Should I Become a Mathematician?

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  • #1,951
mathwonk
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in my opinion, your advisor's opinion as to where you are likely to get in is more reliable than a magazine's data. this is more accurate than what is obtained from fact sheets. basically we believe each other when someone tells us. "this is someone that you will be glad to have in your program."

My experience on this forum is one of saying over and over to students that real qualifications matter more than paper qualifications, and not being believed.
 
  • #1,952
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I see. Well, I had intended to ask one of the two professors who I feel have had the best chance to gauge my abilities/weaknesses and I will do so soon. It really isn't so pressing as I am still a sophomore, however, its good to know what is realistic to expect.
 
  • #1,953
mathwonk
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well I'll tell you my own experience. I was a math major at an Ivy league school, and kept pretty much to myself, never hanging around the math department or meeting the profs, and didn't even know my own advisor, as he was a grad student who declined to respond to my phone calls.

When junior year came around I worked hard to reposition myself by taking advanced honors calculus from the clearest lecturer in the department, and I studied hard and memorized proofs and did homework, and got one B+ and one A-. This was pretty good in those days, but not enough to put me in the company of people getting into top grad schools like Princeton or Harvard.

My advisor at the time, a full prof, advised I take two grad courses as a senior, to impress admissions people at grad schools as much as possible. So I signed up for these two grad courses. But the work load was more than I was ready to gear up for so I dropped one, and focused on the other.

I was still not a real student who tried hard to understand everything and spent lots of time thinking abut the material. i still had my social life as main priority. So I left studying to the end and tried to do a lot of memorizing at the last minute.

So I got one A in one grad course and went for advising. The prof was disappointed that I had only taken one grad course, and said I might try applying to Columbia and Maryland, and he recommended Brandeis, a school I had barely heard of.

When Eilenberg interviewed me at Columbia he was shocked that I had not taken the algebraic topology course from Raoul Bott, whom he regarded as a "God". He asked me to generalize the one thing I had learned from Bott on the first day of class, the definition of the zeroth homology group, and wanted me to guess the definition of the nth singular homology group (something I later learned was introduced by him).

I failed miserably and he told me that Columbia students knew a lot more than me, but he thought I might make it at a big state school like Maryland, and that Brandeis was probably going to expect too much too soon from me.

So I got accepted at Brandeis, and when I asked him for advice, he said grab it. So I went to Brandeis, and I think I did know less than my peers, but to be fair they regarded me as one of the stronger students in ability.

So I probably could have done well there but lost focus again during the vietnam war and wound up sliding down the greasy pole again.

Finally, years later, after a 4 year teaching career, and with new found commitment from supporting a young family, I enrolled at Utah, and again found it very challenging, even though again they regarded me as one of the top students.

So even for students regarded as good, grad school is still very hard, but professors who speak to you for a while do feel they have a sense of how strong you are and where you can succeed.

My story has an afterword as well, in regard to Eilenberg. After a 5-10 years of hard work, I became a postdoctoral student at Harvard, and when Eilenberg came nearby to speak, I introduced myself as someone he had interviewed years before. When he learned I was then at Harvard, he immediately apologized for not admitting me, as if he assumed my presence at Harvard 15 years later was evidence he had assessed me incorrectly!

So you see there is no cut and dried process of deciding where to go, or if there is I have not been part of it. The point is to prepare well, commit sincerely to hard work, and make the acquaintance of some professors who will know when they talk to someone how sharp and/or knowledgeable he/she is.

Nowadays there is also the existence of material on the web, such as the advice on terrence Tao's page and that specifically for grad students on Ravi Vakil's page at Stanford. Ravi e.g. said a few years ago that any potential students wanting to work with him should have worked pretty much all the exercises in Hartshorne's algebraic geometry book, which gives you an idea of how much time you should have spent preparing to go to Stanford. The one person who told me he had done those exercises was a grad student at Harvard and finished there and went on to become a well known mathematician.
 
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  • #1,954


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?
 
  • #1,955
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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.
 
  • #1,956
mathwonk
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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.
 
  • #1,957


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?
 
  • #1,958
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I recommend What is Mathematics? by Courant. It also covers a lot more topics, but definitely more than worth every cent.
 
  • #1,959
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You say you don't like Dummit and Foote mathwonk, so what book(s) would you recommend instead?
 
  • #1,960
mathwonk
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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.
 
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  • #1,961
mathwonk
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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.
 
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  • #1,962
morphism
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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.
 
  • #1,963
mathwonk
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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.
 
  • #1,964


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?
 
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  • #1,965
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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.
 
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  • #1,966
mathwonk
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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.
 
  • #1,967


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?
 
  • #1,968
mrb
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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.
 
  • #1,969
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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.
 
  • #1,970
mathwonk
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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
 
  • #1,971
mathwonk
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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.
 
  • #1,972
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is that from the general GRE, or the Math GRE?
 
  • #1,973
mrb
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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.
 
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  • #1,974
mathwonk
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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.
 
  • #1,975
mathwonk
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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 [Broken]

"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.
 
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