Other Should I Become a Mathematician?

[zarathustra00]

Im an undergrad math student struggling with proper motivation and now getting low marks. Like a lot of people, during high school, I got bored and picked up bad habits. Now, I think I just lack discipline and fail on every level.

I do not wish to give the wrong impression. you asked about the negative period of my career, and i related it, but it is very hard and unusual to come back from such excesses and neglect. it is better to avoid them.

I worked very very hard for years after that, beginning roughly in 1970, to retrain myself, in mathematics and self discipline. I managed to do so intensively over the period from 1970 to 1981, working almost every day, sometimes up to 20 -30 hours at a time

If I may ask, how did you pull it off?

 

[Bourbaki1123]

Mathwonk,

You bring up a point that may be a weak point for me: many students take grad courses as undergraduates, however; my school does not offer graduate math classes. I would certainly be able to take some next year(I'll be a junior), but my school simply doesn't offer them. I will do independent study in algebraic geometry, but I would like to get some grad courses under my belt.

Is there any way to do grad courses by correspondence? If not I will check with my professors and see if they know of anything. There is a nearby school with a decent graduate math program that has some sort of reciprocity agreement with my school, so I would imagine there might be a way to take courses over there.

I am studying from Dummit and Foote in my undergrad abstract class and I will have thoroughly been acquainted with the materials in Chapters 1,2,3,4,5,6,7,8,9,13,14(mostly through class exposure, some topics are skipped over and I go back and make sure I understand them and can do the exercises they cover basic finite group theory, sylow p-subgroups, fundamental theorem of finitely generated abelian groups, rings, fields, galois theory ect. ) and I will cover the material on commutative algebra and Category theory on my own/with the professor. I will probably make a point of going back and learning Ch.10-12(mostly matrix theory from a group theoretic perspective). So I feel that I will have a good base for a grad level group theory/abstract algebra course by the end of this school year.

 

[mathwonk]

if you just complete dummitt and foote you will have a graduate algebra course in my opinion. that is not my favorite book, but to be honest it really covers a lot of good stuff and has great problems, so do huge numbers of the problems, and you will be better prepared than most beginning grad students.

 

[Wretchosoft]

Hi,

At the moment I'm a college math student with too much time on my hands. I'm taking the hardest math course available to me (as a freshman) and, while it is challenging, I want to do more. I'm essentially locked out of upper-division courses for the remainder of this year, so I want to take the opportunity to cover up some weak spots.

1.) I'd like to brush up on my geometry, but I don't want a dry book that treats the subject with an emphasis on rigor, complete derivation from axioms, etc. That probably sounds like I've precluded the suggestion of any geometry book. I guess what I'm looking for is a book that shows how to approach and solve problems in geometry. Geometry is not my favorite mathematical topic, but I want to be well-rounded.

2.) I've borrowed a book on discrete mathematics primarily for the material on modular arithmetic, recursive sequences, and counting. It's basically a collection of lecture notes, but it's well conceived. Can anyone suggest some more good books aimed at the lower level that cover these topics, as well as some light number theory?

 

[Vid]

I recommend What is Mathematics? by Courant. It also covers a lot more topics, but definitely more than worth every cent.

 

[qspeechc]

You say you don't like Dummit and Foote mathwonk, so what book(s) would you recommend instead?

 

[mathwonk]

as to D-F, i think what i said above was " its not my favorite". its a very good choice for most students.

I happen to be the kind of person who picks on small flaws, even if the book as a whole is excellent for learning.

learning is an organic process, not at all cut and dried, so you should pick the book that speaks to you.

As you probably know, I myself have written four algebra books, available free on my dept web page (notes for math 4000, 4050, 843-4-5, 8000). try those if you want.

I like books that have the stamp of a master, like those by jacobson, bourbaki, mike artin. and there are topics, and honest attitudes, in lang that are hard to find elsewhere.and for good geometry books try hilbert and cohn - vossen, thurston, hartshorne's geometry euclid and beyond. or euclid himself, and archimedes.

 

[mathwonk]

here something i only know how to find in DF: a noetherian ring which is a one dimensional unique factorization domain, is in fact a principal ideal domain.

the converse is easy to prove, the direction stated above not so easy. It is not hard to prove a height one prime ideal in a ufd is principal. [take any element of it not zero. factor that element into irreducibles, i.e. primes. then by definition of a prime ideal, some one of those factors is in the ideal. that factor generates a prime ideal, which by hypothesis must equal the original ideal ,since height one means it equals any non zero prime ideal it contains. thus the opriginal ideal is principal.]

then the result is to prove that if all primes are principal, in fact all ideals are principal, which i guess is where the noetherian hypothesis is used. anyway, this is "proved" as a guided exercise, in D-F, and I recall needing to use the prime decomposition theory to prove this myself in grad school.

also D-F does a good job of laying out clearly what you need to check to know a group is a semi direct, or direct, product of two subgroups. It will say clearly: check that one subgroup is normal, and that..., then you know you have a semi direct product. this kind of thing is good pedagogy, as opposed to what i tend to look for, which is good deep mathematical insight. but after the fact, i admit to myself that their pedagogy also helps me when teaching and learning the topic.

 

[morphism]

mathwonk, I just came across the following set of lecture notes of Artin on noncommutative algebra: http://math.mit.edu/~etingof/artinnotes.pdf. Since you are a fan of Artin's algebra book, I thought you might find them interesting.

 

[mathwonk]

thank you. I never liked non commutative algebra much, so maybe i could learn some from Mike's notes. Of course as I have recently noted, most of algebra is about either commutative groups, or non commutative groups of automorphisms of them, so you cannot really avoid non commutative algebra, matrices e.g.

 

[PhysicalAnomaly]

I have noticed that the GRE includes a wide range of topics including stats. If I don't take any stats units, will it make a big difference?

PS Also, how long ahead of doing a PhD must one apply and take the GRE?

PPS Does a PhD in the US take 3 years or 5 years?

 

[Bourbaki1123]

PhysicalAnomaly;

I can tell you that my professors advised that I take the (I am assuming you are talking about the math subject gre) GRE either Spring of my junior year or fall of my senior year(which seems a bit late, I'm not sure how grad admissions works though).

As far as length, all that I have heard indicates that it varies pretty widely. You might be someone one can break out a good idea in two years or it might take close to seven or eight. I have heard of people taking various times within the two to ten year range to complete their phd's. I believe that the average is around five or six though.

mathwonk,

One of my professors recommended Goursat's Complex Variables text in order to gain a view of complex geometry more suited to Algebraic Geometry. I was wondering what you think of Goursat's texts and whether there might be additional texts that I might look at that focus on the aspects of complex analysis that carry over to modern algebraic geometry.

 

[mathwonk]

goursat's texts are superb. the great mathematician arnol'd has recommended them in recent years and as a result i bought all three volumes. i have not read much so far. these are very old and very high level.

among modern books (i.e. only 50 and not 100 years old) i like cartan.

 

[PhysicalAnomaly]

But I've noticed that the GRE involves topology, analysis, algebra, pde's and stats. Do students usually have a firm grasp on all of it by their junior year? And what if they don't do any pde's or stats?

 

[mrb]

PhysicalAnomaly:

I've seen 2 former Math GRE exams and several practice exams and I don't think I've seen PDEs on any of them. The topology and algebra on the tests seems pretty basic. I don't recall any real statistics questions. As for whether juniors have a firm grasp on those topics, at least at my school, the answer is no, not even remotely close. But then again neither do the seniors. FWIW, I freaked out at the beginning of this semester when I first saw a sample Math GRE exam and realized I was not at all prepared (and I posted about it here...). After this semester including an Analysis class, Algebra, Topology, and study on my own, I feel much, much better about it. Just make sure you are staying on top of things on your own whether your classes are or not.

 

[morius]

I'm an undergrad game programming student and I would like to get a maths degree as well (online, since I'm actually studying abroad and it would be just too complicated for me to try and get an on-campus degree). This is since I really like learning, mathematics, and learning mathematics.
I haven't actually found too many online maths degrees, so I was wondering if you could recommend me one or at least tell me what I should be trying to find in it.

 

[mathwonk]

look guys, here is an extract from the official site for the gre. it is utterly trivial looking stuff, nothing like pde:


The Math Review is designed to familiarize you with the mathematical skills and
concepts likely to be tested on the Graduate Record Examinations General Test.
This material, which is divided into the four basic content areas of arithmetic,
algebra, geometry, and data analysis, includes many definitions and examples
with solutions, and there is a set of exercises (with answers) at the end of each
of these four sections. Note, however, this review is not intended to be compre-
hensive. It is assumed that certain basic concepts are common knowledge to all
examinees. Emphasis is, therefore, placed on the more important skills, concepts,
and definitions, and on those particular areas that are frequently confused or
misunderstood. If any of the topics seem especially unfamiliar, we encourage
you to consult appropriate mathematics texts for a more detailed treatment of
those topics.


TABLE OF CONTENTS
1. ARITHMETIC
1.1Integers..................6
1.2Fractions...................7
1.3Decimals...................8
1.4Exponents and Square Roots............10
1.5Ordering and the Real Number Line............11
1.6Percent...................12
1.7Ratio...................13
1.8Absolute Value................13
ARITHMETIC EXERCISES.............14
ANSWERS TO ARITHMETIC EXERCISES.........17
2. ALGEBRA
2.1Translating Words into Algebraic Expressions.......19
2.2Operations with Algebraic Expressions.........20
2.3Rules of Exponents...............21
2.4Solving Linear Equations.............21
2.5Solving Quadratic Equations in One Variable.......23
2.6Inequalities..................24
2.7Applications.................25
2.8Coordinate Geometry...............28
ALGEBRA EXERCISES...............31
ANSWERS TO ALGEBRA EXERCISES..........34
3. GEOMETRY
3.1Lines and Angles................36
3.2Polygons..................37
3.3Triangles..................38
3.4Quadrilaterals.................40
3.5Circles..................42
3.6Three-Dimensional Figures.............45
GEOMETRY EXERCISES ............... 47
ANSWERS TO GEOMETRY EXERCISES ......... 50
4. DATA ANALYSIS
4.1Measures of Central Location............51
4.2Measures of Dispersion...............51
4.3Frequency Distributions..............52
4.4Counting..................53
4.5Probability..................54
4.6Data Representation and Interpretation.........55

 

[mathwonk]

i never took the gre myself, but back when i was an undergrad, a friend took it and told me there was one question on abstract algebra, this one:

" which of the following is possible for the order of a subset of a group of order 12:
i) 5, ii) 6, iii) 7, iii) 8?"

everyone i have asked this question of has gotten it right, and none of them has known what a group was.

the only question on topology was this: " which of the following subsets of the real line is connected?"
i) the two point set {0,1}, ii) the set of rationals, iii) the interval (0,1), the set of all positive and all negative reals?

same result here. everyone i have asked this of has got it right and none of them knew what "connected" meant in the topological sense, indeed no one i have asked had studied advanced math at all.so in my experience these tests are for mathematical imbeciles. maybe they are harder today, but i doubt it, since the official review above concerns only precalculus. but you see why they have little use in determining who can get a phd in math.

 

[Proton Soup]

is that from the general GRE, or the Math GRE?

 

[mrb]

What mathwonk quoted in his "look guys" post above is most definitely NOT from the Math Subject GRE. In fact it even specifically says "General Test." These are two very different tests.

Moreover, while someone could reasonably say that the algebra and topology on the math GRE is basic (in fact, I just did a couple posts ago), there are 4-5 questions from each topic on my sample exams, not just one, and none of them are trivial in the way mathwonk's questions listed there are.

 

[mathwonk]

they are not MY questions, they come directly from the gre website, or from my memory. you never said math gre or subject gre, just gre. please show me the questions you think are non trivial from whatever gre test you are interested in. i am not trying to trick you. i will be happy to learn as much as possible about the gre. it just doesn't impress me much from what i know so far.

 

[mathwonk]

ok guys thanks for wising me up, i have now found the math test i think, and here is a question in abstract algebra from

http://www.ets.org/Media/Tests/GRE/pdf/Math.pdf

"which of the following is a subgroup of Z:

i) {0}, ii) {n: n≥0}, iii) n even, iv) n divisible by both 6 and 9, v) Z.

now that really isn't much harder than the one i recalled from 45 years ago is it?

or am i still on the wrong test?

the other questions were mostly basic calculus questions.

ok i finally found a complex contour integral, and some basic little herstein problems like showing a group with x^2 = x for all x is commutative (which takes about 3 minutes to do in your head by squaring xy), but there ARE A LOT OF PRETTY EASY things like the high school rational roots theorem. it really is a mix but mostly elementary stuff lots of high school students could do.

 

[mrb]

Sure, the material on the test is not extremely advanced. But it is also not a cakewalk for most math undergrads. In a perfect world every undergrad would have decent classes in abstract algebra, topology, and analysis before taking the test. But it doesn't always work out like that... What tends to be difficult for me is changing gears quickly. "Compute this line integral. Now find the error in this proof about compactness. Now determine when this sequence converges. Now answer this question about complex analysis, which you haven't taken. And do each in about 2 minutes."

 

[mathwonk]

heres another one from the website above: if the domain of a continuous function is a finite open interval, then the range is which:
i) an interval, ii) an open interval, iii) a finite interval?this is again something any layperson who has seen trig (graph of sin and tangent) could do.

but my point is I believe these are not taken seriously by good schools as having much bearing on readiness for grad school so don't stress out over it. when i applied to columbia, brandeis and maryland, none of them required this test so i didnt even take it. i presume it was because they considered it irrelevantly trivial.

well things have changed, columbia now requires the gre general and math subject. what used to be trivial is now required. or maybe i just got a pass somehow.

 

[Proton Soup]

my old school did not require GRE or even engineering GRE to get in engineering grad school. you could take the Miller Analogies Test. I'm not even sure that was required if your GPA was high enough, but my memory is fuzzy now.

 

[mathwonk]

well of the three schools i applied to in 1965, two now require the gre and subject test, columbia and maryland, but not brandeis. maybe i got into brandeis with a nice fellowship and just skipped taking the gre.

but really, i was talking to an undergrad the other day who is planning to apply to some good schools, and he is taking our graduate analysis course and knows it well enough to explain things to me that i do not know well.

this is the kind of student who will get into a good program, certainly not someone for whom mickey mouse questions about first year calculus area calculations are a challenge, such as one sees on the gre.

when i interviewed at columbia i was thought ignorant and denied entrance because i did not know and could not recreate spontaneously the theory of singular homology, not based on some trivial first year undergraduate topic, or worse yet some high school topic like which of these rational numbers could be a root of this equation.

if you want to go to a good math grad program, you should be able to ace gre's, but they will not determine much about your readiness, nor your competitiveness with strong applicants.

one of our best young products a few years ago was a high school student who took our grad courses as a high school student, finished high in the putnam competition, and then went to an ivy league school for college and aced the most advanced courses.

those kids are the competition at top grad schools like princeton, not people who are puzzled by questions on "which of these multiplication tables gives a group of order 4?".

still there are a lot of programs which do need all the applicants they can get, and even at columbia there is a reason the tuition is free for phd candidates.

the fundamental tools of a mathematician remain linear algebra and advanced calculus, so try to learn those well. mastering hoffman and kunze , and spivak's calculus on manifolds is a good project.

gre's are there to weed out the totally unqualified i imagine, not to determine the top candidates.

 

[mrb]

So the message I've gotten from you over the past few posts is that anyone who cannot currently dominate the Math GRE is a moron and should not even bother trying to become a mathematician. Could you possibly be any more elitist and condescending?

Here's the reality for me: math was extremely easy for me in elementary & middle school and after a lot of begging my mother convinced the school to let me advance a grade... which turned out to also be really easy. Nobody was around to show me more advanced math or point the way (and this was in the early/mid 90s, so looking online wasn't an option) so I just stopped going to school and barely graduated. ~10 years later I started undergrad as a math major, and just recently realized how weak the math program is at my school and how unprepared I was. So now I'm doing a lot of study on my own and have learned a lot in the last few months (3 months ago I didn't know what a vector space was... because vector spaces were only mentioned once in my linear algebra class). So no, I cannot ace the Math GRE at the moment but I am very confident in my talent and know I will be able to before too long.

You should be a little more careful throwing around terms like "mathematical imbecile" because you are insulting probably almost everyone who has posted in this thread (look at the table of data on Math GRE scores and realize that most prospective math grad students get only about half or fewer questions correct).

 

[mrb]

Since you did undergrad at an Ivy League school, perhaps you are not aware that most undergrad programs are less than ideal and the fact that students come out of them with a weak background does not necessarily imply the students are stupid.

 

[JasonJo]

I just got used to the elitism in math and physics. I don't accept it, but these are field where this kind of thing is prevalent.

If you like doing math, keep trying to be the best possible mathematician. That's all I try to stick to now.

 

[PowerIso]

Math GRE is over hyped. I tend to agree with mathwonk's assessment of the test . It is essentially a cutoff point.

mrb if you feel you will one day be able to dominate the GRE than you should relax a bit. I believe Mathwonk was simply saying that if you can't do well on the GRE then you probably are not ready for graduate school and you have to admit right now at your level you are not ready for graduate school. One day, sure, but not right now.