Forum index » what are the best textbooks to study from? + study advice?

i want to go for a PhD in applied/interdisciplinary mathematics. i have a bachelors in math and a masters in math education. i wish to be a professor so that i may both teach and research math.

i have applied to 11 programs for fall of 2009. so far i've got 3 rejections, and some waitlists, and a few more no-responses that i'm waiting on. but overall, its just not looking too good at this point. a couple more of those are almost certainly rejections, and the state of the economy has: 1) decreased the available funding to programs and hence decreased the number of spots available, and 2) it has increased the number of applicants because of so many people wishing to ride out the economic crisis in grad school. 1 + 2 = harder to get into a good program for everybody except for the absolute top candidates.

i took the subject test once in november. i did very poorly. my score was below 40%. (a very small consolation is the fact i did get an 800 on the Q part of the general GRE.) i had studied for a solid 2 months. and i used the "cracking the gre math test" as well as my old calculus and linear algebra textbooks from my undergrad. i spent about half my time studying calculus and the other half studying a mix of linear algebra, a little bit of number theory and a little bit of abstract algebra.

truthfully, only 4 of the programs i applied to required the subject test. still, i get the feeling i may have to re-apply all over again next year. and i think i should retake the test.

a couple of problems that i had with the GRE subject test:

1. i didnt have NEARLY enough time. i answered only about 2/3rds of the questions.

2. i did not study my abstract algebra enough. in fact, i barely studied it and i didnt realize how much of the test would be about this topic. i also didnt even have a good text on this subject to study from.

3. so much uncertainy. it felt like so many of the questions were "trick" questions that required the memorization of one little odd fact that would cut the time necessary to do the problem substantially.


i really like my calculus book from college and it feels like such a great book, but it seemed like there were a number of 'trick' calculus questions on the GRE that i don't feel are really emphasized in my calculus textbook.


could someone recommend some actual textbooks (not study guides) to study from that cover the kind of questions that the GRE asks? (particularly abstract algebra)

and what are the best approaches to studying and preparing for this exam?

whatever i did the first time didn't seem to work, and i believe i have the capability to do so much better than what i did. (and hell, i already did one masters degree, so i know i'm not as stupid as the test results made me feel)

Since people on this forum are busy so I would like to give some suggestions :

could someone recommend some actual textbooks (not study guides) to study from that cover the kind of questions that the GRE asks? (particularly abstract algebra)

1. Cracking the GRE maths
2. Barrison's book which includes 6 practice test
3. REA GRE math practice tests http://www.amazon.com/GRE-Mathematics-REA-Best-Preps/dp/0878916377- If you can solve all the problems in this book you will definitely get good result
4. Some sample tests posted on this forum - you can find them out

For algebra section I strongly recommend the following book
http://www.amazon.com/Abstract-Algebra-2nd-David-Dummit/dp/0471368571

what are the best approaches to studying and preparing for this exam
it depends on yourself, if your maths background is good you should head to >90%

This forum is the greatest forum for GRE maths, discussing if you have questions

Thanks for the tip about Dummit's book. I haven't seen an abstract/modern algebra textbook this well organized. I was studying from Gallian's "Contemporary Abstract Algebra" and we used Joseph Rotman's introductory book which was really abstruse for the sake of being abstruse (and for the sake of using the word abstruse, the General GRE is on its way for me this summer!)

Anyways, here's a link I happened to come across

http://www.ebookee.com/Abstract-Algebra_134456.html

Dummit-Foote is MUCH more detailed than you really need for this exam. Gallian is sufficient and, actually, preferable, as it doesn't bog you down in details and actually provides solutions to its exercises.

Come on, you're preparing for a simple multiple-choice exam. Save Dummit for your phD qualifying exam preparation.

Ok, I got 97% in the test and got into some top 10 programs and this is my take on the test. I assume you have taken a course in real analysis at the level of Rudin's Principles, a basic course in topology and a course in abstract algebra? I would recommend the following reading (and doing most problems):

1. Dummit and Foote, Abstract Algebra Part I-II

2. Munkres, Topology, Chapters 1-3

Then you need to hammer daily on exercises from your thick Calculus brick. You need to be able to do integrals and solve basic differential equations in your sleep -- the test is mostly about speed as you probably know. Another option for topology is going through the first 30 pages or so of Bredon's Geometry and Topology, but I would recommend this only if you actually understand topology and just need to remember the details (it does in 50 pages what Munkres does in a few hundred, then jumps to algebraic topology).

For Lebesgue theory, it would be a good idea to read up to chapter 4 of Royden's Real Analysis. It also covers some interesting set theory regarding the axiom of choice that has actually been on the test a few times. This is a tougher text though if you haven't seen the math before. However, you should be able to do most possible test problems requiring Lebesgue theory by just remembering when a Riemann integral equals a Lebesgue integral and when you can exchange integrals and limits under Lebesgue theory (the exercises are always about changing to Lebesgue, doing the limit, converting back to Riemann plus they're usually trivial if you know this).

Finally, as you said the test is about tricks (at least to some extent). One of the tricks is to be able to come up with counterexamples quickly in order to eliminate answers. This means that you need to make a list of topological spaces, algebraic structures and functions with some "special" properties. These are often mentioned in text books as "canonical" examples. These examples should be memorized so well that you can immediately recognize them when they show up.

You should also think about general principles of how you should do certain classes of problems as fast as possible. For example when checking if something is a vector space, it's often fastest to first check if it has a zero element. This means that you should not test additivity first, because it generally takes more time. This also applies to abstract algebra where certain axioms are always quicker to check. You should think about the right order for all possible structures on the test and get your brain wired to automatically do the problems in this order.

I wrote myself with LaTeX a 10 page list of basic formulas and examples for matrices, eigenvalues, determinants, traces, differential equations, integrals, power series, topological spaces, algebraic structures and theorems that should be remembered. I was only able to spend about two weeks in total doing full time studying for the exam and spent most days just staring at the list and it proved to be a pretty effective approach.

Other than that type of test prep, did you have any particular strategy going into the test, or did you grind through the problems one after another?

On the practice tests, it appears obvious that the GRE people intentionally disperse and intersperse the calculus and algebra problems... and who knows when a combinatorics or topology problem will arise suddenly to scramble the mental circuitry after an easy calc deduction. I'm considering a strategy of grouping the problems together by calc/algebra/other on the actual test to maintain focus. Wise? Unwise?

I did first all the theoretical questions that could be answered in just few seconds. I got around 20 such questions done in 8 minutes. This way I knew that I could devote way more than 2 minutes on the other exercises. I don't think it really made much of a difference, but at least it boosted my confidence during the test.

With the level of theory required, you really shouldn't need to switch over from algebra to topology mode in your brain. Seems like you need to brush up on the material from your undergrad.

The topology problems were pretty much the easiest on the test. You only need to think about some examples of different topological spaces. Otherwise you pretty much only need to know that the preimage of an open set is open under a continuous map, and the image of a compact set is compact.

I think your strategy is the reason you did so well. I may follow suit.

I don't anticipate any questions relating to Imbedded Manifolds or Riemann Geometry (beyond the Riemann integral).

I had a 6 year gap between Calc III and Differential equations (2000-2006), so I've spent the last year reviewing multivariate calculus among other mathematical subjects. Most of the undergrad material is fresh in my mind.

I would say that an elementary knowledge of Topology is satisfactory. Munkres wrote a good book, but my school infrequently offers their Topology course, so I read Bert Mendelson's Introduction to Topology and a few introductory books on Chaos. I'm sure I'll get an opportunity to read Munkres inevitably though. His book appears to be the standard.

The algebra questions I've read from the practice tests are sort of "hit and miss" type questions. Either the answer is easily deduced from the choices or it requires finding the sets which satisfy group/ring/field properties.

eof: if you are still around, would you mind posting that tex file?

thanks!

Yes eof. Please post the tex file.
Pleeaaaaase.

Hello, everyone! Could somebody recommend a good book on Linear Algebra?

Oh come on, no ideas at all?

Well, there is Linear Algebra Done Right, which is okay.
From a "drill and skill" view point, Linear Algebra by Stround and Booth might be nice (having working over some of their Vector Analysis text).