Should I Become a Mathematician?

In summary, to become a mathematician, you should read books by the greatest mathematicians, try to solve as many problems as possible, and understand how proofs are made and what ideas are used over and over.
  • #1


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I am interested in starting this discussion in imitation of Zappers fine forum on becoming a physicist, although i have no such clean cut advice to offer on becoming a mathematician. All I can say is I am one.

My path here was that I love the topic, and never found another as compelling or fascinating. There are basically 3 branches of math, or maybe 4, algebra, topology, and analysis, or also maybe geometry and complex analysis.

There are several excellent books available in these areas: Courant, Apostol, Spivak, Kitchen, Rudin, and Dieudonne' for calculus/analysis; Shifrin, Hoffman/Kunze, Artin, Dummit/Foote, Jacobson, Zariski/Samuel for algebra/commutative algebra/linear algebra; and perhaps Kelley, Munkres, Wallace, Vick, Milnor, Bott/Tu, Guillemin/Pollack, Spanier on topology; Lang, Ahlfors, Hille, Cartan, Conway for complex analysis; and Joe Harris, Shafarevich, and Hirzebruch, for [algebraic] geometry and complex manifolds.

Also anything by V.I. Arnol'd.

But just reading these books will not make you a mathematician, [and I have not read them all].

The key thing to me is to want to understand and to do mathematics. When you have this goal, you should try to begin to solve as many problems as possible in all your books and courses, but also to find and make up new problems yourself. Then try to understand how proofs are made, what ideas are used over and over, and try to see how these ideas can be used further to solve new problems that you find yourself.

Math is about problems, problem finding and problem solving. Theory making is motivated by the desire to solve problems, and the two go hand in hand.

The best training is to read the greatest mathematicians you can read. Gauss is not hard to read, so far as I have gotten, and Euclid too is enlightening. Serre is very clear, Milnor too, and Bott is enjoyable. learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

If these and other top mathematicians are unreadable now, then go about reading standard books until you have learned enough to go back and try again to see what the originators were saying. At that point their insights will clarify what you have learned and simplify it to an amazing degree.

Your reactions? more later. By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me. Please correct me on this point, since nothing this general is ever true.:wink:

Remark: Arnol'd, who is a MUCH better mathematician than me, says math is "a branch of physics, that branch where experiments are cheap." At this late date in my career I am trying to learn from him, and have begun pursuing this hint. I have greatly enjoyed teaching differential equations this year in particular, and have found that the silly structure theorems I learned in linear algebra, have as their real use an application to solving linear systems of ode's.

I intend to revise my linear algebra notes now to point this out.
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  • #2
I probably want to become a mathematician. I am not sure whether to go into pure or applied math. I will probably opt for the latter, as I like being able to develop ideas useful for the world. Mathwonk, I am currently reading and doing problems from Apostol's vol. 1 Calculus. I realized in the past years, that I was very obsessive compulsive about doing every single problem. If I got stuck on one problem, I had to finish it. But now I just take the problems that really pertain to the material (i.e. not plug and chug problems), and if I get stuck, I just move along and post the problem here.

If I want to become an applied mathematician, is studying the book by Apostol ok? I want to really understand the subject (not some AP Calculus course where I just "memorized" formulas). Last year, I tried reading Courant's Differential and Integral Calculus, but it seemed too disjointed. I like Apostol's rigid, sequential approach to calculus.

Also, if I want to become an applied mathematician, should I, for example, major in math/economics? Here is my tentative plan of future study:

Apostol Vol. 1: Calculus
Apostol Vol. 2: Calculus (contains linear algebra)
Calculus, Shlomo Sternberg
Real Analysis
Complex Analysis

What would you recommend an applied mathematician take? Also, would you recommend me to go back and reconsider the old Courant, as I remember you saying that his book contains more applications? Or am I fine with Apostol?

Thanks a lot :smile:
  • #3
courtrigrad said:
Also, if I want to become an applied mathematician, should I, for example, major in math/economics?

Hell no. Maths and economics majors know jack about maths either pure or applied. Economists struggle to add up, never mind do maths properly (including applied maths).

If you're going to be a good applied mathematician then you'll be able to do Apostol and the purer stuff: you might not see the utility of it a great deal at times, but you will be able to do it, and it might well come in useful later.
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  • #4
I am glad to see Matt is chipping in. Courtigrad, I think Apostol is outstanding and probably more than sufficient for training in any future direction, but at intervals I suggest going back and reconsidering Courant. I also did not like it as a student, but appreciate it more now.

One thing a friend/student of mine said about his career in applied mathematics may be useful: he said it was primarily the difficulty of the pure mathematics he studied that readied him for applied mathematics work, not the specific knowledge. Having had to learn algebraic topology taught him how to learn something hard, and he had a big advantage over others in his field when he needed something new. He knew how to learn.
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  • #5
so I shouldn't major in math/economics if I want to become an applied mathematician? Just apply in pure math?
  • #6
It fits with my friend's experience, but you could ask some applied mathematicians. I agree with Matt though, a math degree is much more appropriate than an economics degree.
  • #7
Becoming a mathematician.

Being a mathematician means doing mathematics, but the activity is not the same as the job. Being a professional mathematician means being a professor, doing research and teaching and writing, or working in an industry using math tools to do things like design cars, or solve turbulence problems for aircraft, or to estimate the actual pollution in streams from samples. I only know about the professor side of it since I have been teaching and working in a university setting most of my life, but the behavior of learning and practicing mathematics is probably not too different for all intended lines of work. Ironically, a professor often has so many duties associated with teaching, grading, evaluating people, recruiting, etc,.. that he/she has to scrounge time to actually do math.

Getting started: Junior high.

Unless your parents will agree to send you to a special math school, and you live somewhere like North Carolina where these exist, or can get into and afford a prep school, you have little control over the training you get at this level. Your teacher may not know much math at all or even like it. But the good side is that school is usually pretty easy at this level so you should have free time to spend reading and learning on your own. A wonderful book to try is What is Mathematics? by Courant and Robbins. If you want a leisurely book for the public about a mathematical triumph that anyone can read, try the book by Simon Singh on Fermat’s last theorem as solved by Andrew Wiles.

math team, SAT’s
If possible join the math team at your school, and participate in math contests, practicing solving problems. This is not only fun, but good experience at test taking, a useful survival skill in the world of SAT’s. Being good at tests is not quite the same as being good at math, but it is the way talent is often measured and rewarded in young people, so it can help you earn scholarship money and admission to top schools. When I won the state math contest in high school in Tennessee I started getting job offers, and a high SAT score earned me a merit scholarship and fairly easy admission to Harvard. (Those were the old days.) Be aware though that the more important SAT test is the verbal one. It is harder, a better measure of your reasoning skill, and a better aid at distinguishing yourself. I am out of date here too, as I have heard they recently dropped the analogies portion, which of course was the most valuable part for detecting reasoning ability.

Caveat: This is a political world, and popular high stakes tests are not entirely intended to measure ability, but also to reward some political constituency, as witness the dishonest recent “recentering” of SAT scores, i.e. SAT grade inflation, making it harder for really good students to stand out. The ridiculous “no child left behind” rules have actually made it harder for some of my best students to get teaching jobs, because the criteria are so stupid that they work against really gifted people. For example some school districts require candidates to have taken certain mickey mouse math courses at the college level, whereas only a very weak student would not have taken them already in high schol or even junior high. But it is useless to protest, just learn to survive. Find out what the rules are for the goal you seek and make sure you qualify even if it seems silly and a waste of time. The books by the late Paul Torrance on gifted education, especially for the creatively gifted, are very helpful. They explain how to find resources for bright kids, and how to help them get credit for what they do.

Giftedprograms, TIP:
One thing that was a great experience for our young kids, is the TIP summer program at Duke University. They admitted bright junior high kids based on SAT scores taken when 12 years old, and offered a wonderful environment of talented students, excellent teachers, and fascinating introductions to topics like physics (delightful book: "Thinking physics" by David Epstein?) and number theory, that 12 year olds do not usually see in school. I presume it is still a good program. Of course before taking the SAT test, you should get hold of some practice SAT books and take a bunch of them to learn how they go. If you have an experienced professor or teacher around, or any parent who is test savvy, they can help you learn the difference between a true response and a correct response. This is hard to quantify, but my bright children often argued correctly that a certain multiple choice was actually true, but I knew from experience with tests that a different one was wanted by the tester.
Many states also have “governor’s honors” programs, and other special opportunities with various qualifications. Try to find out about them and get in on them. But do not despair if this is out of reach. Such things did not exist in my day and I never had any such special opportunities or training, just a mediocre classical high school math education, no calculus or anything advanced, just the added practice of the math team. I did have an excellent basic algebra course and Euclidean geometry, but that’s it. I knew the “root - factor theorem” for polynomials (r is a root if and only if (x-r) is a factor), and I had lots of practice trying to prove geometry facts. I didn’t even know trig.

more later.
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  • #8
part 2 of who wants to be a mathematician?

2) Basic Preparation, High school:

In high school, it is usual nowadays to take AP calculus. More important for mathematical background, is to get a good course in polynomial algebra and Euclidean geometry, with thorough treatment of proofs. A course in logic would help as well if it is available. Again, one must make do with what is available, but be aware that courses like AP calculus are more designed to please parents and impress admissions officials than to train mathematicians. Most of the people making decisions about what to offer are completely ignorant of the needs of future scientists, and are only concerned with entrance to prestigious schools. Again one must play the game successfully, so even though these people have no idea what you need to become mathematician, they still are able to make decisions on who gets into top schools, so it is prudent to impress them, while also trying to actually learn something on the side.

So what I am saying is this: in order to succeed in college calculus, one absolutely MUST have a solid grasp of high school algebra and geometry, although most high schools shortcut these subjects to offer the more prestigious but less useful AP calculus. Thus it is wise to work through an old fashioned high school algebra book like Welchons and Krickenberger (my old book), or an even older one you may run across. A wonderful geometry book is the newer one by Millman and Parker, Geometry: a metric approach with models, designed for high school teacher candidates in college. If you can find them, the SMSG books from Yale University Press, published in the 1960’s are ideal high school preparation for mathematicians. These were produced by the movement to reform high school math in the early 1960’s but the movement foundered on the propensity to put profit before all else, the lack of trained teachers, and the unwillingness to pay for training them.

e.g. here is a copy of a precalculus book from that era:


Bookseller: Lexington Books Inc
(Garfield, WA, U.S.A.) Price: US$ 35.00
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Book Description: Yale University Press., 1961. Good+ with no dust jacket; Contents are tight and clean; Ex-Library. Binding is Softcover. Bookseller Inventory # 41816

and an algebra book:

Mathematics for High School First Course in Algebra Part I Student's Text
Bookseller: Bank of Books
(Ventura, CA, U.S.A.) Price: US$ 14.25
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Book Description: Yale University Press. Soft Cover. Book Condition: ACCEPTABLE. Dust Jacket Condition: ACCEPTABLE. USED " :-:Fair:-:Writing on first page, covers bent and creased, a little water damage, corners bumped, covers dirty, page edges dirty, spine torn.:-:" Is less than good. Bookseller Inventory # 19620

another algebra book:

Clarkson, Donald R. Et. Bookseller: Becker's Books
(Houston, TX, U.S.A.) Price: US$ 15.00
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Book Description: Yale University, 1961. Book Condition: GOOD+. wraps School Mathematics Study Group Studies in Mathematics Volum V111. Bookseller Inventory # W040215

and one on linear algebra:

Introduction to Matrix Algebra. Student's Text. Unit 23.
School Mathematics Study Group Bookseller: Get Used Books
(Hyde Park, MA, U.S.A.) Price: US$ 25.00
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Book Description: Yale University Press. Paperback. Book Condition: VERY GOOD. USED 4to, yellow wraps. Slightly skewed; wraps sunned and a little worn at spine; text fine. Bookseller Inventory # 44146

Here are some books I use currently in teaching math ed majors, which would be bettter used in high school:

An Introduction to mathematical thinking, by William J. Gilbert and Scott A. Vanstone. paperback, ISBN 0-13-184868-2, Pearson and Prentice Hall.

also: (better) Courant and Robbins, What is Mathematics?

After mastering basic algebra and geometry, there is no harm in beginning to study calculus or (better) linear algebra, and probability. A good beginning calculus book is Calculus made easy, by Silvanus P. Thompson, (ISBN: 0312114109)
Bookseller: Great Buy Books
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Book Description: St. Martin's Press, 1970. Paperback. Book Condition: GOOD. USED Ships Within 24 Hours - Satisfaction Guaranteed!. Bookseller Inventory # 2397224 .

I love his motto: “what one fool can do, another can”

Do not laugh, this is a good book. And therefore his book on electricity and magnetism is probably also good (he was a fellow of the Royal Society of Engineers).

Elementary Lessons in Electricity & Magnetism. New Edition, Revised Throughout with Additions
Thompson, Silvanus P. Bookseller: Science Book Service
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Book Description: MacMillan Company, New York, NY, 1897. Hard Cover. GOOD PLUS/NO DUST JACKET. Red cloth covers are clean and bright with some wear at the tips and the head and foot of the spine; gilt lettering on spine is bright and easy to read; institutional lib book plate on inside front cover and lib stamp on copyright page; owner's signature inked on front flyleaf; binding cracked between front and rear endpapers and has been reinforced with clear tape; inside pages clean, bright and tight throuhgout. Overall, still a very useful, solid and clean working or reading copy. Bookseller Inventory # 008802.

Learn right now: the price of a book is unrelated to the value of the book as a learning tool, only to the scarcity of the book, and its popularity. [Notice how cheap these wonderful books are compared to the **&^%%$$! books that sell for $125. and up, that are required for college courses.]

Finally, if you are a very precocious high school student, and have learned algebra and geometry, you may profitably study calculus. In fact, to play the game of college admissions, you may need to take AP calculus, even if the teacher is an idiot, just so the admissions officials will believe you have “challenged yourself”. There are many good calculus books, beyond the humorous (but valuable) Silvanus P. Thompson, although that may already suffice for an AP course. The delightful math book I had as a high school senior, was a combination of logic, algebra, set theory, analytic geometry, calculus, and probability, called Principles of Mathematics, by Carl Allendoerfer and Cletus Oakley. This was a wonderful book, and opened my eyes to what was possible after a long period of boring mathematics courses at the dull high school level.

here is a copy:

Allendoerfer, Carl B. & Cletus O. Oakley Bookseller: Adams & Adams - Booksellers
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Book Description: McGraw-Hill, N.Y., 1963. Hard Cover. Book Condition: Very Good. No Jacket. 8vo - over 73⁄4" - 93⁄4" tall. xii + 540pp. name on front endpaper. Bookseller Inventory # 014846.

I still have a copy of this book on my shelf.

A lovely calculus book, for beginners, with delightful motivation, is Lectures on Freshman Calculus, by Cruse and Granberg. They motivate integration by the “Buffon’s problem” of computing the likelihood of a needle dropped at random, falling across a crack in the floor. (I reviewed it in 1970, and criticized the flawed discussion of Descartes’ solution of the problem of tangents, but I wish now I hadn’t, as it might have survived longer.)

here is a copy:

Lectures on Freshman Calculus
Cruse, Allan B. & Granberg, Millianne Bookseller: Hammonds Antiques & Books
(St. Louis, MO, U.S.A.) Price: US$ 18.00
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Book Description: Addison Wesley 1971, 1971. Hardcover good condition with minor soiling, no dustjacket xlibrary with usual markings ISBN:none. Bookseller Inventory # LIB2958010770.

As before, participate in the math team, and practice your vocabulary, to pass high on the verbal SAT. And read lots of books. Mathematicians have to also describe what they do to literate folk, and of course also need to “woo women” (or your choice), as observed in dead poets society.

more later.:blushing:
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  • #9
Matt, feel free to jump in here and describe your possibly more helpful or normal path to becoming a math guy, or give any advice you want, or counter any goofy advice I have given. I kind of hung in there in spite of everything going hooey for a time in the 60's, and may not have as much to offer the average person. My motto was sort of "never give up" no matter what, and may not synch perfectly with the readers of this forum. Best, roy.
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  • #10
Similar to Courtrigrad, I'm reading Apostol Volume I. Currently, it's extremely entertaining. I've been working on proofs, and I think I'm getting okay at writing them (for my Linear Algebra book I'm reading.) I've been trying to get a list of books to begin studying. So far, my plan is



Apostol Volume I
Apostol Volume II (Unfortunately, very expensive...)

Differential Equations

Zill- This is the book I'm using now in my DE class. Unfortunately, my class doesn't focus on very much theory at all...Bunch of applications...boring.

Linear Algebra

Kolman (1970's, old, but easy to read and I like the Theorem-Proof-Example layout. I like to read the theorem, and try to prove it before I read there proof. So far, that's been going well.)
Friedberg (2nd Edition).

Shilov- Elementary Real and Complex Analysis
Rudin- Principles of Mathematical Analysis (Also expensive...)

Modern/Abstract Algebra
No clue. Any suggestions?

Since I just graduated High School, I've been trying to spend my time productively studying Linear Algebra and Apostol. I really love Linear Algebra in the chapters on Vector spaces, subspaces, Linear Transformations, Isomorphisms, etc. However, manipulation of matrices (solving systems using boring matrix algebra) is a tedious process that doesn't interest me as much. I thoroughly enjoy proving the theorems the book provides.

I hope to be a mathematician and teach as a professor. Any recommendations for textbooks? Also, after thoroughly studying Linear Algebra, would it be wise for me to begin reading a text on Abstract Algebra? Or is there more mathematical preparation required?

Thank you, and thanks for making a topic about becoming a mathematician!
  • #11
However, manipulation of matrices (solving systems using boring matrix algebra) is a tedious process that doesn't interest me too much. I thoroughly enjoy proving the theorems the book provides.
This is actually a very interesting problem... provided you don't actually have to execute the tedious details by hand. :smile:

On the one hand, there is all sorts of challenging work in trying to figure out how to compute these things efficiently, if you like that sort of thing.

On the other hand, manipulation of matrices is useful for solving all sorts of problems, and it's interesting to figure out how to set up the problem correctly, and the right algorithm to get the information you need!

For example, suppose that you're working in 16-dimensional space. You have two 12-dimensional planes which are denoted with the following data:

A point in the plane.
A basis for the vector space of displacements from that base point.

Or, equivalently, your planes are the images of maps of the form:

T : R^12 --> R^16 : x --> Ax + b

for some rank-12 matrix A and vector b.

Your challenge: figure out how one would compute the intersection of those two planes!

(Warmup problem: suppose that your planes are actually vector spaces: that is, your map is an actual linear transformation)
  • #12
thanks for pitching in, hurkyl! i suspect you are a physicist by training (?), but you are obviously a very strong mathematician by inclination and talent.
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  • #13
as to those planning a career in math, here is a relevant joke i got from a site provided by astronuc:

Q: What is the difference between a Ph.D. in mathematics and a large pizza?
A: A large pizza can feed a family of four...
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  • #14
here are the ode books i used in my spring 2006 course:

1. title: An Introduction to Ordinary Differential Equations
by Earl A. Coddington
ISBN: 0486659429
Dover Publications

2. title: A Second Course in Elementary Differential Equations
author: Paul Waltman
ISBN: 0486434788
Dover Publications

3) Differential Equations and Their Applications: An Introduction to Applied Mathematics
Martin Braun, Martin Golubitsky?,
Jerrold E. Marsden (Editor), Lawrence Sirovich (Editor), W. Jager (Editor)

4) Ordinary Differential Equations, by V.I. Arnold.
Paperback: 270 pages, Publisher: The MIT Press (July 15, 1978)
ISBN: 0262510189.

The one with the most to offer a beginner is Braun. The one I liked best with the most to offer me, i.e. the most sophisticated (try it if you want to see what I mean) was Arnol'd. The easiest one, that I had in college at harvard, was Coddington.

a great ode book that i did not appreciate until recently was by hurewicz. here is a copy:

Lectures on Ordinary Differential Equations.
Hurewicz, Witold.
Bookseller: Significant Books
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  • #15
here are cheap copies of rudin:

24. Principles of Mathematical Analysis 1ST Edition*(ISBN: 1114135615)
Rudin, Walter
Bookseller: Powell's Books
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Book Description: MCGRAW HILL PUBLISHING COMPANY. HARDCOVER Mathematics-Real Analysis. USED, Less Than Standard. Bookseller Inventory # 04111413561502

RUDIN, Walter.
Bookseller: Robert Campbell Bookseller
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Book Description: New York: McGraw-Hill, 1964., 1964. Second edition. Hardcover. Very good in very good dust jacket. 270pp. Bookseller Inventory # 26517

i recommend this book as easier to read than rudin:

42. Introduction to Topology and Modern Analysis (International Series in Pure and Applied Mathematics)
Simmons, George F.
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  • #16
mathwonk said:
thanks for pitching in, hurkyl! i suspect you are a physicist by training (?), but you are obviously a very strong mathematician by inclination and talent.

Im sure Hurkyl will respond, but I thought I remembered seeing him state in some thread that he was a mathematician pursuing some physics (?)
  • #17
Stands up...

My name's J77, and I'm probably a mathematician...

Well, I have a degree in mathematics, a masters in mathematics and a PhD which was heavily maths related.

I'm probably even an applied mathematician! :tongue:

If I were to narrow it down a bit, I do nonlinear dynamics. However, to quote a recent survey by Philip Holmes
Nonlinear dynamics, more grandly called "nonlinear science" or "chaos theory", is a rapidly-growing but still ill-defined field...
I publish in all many of journals from linear algebra, through physics, into aerospace engineering.

To become a mathematician... That's a hard one.

I was always good at maths tests from an early age - I think some people have a natural ability at maths. I certainly don't think that tests are a good indication of what makes a good mathematician though - and it sometimes saddens me to see so many threads in this forum which are just about obtaining so many points in this or that test. I'd say that a good mathematician should just have the ability to think laterally. Be able to throw a bit of imagination into the mix. Anyone can learn procedures/algorithms for an exam. However, the true test is seeing a new problem and using past experience, or developing new techniques, to solve that problem.

Also, as far as starting out goes, do a maths degree. Only after you start a university course will you see what parts of maths you like and what parts you dislike. Then, of course, be more specific with the masters and finally the PhD.

However, don't set out a course to follow from when you're 18 - just go with the flow :smile:

I hope this thread keeps going with good advice and will add more...

For now, advanced books on nonlinear dynamics/bifurcation theory:

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.

Y. Kuznetsov, Elements of Applied Bifurcation Theory.

And to keep with some of the previous posts:

V.I. Arnol'd. Catastrophe Theory.
  • #18
Perhaps someone could give some advice...As a followup to what everyone has posted, I have been on the lookout for math resources for what I like to think of as "the theoretical engineer". I consider it to be that gray area between engineering, physics, and applied math that has emerged recently, where linear algebra, combinatorics, and discrete mathematics have found ample use.

I have taken several such courses, but the resources for self study seem to be slim. The courses themselves were primarily tought out of lecture notes supplemented by texts by Johnsonbaugh and Strang, which while excellent, didn't address most of the finer details of the courses.

In particular, I have been searching for resources on topics like measure theory, and differential geometry and manifolds, and I haven't found much that is written toward or even accessable to a non-mathematician, mostly due to the prerequisite vocabulary knowledge assumed by the reader.

For example, I recently found an excellent book on introduction to topology by Mendelson, which was very easily readable and understandable by anyone with some basic set theory background; unfortunately I have been unable to find books on other topics written at a similar level. I would certainly appreciate some pointers, thanks.
  • #19
I wanted to specialize in number theory, but then I read a very discouraging book by Guy; now I'm not so sure anymore.
  • #20
Dragonfall said:
I wanted to specialize in number theory, but then I read a very discouraging book by Guy; now I'm not so sure anymore.

You mean Guy's Unsolved Problems in Number Theory? The sheer volume of available problems should be encouraging! It gives many targets to work towards, even if you don't hit the target you might hit something interesting along the way.
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  • #21
It is very frustrating and discouraging that while any of the problems can be understood by a child, I am unable to make the most minuscule contribution to even one.

Et tu, [tex]\mathbb{N}[/tex]?
  • #22
Becoming a mathematician, part 3) AP courses etc..

3) AP courses and college admissions.

Although I have poo poohed AP calc courses as not comparable to college courses, and often taught by unqualified teachers, there are many reasons to take them anyway in high school. At some high schools they may be the best courses offered, and they may be taught by the best teachers available. Politically, they are frequently seen by college admissions officials as evidence that the student has challenged him/herself by taking the most difficult courses available to him/her. Thus trying merely to get a good mathematical preparation and avoiding bad AP courses can backfire at college admissions time, since the admissions officials are not knowledgeable enough to see these ill conceived courses for what they are. The same goes for high school students taking college courses while still in high school. I have advised some very smart kids to stay at their high schools and establish their math backgrounds with good solid high school level algebra and geometry, augmented by extra reading and projects, only to see these same kids turned down by their first choice colleges, in favor of weaker students whose high school resumes featured college calculus courses. It is very hard to detect real ability outside ones own field, or even in it, and college admissions officials are not that great at it. They tend to like easy identifiers, like newspaper editor and AP courses, prizes, and college courses taken in high school. They do not know how deep a thinker a kid is since they have not taught them, and often they are not scientists themselves, and would not know how to recognize a budding scientific mind first hand anyway.

Fortunately after you do get in college this kind of myopia lessens since you are being evaluated by scholars who (hopefully) see your work, but it never fully goes away. When applying to colleges try to get letters from people who will write positive ones, and who understand the game well enough to know how to do this honestly but skillfully. I have unwittingly sandbagged some outstanding kids early in my career, by simply writing truthful, unvarnished letters that appeared weak compared to the overblown and ridiculous ones written by less academically knowledgeable people. As a college math professor and researcher of course I have seen, met, heard, and even worked with some of the smartest people in the world, Fields medalists and so on, so few high school kids no matter how bright, are going to really blow me away. But admissions officials reading my letter do not give me credit for being on a different level from high school teachers and guidance counselors, and so those people may be better letter writers for this purpose. Of course by now I know how to tell the truth more persuasively.

Once you get in college, the people teaching you will usually be actual mathematicians themselves, and will know exactly how well you are grasping the material, and can write letters for grad school that will help you accordingly. Before college, take all the instructional opportunites available to you, just do not expect them to live up to the hype they may enjoy. Spend as much time as possible immersed in the subject itself, and with like minded people, to keep the love alive, but be aware of your resume.:smile:
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  • #23
Thanks J77. Your input is just what Courtrigrad, and no doubt others, seems to be asking for.
  • #24
Jbusc, topology is such a basic foundational subjuect that it does not depend on much else, whereas differential geometry is at the other end of the spectrum. still there are introductions to differential geometry that only use calculus of several variables (and topology and linear algebra). Try Shifrin's notes on his webpage.
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  • #25
I am curious to see how hurkyl describes himself - at times he has also struck me as perhaps a mathematical logician.

maybe i just assumed all the moderators on the "physics forum" were physicists.:smile:
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  • #26
dragonfall, the way to solve problems is to make them easier, try changing the hypotheses of some of those problems, and just keep changing them until you get a problem you can solve. then try to work your way back up a little bit.

25 years ago Bob Friedman and I showed that most Riemann surfaces with involution have different matrices of skew symmetric periods. About the same time Ron Donagi conjectured that if two such gadgets had the same period matrices, then the quotient Riemann surfaces must both be 4:1 covers of the line. No one has been able to show this more precise result yet, but many have tried, and I still hope to.

The problems you are talking about have stumped everyone in the world for decades or longer. Such books are not meant as a problem set for young mathematicians. Of course if you solve one fine, but if not, you are in very good company.
  • #27
is there any difference in majoring in math in a liberal arts college vs. a bigger college. I am going to a LAC this August.

  • #28
I'm a mathematician / computer scientist by training (just BS) and employment. I've just made a hobby out of doing a tremendous amount of self-study! I even sometimes read textbooks as "light reading".
  • #29
Wow. You would do a superb PhD if you have the inclination, but as you are already earning a living that would be a sacrifice.

You have the innate power and creativity of a PhD level mathematician. This is unusual with only a BS.
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  • #30

Well I went to a big university (Harvard) and found majoring in math there stimulating by the exposure to top people and high standards, but discouraging through the impersonality and lack of hands on guidance. So it took me a long time to find my way, but because I never gave up, eventually the high standards were still in my brain and became helpful far down the line.

In my opinion you could possibly do even better and sooner at a LAC (which one?) with some personal guidance from people who actually get to know you. And even if the Fields medalists are not teaching there, still the level of faculty is so high now everywhere, I believe you will be very well served.

The best early teaching I received was at Brandeis, a small research university, much more personal than my undergrad experience at Harvard. Later I went to Utah and got great grad school guidance and finally returned to Harvard as a postdoc, the ideal status for me at Harvard. I.e. Harvard is at such a high level that the instruction was more appropriate for me as a postdoc than an undergrad.

One of the best research algebraic geometers/ topologists in the country (Robert MacPherson) went to a small liberal arts college, Swarthmore.

I think it is hard to find a college in the country now where there is not more offered than one person can easily absorb. What do you take in 4 years, 32 semester courses? and the Harvard catalog contains over 3,000 courses.

It might possibly help to go to a college witha grad program. E.g. Wesleyan is to me a typical liberal arts college, and has 26 undergrad math courses and 29 grad math courses, more than anyone could possibly take.

The difference with going to Harvard is there you will also have the chance to take graduate algebraic topology or algebraic geometry as an undergraduate, but how many people need this at that stage?
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  • #31
Yeah, I was waitlisted by Wesleyan and Oberlin (hopefully will get off). I am currently planning to attend Denison University in Ohio. My current goal is to become an applied mathematician; perhaps to a "3+2 pre-professional program" (i.e. 3 years at Denison and 2 years at Columbia) or stay 4 years at Denison and apply to graduate school. My only concern is whether I will be able to study the core essentials properly (in college). Or maybe I have to do Apostol by myself, and follow the likes of Stewart in college.
  • #32
I looked at their webpage and it looks as if they have a very active department. there was a conference there on group theory and one of their graduates placed first in the nation in a math contest recently.

It looks like an especially strong place in applied math, and also has a presence in groups, functional analysis, and knot theory. The faculty picture is also fun looking. I think you will enjoy it there.

This will be a place where there are not a lot of advanced grad courses, but the treatment of undergraduates should be outstanding. It looks like a very promising place indeed. good luck, and as Bill Monroe told my brother to tell me " tell him, don't hang back, come right up and introduce himself". (My brother was Bill Monroe's fiddler in college.)

keep in touch.
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  • #33
jbusc, here is a gorgeous book on manifolds, from lectures by a fields medalist and great expositor. try it. and give it some time. if you can read this you will really learn something.

Topology from the Differentiable Viewpoint

princeton univ press.

John Milnor

Paper | 1997 | $26.95 / £17.50 | ISBN: 0-691-04833-9
76 pp.

This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.
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  • #34
courtrigrad said:
If I want to become an applied mathematician, is studying the book by Apostol ok? I want to really understand the subject (not some AP Calculus course where I just "memorized" formulas). Last year, I tried reading Courant's Differential and Integral Calculus, but it seemed too disjointed. I like Apostol's rigid, sequential approach to calculus.

Also, if I want to become an applied mathematician, should I, for example, major in math/economics? Here is my tentative plan of future study:

Apostol Vol. 1: Calculus
Apostol Vol. 2: Calculus (contains linear algebra)
Calculus, Shlomo Sternberg
Real Analysis
Complex Analysis

What would you recommend an applied mathematician take? Also, would you recommend me to go back and reconsider the old Courant, as I remember you saying that his book contains more applications? Or am I fine with Apostol?

Thanks a lot :smile:
I'm not familiar with the texts which you name.

During my maths degree, we used Calculus and Analytic Geometry by Gillett, and Calculus by Boyce and DiPrima, the latter I still look at from time to time.

If you're reading off your own back, starting ODEs from scratch, I'd suggest the dynamical systems book by Boyce and DiPrima: Elementary differential equations and boundary value problems, as a good starting point. Possibly coupled with a more application based book like: S. Strogatz: Nonlinear Dynamics and Chaos. (or K. Alligood, T. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems.)

I'm sure others will like and dislike (I've heard B&DiP talked down before) the choices, but they are only entry points, with the Strogatz book bridging the gap between elementary calculus based texts and the books I recommended in my previous post.

If you could be more specific about the content of the courses, that would help. Obviously content varies from institution to institution. Also, you present level of education may help - I presume you've just started university or are finishing high school?

edit: Just to add, if you want a book you'll use time and time again: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table by Milton Abramowitz and Irene A. Stegun
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  • #35
I liked the beginning ode book by martin braun for my class, exactly because it featured applications, and hence entertained and motivated the class. It discussed using ode's to date of paintings and detect forgeries, predict populations of pairs of interacting "predator prey" species like sharks and food fish or hares and wildcats, troop deployment in battles with illustrations from WWII (Iwo Jima), and lots more such as "galloping gertie" the famous tacoma narrows bridge that blew down years ago. It was very well written.

Boyce and DiPrima is a time tested, often used, and well liked standard book at my university too, indeed THE standard book on ode, but I was looking for a good alternative that cost a lot less. Sadly, as soon as a book becomes a standard, the price now shoots above $130. I got my copy of Braun used for $2. Braun is also more entertaining for me, but I think you cannot go wrong with BdP.

I would suggest studying ode sooner than some of the other topics on your list, like reals and complex analysis. Also as wisely mentioned above, it seems prudent to go with the flow, and not be too rigid in your planning at this early stage.

And the calculus book by Sternberg, if you mean Advanced Calculus by Loomis and Sternberg, it is very abstract and advanced, treating calculus essentially as functional analysis. Of course once you have finished Apostol, it will probably be fine, but I suspect the view Loomis gives in the first half of calculus is not essential for an applied mathnematician. I like it though (I took the course from Loomis in the 1960's from which this is the resulting book. The only thing I learned was that the derivative of f at p is a linear map differing from f(x)-f(p) by a "little oh" function, which is of course the main idea.)

There is another newer book by Sternberg and Bamberg, math for students of physics that sounds intriguing, but I have not seen a copy. In the 1960's Bamberg was the absolute most popular and entertaining physics section man in a department which was otherwise bleak and forbidding for its physics instruction. (I still remember his list of useful constants: Planck's constant, Avagadro's number, Bamberg's [phone] number...)
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