The Should I Become a Mathematician? Thread

by mathwonk
Tags: mathematician
 P: 985 I wanted to specialize in number theory, but then I read a very discouraging book by Guy; now I'm not so sure anymore.
HW Helper
P: 1,996
 Quote by Dragonfall 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.
 P: 985 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, $$\mathbb{N}$$?
 Sci Advisor HW Helper P: 9,406 Thanks J77. Your input is just what Courtrigrad, and no doubt others, seems to be asking for.
 Sci Advisor HW Helper P: 9,406 Jbusc, topology is such a basic fioundational subjuect that it does not depend on much else, whereas differential geometry is at the other end of the spectrum.. still there are inrtroductions to differential geometry that only use calculus of several variables (and topology and linear algebra). Try Shifrin's notes on his webpage.http://www.math.uga.edu/~shifrin/
 Sci Advisor HW Helper P: 9,406 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.
 Sci Advisor HW Helper P: 9,406 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.
 P: 1,239 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. Thanks
 Emeritus Sci Advisor PF Gold P: 16,101 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".
 Sci Advisor HW Helper P: 9,406 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.
 Sci Advisor HW Helper P: 9,406 jbusc, here is a gorgeous book on maifolds, 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. P: 1,157  Quote by courtrigrad 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 ODE's 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 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  Sci Advisor HW Helper P: 9,406 I liked the beginning ode book by amrtin 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...)