Should I Become a Mathematician?

[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.

 

[Bourbaki1123]

I think that one can often overcome the weakness of his school's math program. My school does not have the strongest math program, it is really quite weak(actually not as weak as mrb's, we covered everything in chapter 1-6 or so in David Lay's book. You must not have even covered eigenspace, nul space,collumn space, rowspace ect. if you didn't cover vector spaces.). I took the matter into my own hands and started self study in tough books before I even started college before I was really ready for them so I had to put them down,but; if you really love the subject you will not give up on a difficult aspect, you go back again and again until you can muscle through every concept and problem.(I'm talking about books by authors like Rudin, Goursat, Lang, Artin). I also make it a point to get to know my professors and ask less trivial questions that I might have encountered in my self study. I also ask for a lot of advice as far as what I should be doing, what books will prepare me well for grad school, ect.

I don't see why a weak math program would hold back a strong student. I can see how time constraints might, if one does not have the time to supplement their courses. If, on the other hand, you find a weak program too difficult to juggle with outside study, you might not be cut out for grad school, I don't know. Working might be a mitigating factor ect. There are many things which can put a hold on extra study. Hanging out with friends too much is something that might have to be sacrificed.

It seems like it would really depend on how strong the grad school is. For instance, UC Berkeley's math grad program has a very high drop out rate. This is a very difficult program that only people who can ace the Math Subject GRE and were published as an undergrad or something along those lines can do well in.

My personal circumstance is kind of similar to mrb's. I studied algebra in elementary school and became very interested in it and picked up concepts quickly, but much to my dismay, every year of advanced middle school math and basic high school math was essentially the same and I quickly became disinterested. The same thing was true of science. My 7th grade science teacher was a soccer coach or something along those lines and he would have no answers to my questions pertaining to astronomy or physics.

I did poorly in high school(not terribly, but a 2.98 gpa) and had intended to do art or music, which are two other passions of mine, but my interest in science was rekindled by my Honors physics teacher senior year(a friend told me that I should take the class). I started to self study calculus, analytic geometry, and trig, because I had not gone beyond geometry and algebra. I got a 1400 on my SAT but went to a local state university because I was wary of my math skills at that point. I nearly tested out of calc I, and could have skipped the first segment(the split it into two segments Calc I a and b) but decided not to. Instead I took the extra time I had since I knew the material in class to work on less trivial problems and study some theory. I looked at some of Apostle's book and did some problems and that really gave me a bit of an edge. Now I am a sophomore studying Rudin and Goursat on my own and I will have exhausted my school's math curriculum as of next year and will have to try to do courses at a nearby university and independent study.

The point is, if you really love math, and you have any spare time, read math, do math and talk about math to anyone you can. You don't necessarily have to abandon your social life totally, but at the same time, don't spend every waking hour hanging with your homies.

 

[PhysicalAnomaly]

The syllabus was very intimidating. But I just found their sample questions and was quite surprised - at least half the questions could easily be done with high school stuff and the most of the rest were very guessable considering that they furnish us with an explanation before they ask the question. And it's all multiple choice! I haven't had multiple choice maths questions since... I can't remember. :D

 

[mrb]

(actually not as weak as mrb's, we covered everything in chapter 1-6 or so in David Lay's book. You must not have even covered eigenspace, nul space,collumn space, rowspace ect. if you didn't cover vector spaces.).

Vector spaces were maybe not the perfect example. We also used Lay and covered through part of Chapter 5; the big problem was that this was a summer course lasting barely over a month so we got less than a week on Ch 4. A concept you work with for 3 or 4 days and then don't ever hear about again for a year tends not to be retained.

Here's another example, though: I was through Calculus 3 and Diff Eq before I ever heard of the Mean Value Theorem or the Intermediate Value Theorem (which I first read about on this forum, and then on my own from Spivak's Calculus, and finally just recently in my Analysis class).

There really is no avenue at this school for excelling. I realized last semester that my advisor was suggesting the classes she was for me based on the fact that they were easy, despite my clearly telling her I was interested in grad school and wanted to learn and my 4.0 Math GPA. We supposedly have a Math Honors program which involves undergrad research. I have been trying for 3 weeks to find a prof to be my advisor for the program. None of the profs I've asked knew there was a Math Honors program or had ever advised anyone for it. And none have been willing to advise me except possibly one, and he seems reluctant because he has admin duties and worries about his time. So I honestly don't think there really IS a Math Honors program; it's just something they put on their website and other materials because it looks good.

The point is, if you really love math, and you have any spare time, read math, do math and talk about math to anyone you can. You don't necessarily have to abandon your social life totally, but at the same time, don't spend every waking hour hanging with your homies.

I agree 100% but it took me a while to realize how much I should be studying on my own, partly because I previously had not decided with certainty to do math grad school. Social issues are, ahem, not a problem for me. I have no social life except for a gf who is generally tolerant about me spending hours and hours on math.

 

[Bourbaki1123]

Thats pretty bad. I had at least some knowledge of the IMVT and MVT in Calc 1 and definitely by calc 2. It seems necessary to give some basic results, I don't know if the teacher simply glossed over how most formulae involving it are derived? How did you guys go over the fundamental theorems of calculus and taylor series?

I guess it really isn't a big deal since you know it now and grad schools probably assume that most early calc courses are the same or generally don't care.

My school doesn't have a honors math program either.

A couple questions, directed more towards mathwonk,

Do you think that Bartle is a good Analysis text, and do you know much about Paul Dienes Text The Taylor Series (i.e. how would you rate it and why.)

 

[mrb]

Thats pretty bad. I had at least some knowledge of the IMVT and MVT in Calc 1 and definitely by calc 2. It seems necessary to give some basic results, I don't know if the teacher simply glossed over how most formulae involving it are derived? How did you guys go over the fundamental theorems of calculus and taylor series?

Nothing was ever derived. FTC and Taylor Series stuff was presented but not proved or even informally demonstrated.

 

[mathwonk]

I was responding to some posts in which people said essentially: here are my gre scores, where am i going to get in? princeton? washington? etc etc...

i am saying that gre is not a big factor in getting into the best places, that anyone struggling with the gre is simply not going to get in those places.

i hope that is useful time saving information to people thinking of applying to princeton. namely if you think the gre is hard, don't bother. princeton IS an elite school.

some students come into harvard having already read and worked through books like griffiths and harris algebraic geometry. my first advisor came to columbia having already proved the riemann singularity theorem in a rigorous way for the first time by anyone in over 100 years.

this does not mean someone who has to work to do well on gre cannot find a home where they will fit in well, but it won't be at harvard or princeton or mit or columbia.

I think I made it very clear that my post was aimed at people who want to know how to tell if they are going to get into the very best schools. i say that unless you find the gre easy, you are not going to.

when i was an undergraduate at harvard, the only people even applying to harvard grad school had already taken year long graduate courses in algebra (lang), algebraic topology (spanier), real and complex analysis (big rudin, cartan, ahlfors) as undergrads.

the rest of us looked elsewhere.
...sorry i am not familiar with bartle and dienes. bartle is a familiar name though, so probably has a good track record.

 

[PhysicalAnomaly]

Heh, same thing here. Nothing is derived and IMVT and MVT weren't taught in any of the calc units. They present stuff like Stokes's and Gauss' theorems but aren't very clear on what they actually mean much less derive it. They've neglected to mention Green's theorem or Fubini's - we just change the order of integration as we wish. But that's not really important. It'll be covered again in real analysis anyway, right?

 

[PhysicalAnomaly]

when i was an undergraduate at harvard, the only people even applying to harvard grad school had already taken year long graduate courses in algebra (lang), algebraic topology (spanier), real and complex analysis (big rudin, cartan, ahlfors) as undergrads.

Australian unis don't really offer that kind of courses. The most I can get is a year's worth of grad analysis and algebra with a mix of self-study, honours year units and maybe an exchange. Does that mean that the top universities won't be an option?

 

[mathwonk]

not necessarily. those were options at harvard so they expected the harvard undergrads to have taken them. talented people from less rich environments could be cut more slack.

but remember that was long ago. things are different now. but harvard students are still very very sophisticated and advanced.

only one thing is sure however, as a friend reminded me about applying for anything:

"if you do not apply you definitely will not get in."

 

[mrb]

But that's not really important. It'll be covered again in real analysis anyway, right?

At my school the standard undergrad real analysis is a 2 semester sequence, with single variable topics covered first and multivariable covered next. Unfortunately the single variable portion is never completed in the first semester, so the second semester is mostly spent doing what should have been done the first semester.

I don't want to drag this thread away from its purpose, but as long as I'm complaining about my math education, I want to provide this contrast:

When I thought I wanted to go into Bioinformatics, I emailed the coordinator in that department who invited me to come to his office. I met him, he told me about Bioinformatics in general, and about each of the profs and what their research was on and so forth. I contacted 2 of the profs about doing research with them; met with both of them; both offered to let me work in their labs. I chose one of them and had a rewarding semester.

On the other hand, now I want to do some math research. I have talked to four professors about it. One met with me and it went like this:

Prof: "Well, I'm a numerical guy, so you would have to be able to program to work with me, I'm sorry."
mrb: "I can program. I've been programming for years."
Prof: "Oh. Well, you would have to know C, so I guess..."
mrb: "I know C."
Prof: "Oh. Well have you had Calculus 4?"
mrb: "Yes."
Prof: "Have you taken Applied Math? [this is a course only offered every other year]"
mrb: "No."
Prof: "I'm sorry, but any work I would have for you would depend on that material, so I don't think we can do this."

He couldn't just say he couldn't do it, he had to search for some excuse. Another prof stopped responding to my emails after one reply. Another was enthusiastic and agreed to meet with me but then didn't show up and is now incommunicado. And finally there's the last one, who is still a possibility but as I mentioned above seems reluctant and since he hasn't replied to my email from a few days ago he may be going incommunicado as well.

 

[Vid]

Hmm...

I took Calculus 1-3 in high school, and we derived everything. We had a pop quiz on the formal definition of a limit for a quadratic. These wasn't an honors class this was one of the many sections of calculus taught at this school. If you didn't cover these things in a college course, something is very wrong.

 

[mrb]

Well, yes. I agree. There are some good people in our math department but something is broken at a very fundamental level.