale from 1 to 5, with 1 being "easy" and 5 being "unsolved". That's the way combinatorics problems tend to be -- it's hard to tell at a glance how hard a problem is.

[cambraceres]

For your physics degree

Mathematics of Classical and Quantum Physics

Byron and Fuller

Cam

[wooly_chicken]

For combinatorics, "A course in Combinatorics" by Van Lint and Wilson is good. It has a ton of material from very basic to somewhat advanced and has exercises with hints. I haven't read Stanley's books, but my impression is that Van Lint and Wilson has more of a "geometric" flavor than Stanley.

I think for algebra, Dummit and Foote is a great book to learn from. It has a ton of examples and interesting exercises that make it more conducive for self-learning than other books (i.e. Lang's Algebra is very good, but probably not something you want to teach yourself from.)

I would not suggest reading Munkres. While any that any mathematician needs to know point-set topology, it does not seem to be the kind of subject that would inspire me to want to do math (not interesting on its own IMO).

If you want to learn some more analysis, I would recommend the Princeton Series in Undergraduate Analysis (or something like this). There are four books--a Fourier analysis, a complex Analysis book, a real analysis book and a fourth one I don't know. They call themselves "undergraduate", but this is undergraduate at Princeton, which means equal to graduate level many other places.

Other more "graduate" level books that would be easy for self-study are Allen Hatcher's "Algebraic Topology" (probably want to be comfortable with the material in Munkres point-set topology book or the equivalent to do this). This is available on his website linked to from www.math.cornell.edu. (http://www.math.cornell.edu.) Serre's "A course in arithmeitc" also comes to mind. It does not require that much background knowledge is beautifully written (as it seems is most material by Serre).

[Meromorphic]

A different approach you might try is to identify some theorems you're interested in and using that to guide your selections, with an eye towards choosing things that will cause you to pick up some substantial, useful mathematics along the way. Say, the Jordan curve theorem or the prime number theorem or the unsolvabilty by radicals of the quintic (topology, complex analysis, and algebra respectively), for example.

[srjunkacct]

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I would not suggest reading Munkres. While any that any mathematician needs to know point-set topology, it does not seem to be the kind of subject that would inspire me to want to do math (not interesting on its own IMO).

(snip)

Other more "graduate" level books that would be easy for self-study are Allen Hatcher's "Algebraic Topology" (probably want to be comfortable with the material in Munkres point-set topology book or the equivalent to do this). This is available on his website linked to from www.math.cornell.edu. (http://www.math.cornell.edu.) Serre's "A course in arithmeitc" also comes to mind. It does not require that much background knowledge is beautifully written (as it seems is most material by Serre).

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I took Munkres's class as an undergrad, and I agree that most of the book is kind of boring. However, it is still probably the best exposition of the foundations of point-set topology that exists.

I think people who are more into real/functional analysis might appreciate Munkres more, as there's a nice chapter on function spaces in the later part of the book.