From: frandag@math.ksu.edu (Francis Fung)
Newsgroups: sci.math
Subject: Re: Mathmatics
Date: 23 Dec 1992 01:21:35 -0600
Organization: Dept. of Mathematics, Kansas State University
NNTP-Posting-Host: hilbert.math.ksu.edu

jasonp@bungie.prime.com (Jason Pascucci) writes:

>Hi,

>Yes, this might be a strange request, however please
>indulge me.

>I would like to obtain the firm grasp of 'higher-math'
>without spending time in school for a Doctorate in Mathmatics. 

>I would like suggestions and commentary as to a fairly 
>straightforward, and (as much as possible) all-encompassing
>'coursework' for an individual, starting, basically,
>from Calculus (which I have a fair grasp of, but
>some review I expect will help), and going through
>the same topics as would a (for example) Doctorate of 
>Mathmatics, without the 'overhead'.  

>I'm looking for easily obtainable books and papers, 'standard'
>college textbooks only as a last resort, unless they are 
>exceptionally well written. 

>Mostly, I'm looking for direction, I guess. I would like to 
>expand my knowledge base, and don't have the patience to 
>go through a degree program. I would also like to work at my 
>own pace, which is why I'm interested predomenantly in books. 
>My learning curve is fairly strong, with high comprehension 
>from reasonable reading speeds and minimal need for repetition 
>to leave an impression of an idea. It helps if there
>is a good background with material...such as the thought-process
>which lead to it's discovery and subsequent development...
>as opposed to the 'it works this way, memorize it' method. 
>(Admittedly, it's quite possible that there is nothing out
>there which uses this learning curve...but it would be nice.
>I also realize that there is a lot of 'it works this way, memorize
>it' involved in mathmatics, but a good presentation might seek
>to soften this...)



Well, I'm glad you're wanting to embark upon some studying.
The first thing, though, is that a Ph. D. usually requires 
a focus of some sort. Without some sort of focus, you could
study forever and still not be in a position to research 
anything. But you can attain a decent level of competence
that if and when something strikes your fancy, you will be
able to get yourself up to speed; I guess that actually
doing this once is what is involved in the "post-Masters"
part of the Ph. D. 
The other question I have is why you stated an aversion to
college textbooks. After all, in an ideal world colleges
would use the texts which would get you prepared for a 
Doctorate. However, it is true that many such books are
mediocre. 
In any case, you want to learn stuff, and you want 
motivated, hopefully "genetic" (in the sense that it 
teaches you how to think about the material) material.
Let me think. You have Calculus, which is good.
The major areas of knowledge which you will want
to gain understanding in are Algebra, Analysis, and Topology.
Actually, to see calculus done right, look at Spivak's
"Calculus" which is awesome, fun to read, and in the
back has a lot of hints for further reading, some of which
are still quite relevant after 25 years.
First, Analysis. It will be Calculus done right, basically.
A good book to learn things Rudin's "Principles of Mathematical
Analysis". It is not a cakewalk, and you may want to supplement
it with something else, like maybe Ross' "Elementary Analysis",
 but I am budget-busting here. Then you will want to learn some
 multivariate calculus, say Spivak's "Calculus on Manifolds".
At this point, you will want to learn some complex analysis, 
which is amply done out in Conway's "Functions of One Complex
Variable" or Ahlfors "Complex Analysis". Another nice book set, 
not quite complete but very pleasant, is Knopp's "Theory of
Functions" series from Dover. 
Oh yes; before I go on I should mention Dover, if you don't already
know. They are a mainly reprint publisher, and have a lot
of good, cheap, textbooks. If you are on a budget crunch, you
can get up to speed on almost all the subjects I have or will discuss
in this letter from Dover at a decent price (6 to 10 dollars a book).
They are definitely good as ancillary texts, to go look up something 
that you didn't understand somewhere else.
You will likely also want to learn some measure theory. A good place
to learn this is Cohn's "Measure Theory", or Rudin's "Real and Complex
Analysis", though the latter requires a lot of thought and is not 
genetic to any great degree. Also there are references in the books
above.
Ok, on to Algebra. This is a big subject, and has all sorts of
neat nooks and crannies. You will probably want to learn some elementary 
number theory in order to get started. There are lots of books here, 
Niven and Zuckerman (and Montgomery) "An Intro. to the Theory of Numbers"
being a nice, though quite packed, text. Another nice one is Davenport's
"The Highe Arithmetic", a Dover book, and another Dover one by Beiler called 
"Recreations in the Theory of Numbers" which is very elementary but fun
reading.
Then on to algebra proper, which is usually ordered Groups, Rings, Fields, 
Modules, or something like that. With some experience in number theory 
it will be fairly easy to get a feel for the abstractions involved in
those topics above. One way to start would be with an introductory text
like, say, Clark's "Elements of Abstract Algebra" (a Dover book) or Herstein's
"Abstract Algebra". Then you would move up to a graduate text, like
Hungerford's "Algebra" or Grove's "Algebra" or Jacobson's "Algebra", the last
of which is probably the most detailed (in the sense that every detail is 
spelled out, which is nice the first time through but can obscure the essence
at times; each, however, leaves out something which you probably should know.
This will probably take care of you in the algebra department unless you find
something that strikes your fancy, at which point you can start looking for 
some other references.
You probably will want to learn a little more number theory too, but that
can come later. 
The other major thing is that you probably want to learn some linear algebra
from the easy point of view, before tackling it from the "modules over a PID"
point of view that the books above will do. A nice, easy book to read is 
Smith's "Linear Algebra", and a classic of exposition is Halmos' "Finite
Dimensional Vector Spaces". 
Then there is Topology. You will probably want to know some point-set topology
to get started; some is covered in the analysis books. For more, there is 
Munkres' "Topology" and an assortment of Dover books. Be warned that no 
topology book is perfect, and a lot of them are pretty mediocre.
Then you will probably want to learn some differential geometry, manifolds, 
etc. A good thing to look at is Spivak's "Comprehensive Intro. to Diff. Geom"
which is huge but nice. A shorter thing, with which I have little experience,
is Warner's "Intro. To Diff. Manifolds and Lie Groups". For more
classical material, which is good to be a little acquainted with, there
is O'Neill's book, whose exact title I forget but is something like "An Intro.
to Diff. Geom.", and also Struik's "Intro. to Classical Diff. Geom.".
Then there is algebraic topology, which is really nice stuff. Good references
are not easy to find because the material is technical, though the idea is
not so hard usually. Greenberg and Harper's "A First Course in Alg. Top." is
decent, and Massey's "Alg. Top.: an Intro." and its revision, "A Basic  Course
in Alg. Top." are also very nice. 
Come to think of it, Henle's "A Combinatorial Introduction to Topology"
struck me as a decent book when I looked at it, and was fun. 

I'm sure I have been quite "traditional" and if I thought about it I could
think of more non-traditional but educational books like Bruce and Giblin's
"Curves and Singularities" of stuff like Arnold's "Ordinary Differential 
Equations", but that's more specific stuff I guess.
I think I have covered the ground of a decent program up to about a Master's
level. Above that, it becomes a matter of what you want to do. That will
dictate your choice of texts and papers to read. Usually an advisor with
knowledge of the literature becomes very, very helpful at this point.

By the way, all of the above is very IMHO, but opinions on books are always
welcome; it's one of my favorite topics! Hope this helps!
Francis Fung
Xref: sparky sci.math:17458 sci.physics:21815