Should I Become a Mathematician?

[mathwonk]

maybe not essential, but it helps. i myself would rather discuss math with friends than almost anything. it motivates me also to learn something to bring to the discussion.

having insights is the most exciting, but sharing them is big too. even perelman came over to the US to speak on his work. then he turned down the fields medal, so you might say all he cared about was doing it and talking on it.

 

[camalam]

Hey everybody, I'm just hoping for a little bit of guidance.

I'm currently a grade 12 student and am going to be applying to universities within the next few months. Math and physics have got to be my favourite subjects right now and I'm kind of interested in pursuing a math degree, but I have some reservations.

First, I've never been exposed to any university-level math; I've never done a proof in my life. Do you know of any texts that could introduce me to math beyond the high school curriculum?

Second, I'm interested in the University of Waterloo's math program, and would appreciate anyone's opinion of the faculty and of the program I want to apply to: http://www.math.uwaterloo.ca/navigation/Prospective/programs/math.shtml

Thanks!

 

[PhysicalAnomaly]

I find it to be essential. Textbooks tend to assume that students will have someone to guide them and to discuss problems with. And hearing it discussed really makes one enthusiastic. I'm being inspired completely by talking to one really good tutor who gets so excited everytime we discuss maths topics.

 

[arshavin]

Amongst people around me, there is no culture of intellectual inquiry. Of course this is true for society, in general. I guess if you had a group of friends with deep passion for math, then good things happen.

This is probably the downside of being educated in a place like Australia, where unless you're super gifted and people notice you, otherwise its hard to make mathematically minded friends.

 

[eastside00_99]

arshavin,

when i was an undergrad, I had few friends who studied mathematics. most of my friends were intellectually curious in other was. for instance, they cared about the enviroment, or learning a foreign language, or reading literature. its good to know people like this...one learns a lot from seeing what others really care about. but, when i got to grad school there was no shortage of people to talk about math with. in fact, that is what we spend most of our time at school doing...

 

[majesticman]

I pretty much rely on internet forums and a professor to get all my motivation from these sources...to find students interested in math on a deeper level is hard plus finding the time to sit down and chat about maths...so maths becomes one of many things that are chatted about including current affairs, sports (that's a BIG topic in Oz at all times) etc etc.

But i am pretty much used to it from school so i don't feel deprived but just take it as a fact of life. I only do maths because i feel good doing it anyway so my motivation i guess is more intrinsic.

 

[eastside00_99]

well, math research, at least today, seems to be increasingly collaborative. i feel like whenever i look a people's past papers, unless they are the top guy in the their field, half of them have two or three coauthors. it seems having some smart people around to talk to really does help.

 

[majesticman]

easside00_99 i don't think this phenomenon is restricted to math only...i was looking at some biomedical papers today (and certainly most of a papers i have seen on any subject) usually involve 2-3 authors if not more.

I guess that's where the power of internet comes in...now people can collaborate on projects without being in the same continent

 

[qspeechc]

A bit late on the topic, but I don't rate Halmos' Naive Set Theory too highly. Raher get a book that sets it all out in symbolism, in the standard defintion-theorem-proof format. Then if you have some cash doing nothing perhaps ge Halmos' book, but I wouldn't have it as a primary resource.

 

[PhysicalAnomaly]

I'm guessing that collaboration means they can come up with more new stuff more quickly, especially with the help of minds that think differently. If they don't do it fast enough, someone else might come up with it. And there's probably the pressure to publish tonnes every year.

But yeah, Australian university students by and large don't really care about their studies. What's that? Learning something that won't be covered in the exam? I can hear their unspoken (and spoken) vehement objections.

 

[qspeechc]

Here's a site with a guy listing good maths books by topic and level, naming plenty of free online books:

http://www.chinesepdf.com/redirect.php?tid=54695&goto=lastpost

I like the one bellow for a very short introduction to set theory, I found it exceptionally well written with helpfull diagrams and examples. Contains solutions to excercises:

http://www.cosc.brocku.ca/~duentsch/papers/methprimer1.html

 

[PhysicalAnomaly]

That's a great list. I've got some questions now:

1. Is Artin or DummitFoote better, especially for self-study? Or Fraleigh?
2. Is Simmons a topology or real analysis book and is it better than Munkres?
3. Rudin or Tao for a first course? Or both?
4. Good linear algebra text? Too many listed there... including AntonRorres which looks like the equivalent of Stewart's Calculus :(

Also, is Stewart Tall good for complex analysis?

 

[qspeechc]

Well here's my humble experience:

Artin is a wonderful book, with lots of insight, which is it's strong point. I haven't read Dummit & Foote, but it does cover more topics, and is in a more standard format. Fraleigh is a weaker book than both Artin and D&F. I own the third and seventh editions but I never read them, I don't like the way you're treated like a baby. Fraleigh does have some answers and hints at the back (3rd edition does, not sure about 7th ed.) Another good algebra book is Birkhoff & Maclane, I own the first edition, and it is not as difficult as Artin, but not as simple as Fraleigh, but provides good motivation for the topics and some hints/answers, but the questions are generally quite simple.

I own Simmons, and it doesn't cover all the usual topics one would cover in a real anaysis book like Rudin, it does cover some, but not all. It's sort of like a mixture of real analysis and topology and functional analysis I think. Munkres is a straight topology book aimed at undergrads or 1st year grad students.

Tao's books on real analysis are quite lenghty, which is why I never read them. Rudin is too dry for my liking, but some people love it. I much prefer Pugh's book "Real Mathematical Analysis" which is at a similar level to Rudin, but I believe much better.

For linear algebra I really like Axler "Linear Algebra Done Right" and Hoffman and Kunze. Ok I haven't read any other linear algebra books (besides my course notes), but I still really like those two books. I like Axler's determinant free approach, which I think helped me understand linear algebra more. Hoffman & Kunze is just a legendary book.

 

[morphism]

Simmons is at a higher level than Rudin - both in content and in style. I would classify baby Rudin as a calculus text, and Simmons as a real analysis text. So in this respect you can't really compare the two. In fact, most of the material in Rudin is a prereq for Simmons. Similarly, you can't compare Simmons to Munkres. While Simmons does have all the essential point-set topology results (in the first third of the book), this is only the first half of Munkres (and Munkres has a few extra topics here as well). The second half of Munkres is devoted to algebraic topology, which is something you won't find in Simmons; and the final two thirds of Simmons are devoted to functional analysis and elementary operator theory, i.e. topics you won't find in Munkres.

 

[mobiusdafrost]

is it possible to be an applied mathematician and do work related to physics and engineering? also does applied math at the grad level use more programming than physics and engineering?

 

[james007]

Thank you mathwonk for a very helpful and informative thread. I should have seen this thread earlier so that I can ask you questions about algebra. I have no intention to become a mathematician because I think it's too hard for me, but I plan to obtain a master degree in pure/applied math. I will be graduating at the end of Spring 2009 with a degree in pure math, and I want to go to a graduate school in California, a CSU system school. The problem is that I did terribly poor in my abstract algebra course last year (I passed with a C), and now I don't remember anything taught in that class. Can you give me some advices whether to retake the class or to study on my own? I remember taking the class and I was very frustrated because I didn't understand the material. My professor was sick and we had substitutes once a week. I used Fraleigh book back then.
I did fairly mediocre in my classes: Real Analysis (B+), Linear Algebra(B), Abstract Algebra(C), Complex Analysis(A-), Number Theory(A-), Numerical Analysis(in progress, not doing good), Set Theory(in progress, have B now). Should I even consider graduate school if I am not getting mostly A's in my classes? Some of my classmates are applying to big schools like UCLA, Princeton, MIT, CalTech...and when they ask me where I will go to grad school, I told them I might go to a California State University. Now I'm even afraid that I'm not prepared and qualified for such schools even though they have very low requirements. I don't know whether I should just get job at some high school or continue to pursue the goal I had, getting a master degree. I never tried to take a second course in analysis and algebra at my school because I thought I was not well-prepared for them. All the courses I had are introductory courses, and I didn't even do well in those.

 

[PhysicalAnomaly]

Would Chih-Han Sah be a better book than Dummit-Foote?

Does Artin also cover linear algebra?

Is Hoffman-Kunze the type of book that's suitable for a first course yet covers all essential ugrad material?

Would not learning how to write really short proofs as featured in Rudin be a disadvantage? Would doing proofs more in Pugh's or Tao's wordier manner be a bad thing? Also, does Rudin cover more than Pugh?

Is Ahlfors good for a first course? Stewart and Tall? Recommendations?

 

[qspeechc]

never heard of Chih-Han Sah. Dummit & Foote is still a good book though. Most Algebra books cover some linear algebra I believe, and Artin does cover much linear algebra.

Hoffman & Kunze is used as the MIT linear algebra book, which says pretty much everything you need to know about it. It is comprehensive, and you could use it as a first linear algebra book.

Rudin and Pugh cover pretty much the same material. Pugh isn't necessarily chatty (well, compared to Rudin...) I just think it's the better book: it provides some diagrams, and the questions range from doable to very challenging. Nevertheless, Rudin is a very good book, and either book would make you a theorem-proving machine.

 

[Werg22]

Mathwonk, how much of your mathematical knowledge is implicit? I mean how much of it do you understand without nevertheless being able to formulate it precisely? I find that a lot of mathematics is understood through images and not words. Do you feel the same?

 

[mathwonk]

its hard to put into words.

 

[Tom86]

Would Chih-Han Sah be a better book than Dummit-Foote?

Does Artin also cover linear algebra?

Is Hoffman-Kunze the type of book that's suitable for a first course yet covers all essential ugrad material?

Would not learning how to write really short proofs as featured in Rudin be a disadvantage? Would doing proofs more in Pugh's or Tao's wordier manner be a bad thing? Also, does Rudin cover more than Pugh?

Is Ahlfors good for a first course? Stewart and Tall? Recommendations?

I have a copy of Stewart and Tall and I found it definitely helped to read alongside my undergrad complex analysis course. Lots of people swear by Ahlfors though I prefer Cartan's treatment of the topic ("the Weierstrass point of view"), but I would recommend neither If you've done little in the way of analysis or topology before.
I can't offer much help on introductory algebra texts as I got by with my lecture notes as an undergraduate. These days I keep a copy of Samuel and Zariski handy if I need to look something up.

 

[adnan jahan]

hello, my name is adnan and i want to ask about the field of aplied mathematics i m having three offers for phd one in ''numerical anlysis'' and second one is ''fluid dynamics'' and third one is the offer of ''looping related with programming in the field of algebra'' can u brother sort it out the problem which one is good for future and which one is more interested...thanks for such an interesting topic.

 

[Bourbaki1123]

I was wondering if anyone here has any insight into Ohio State's Phd program in Algebraic Geometry? I'm currently a sophomore, but OSU is a place I have been considering as I find algebraic geometry particularly fascinating. Another school I have been considering, though I think that it is more of a reach, it U of Michigan. I would greatly appreciate any advice/experience anyone can share as far as these schools.

 

[mathwonk]

ohio state only has three mathematicians listed in algebraic geometry, and they seem very very good, but michigan has many more and they are also extremely good.

there is no comparison between the two programs, in numbers michigan is much stronger. This is not to say you cannot do very well at ohio state, but michigan is one of the premier programs in the country in algebraic geometry.

you should visit both and decide based on your own opinion, since in the end it will boil down to your interaction with one advisor.

 

[Bourbaki1123]

Thanks for the advice. I feel that OSU would be a bit of a safety school, and since U of Michigan is probably going to be a bit of a reach I have taken an interest in U of Wisconsin,Madison as another alternative.

Really it would be helpful if there were a way to see the profiles of the students that are accepted into various programs, however; something like the US news and world report isn't really quite specialized enough to do this, unless there is a more specialized issue that I should look at.

If there is some sort of database/book that makes this data available I would very much like to see it. There are a great many individual threads/posts all consisting of the same manner of inquiry(namely the chances one might have of getting into graduate school of varying caliber), but I have yet to see any comprehensive and reliable source of information.

 

[mathwonk]

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.

 

[Bourbaki1123]

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.

 

[mathwonk]

well I'll tell you my own experience. I was a math major at an Ivy league school, and kept pretty much to myself. After doing poorly the first 2 years, I worked hard and did well the last 2, including an A in a grad analysis course.

I was not ready for a top school but I got accepted at Brandeis, a very good school but not that popular. I did know less than my peers, but they regarded me as one of the stronger students in ability. I was distracted by the vietnam war and did not finish.

After a short teaching career, I enrolled at Utah, and again found it very challenging, even though again they regarded me as one of the top students. I managed to finish by working as hard as I could.

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.

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.

 

[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.