Other Should I Become a Mathematician?

[courtrigrad]

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.

 

[mathwonk]

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.

 

[mathwonk]

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.

 

[J77]

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

 

[mathwonk]

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...)

 

[mathwonk]

Becoming a mathematician part4) Starting College

Becoming a mathematician, part 4) College training.

I suspect it does not matter greatly which college you go to, as they all have their strengths and weaknesses. Places like Harvard or Stanford or Berkeley offer famous lecturers on a high level, incredibly advanced courses, and brilliant highly competitive students. For many of us, this can be more intimidating than inspiring. And often the famous professors are simply unavailable for conversation outside of class. In the early 60’s at Harvard, I found the lectures were wonderful, if I got the best professors, and then they walked out and I never saw them again until next time. Office hours were minimal and if I tried to see some of them, they were frequently busy or uninterested. Even intelligent questions in class seemed as likely to be met with sarcasm as a helpful answer. I suspect things have changed now with people like Joe Harris and Curt McMullen there, who are great teachers as well as researchers, and who enjoy students. Of course there were outstanding teachers like Tate and Bott there in the old days too, but not everyone was like them. As a result, I had to go away and get back my enthusiasm for math at a more supportive place.

It is helpful to go somewhere where you will enjoy your time, enjoy the courses and the other students, and get help from professors who think students matter. Today this is more common everywhere, even at famous universities, than it was long ago, but ask around among the student body. And be prepared to work very hard. Some if not most of my own undergraduate frustrations could have been lessened, possibly solved, by better study habits.

As to what courses to take, this is tricky and complicated by the almost worthless AP preparation most kids get today in high school. In general an AP class is a class taught by someone with nowhere near the training or understanding of a college professor, although they may be a fine teacher. But to expect a calculus course taught by an average high schol math teacher to substitute for a honors introduction to calculus taught by Curt McMullen or Wilfried Schmid or Paul Sally, is ridiculous. Nonetheless, so many students have bought this ridiculous idea that Harvard and Stanford do not even offer an honors introduction to calculus anymore for future math majors. There simply are none out there who have not had AP calculus in high school. Thus the student entering from high school is faced with beginning in one of many choices of several variable calculus courses. The most advanced one, the one taught a la Loomis and Sternberg, realistically requires preparation in a very strong one variable course a la Apostol, but which Harvard does not itself offer. So the only students prepared to take it are those elite ones coming in from Andover or Exeter or the Bronx high school of science, but not the rest of us coming in with our inadequate AP courses from normal high schools.

Thus the jump from high school to college has been made harder by the existence of AP courses. So in my opinion, even with AP calculus preparation, it is often helpful for a prospective mathematician to try to begin college in an introductory, but very challenging, one variable calculus course, modeled on the books of Spivak or Apostol, if you can find them. These do exist a few places, such as University of Georgia, and University of Chicago, which still offer beginning Spivak style calculus honors courses. To quote the placement notice from Chicago: “The strong recommendation from the department is that students who have AP credit for one or two quarters of calculus enroll in honors calculus (math 16100) when they enter as first year students. This builds on the strong computational background provided in AP courses and best prepares entering students for further study in mathematics.”

(I am not positive, but I assume that 16100 is the spivak course. But do your own homeworkl to be sure.)

The point is that AP preparation provides no theoretical understanding, so plunging students into advanced and theoretical calculus courses of several variables, as they do at Harvard and Stanford, by beginning in Apostol vol 2, or Loomis and Sternberg, without background from Apostol volume 1 or Spivak, is academic suicide even for most very bright and motivated students.

If you go to a school where there is no Spivak or Apostol vol. 1 type course, where the calculus preparation is from Stewart, or some such book, you are perhaps getting another AP course, only in college. Then you have to choose more carefully. Many such college courses will indeed be no more challenging than a high school AP course, and should not be repeated. Just ask the professor. They know the difference, and will help you choose the right level course. Either get in an honors section, or an advanced course suitable to your background. And join the math club. Try to find out who the best professors are, and do not be scared off if weak students say a certain professor is tough. You may not think so if you are a strong student. Once you get there, try to sit in on courses before taking them, to see which professors suit you. Student evaluations are notoriously hard to interpret correctly. The professor with the worst reputation among students, Maurice Auslander, was in my grad school days at Brandeis my absolute favorite professor. He cared the most, offered the most, and taught us the most. He also worked us the hardest.

Once you get a semester or two under your belt, it will get easier to find the right class, as hopefully the colleges own courses prepare you for their continuations, although this is not guaranteed! There is no way to force one professor to included everything the enxt one expects, nor to exclude material he/she loves that is outside the curriculum. Do your own investigating. Ask the professor what is needed for his/her course and try to get it on your own if necessary. After leaving the honors program temporarily as an undergraduate, I got back in by studying on my own over the summer from an advanced calculus book (David Widder), to make up my theoretical deficiencies and survive the next course.

Everyone should study calculus, linear algebra, abstract algebra, ode, and some basic topology. If you have no background in proofs from high school, you will need to remedy that as soon as possible. It is best to do this before entering, even if they offer a “proofs and logic” course. Such courses are often offered to junior math majors, whereas they are needed to understand even beginning courses well. For this reason it is extremely helpful to read good math books on your own that contain proofs. Today especially it is important to know some physics even if if you only plan to do math. Much of the inutition and application of math comes from physics. Even if you only want to do number theory, sometimes viewed as the purest and most esoteric branch of math, many of the deepest ideas in number theory come from geometry and analysis and even statistics, so nothing should be skipped. Work hard, read good books, seek good teachers, and try to have fun. College is potentially the most exciting and fun time of your life, and the one where, believe it or not, you have the most freedom and free time.

 

[Hurkyl]

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.

When I thought I was going to look for a programming job, my plan was to go back to school and learn more math.

But since I'm actually employed as a mathematician (and have become fairly good at self-study), I don't feel as much need. OTOH, my employer will pay for some full-time schooling (both the classes, and giving me my full pay!), so I really ought to take advantage of it. My buddies keep trying to tell me to go and get a masters in logic.

 

[mathwonk]

Becoming a mathematician part5) some good books

Some recommended undergraduate books for future mathematicians.

Introductory calculus.
1. Calculus (ISBN: 0521867444)
Spivak, Michael Bookseller: Blackwell Online
(Oxford, OX, United Kingdom) Price: US$ 53.66
Shipping within United Kingdom:FREE

Book Description: Cambridge University Press, 2006. Hardback. Book Condition: Brand New. 3Rev ed. *** CONDITION NEW COPY *** TITLE SHIPPED FROM UK *** Pages: 672, Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis. Preface; Part I. Prologue: 1. Basic properties of mumbers; 2. Numbers of various sorts; Part II. Foundations: 3. Functions; 4. Graphs; 5. Limits; 6. Continuous functions; 7. Three hard theorems; 8. Least upper bounds; Part III. Derivatives and Integrals: 9. Derivatives; 10. Differentiation; 11. Significance of the derivative; 12. Inverse functions; 13. Integrals; 14. The fundamental theorem of calculus; 15. The trigonometric functions; 16. Pi is irrational; 17. Planetary motion; 18. The logarithm and exponential functions; 19. Integration in elementary terms; Part IV. Infinite Sequences and Infinite Series: 20. Approximation by polynomial functions; 21. e is transcendental; 22. Infinite sequences; 23. Infinite series; 24. Uniform convergence and power series; 25. Complex numbers; 26. Complex functions; 27. Complex power series; Part V. Epilogue: 28. Fields; 29. Construction of the real numbers; 30. Uniqueness of the real numbers; Suggested reading; Answers (to selected problems); Glossary of symbols; Index. Bookseller Inventory # 0521867444

2a. Calculus. Volume I. One-Variable Calculus, with an Introduction to Linear Algebra. Second Edition
Apostol, Tom M Bookseller: Paper Moon Books
(Portland, OR, U.S.A.) Price: US$ 20.00
Shipping within U.S.A.:US$ 4.50
Book Description: New York John Wiley & Sons, Inc. 1967., 1967. Fine. 666pp. Clean and bright book. No previous owner's markings. 2nd.Edition. Binding is Hardback. Bookseller Inventory # 068435

2b. Calculus. Volume II. Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probabil
Apostol, Tom M Bookseller: Paper Moon Books
(Portland, OR, U.S.A.) Price: US$ 20.00
Shipping within U.S.A.:US$ 4.50
Book Description: New York John Wiley & Sons, Inc. 1969., 1969. Fine. 673pp. Clean and bright book. No previous owner's markings. 2nd.Edition. Binding is Hardback. Bookseller Inventory # 068436

3a. Introduction to Calculus and Analysis (Volume I)
Courant, Richard; Fritz John
Bookseller: Harvest Book Company
(Fort Washington, PA, U.S.A.) Price: US$ 9.95
Shipping within U.S.A.:US$ 3.95
Book Description: Interscience Publishers/ New York 1965, 1965. First American Edition, 1st Printing Hardback in Decorated Boards. 661p. Very good condition. Very good dust jacket with one small closed tear and sunned jacket spine. Satisfaction Guaranteed. Bookseller Inventory # 515288

3a, alt. Introduction to Calculus and Analysis Volume 1 (ISBN: 0470178604)
Richard Courant
Bookseller: Frugal Media Corporation
(Austin, TX, U.S.A.) Price: US$ 10.00
Shipping within U.S.A.:US$ 3.70
Book Description: Wiley, John Sons. Hardcover. Book Condition: VERY GOOD. USED Ships within 12 hours. Bookseller Inventory # 873302

3b. Differential and Integral Calculus Volume 2
R. Courant
Bookseller: Pioneer Book
(Provo, UT, U.S.A.) Price: US$ 13.50
Shipping within U.S.A.:US$ 3.50
Book Description: Interscience Publishers, 1947. rebound Hard Cover Good. Bookseller Inventory # 481571

4. ANALYSIS 1
Lang, Serge
Bookseller: The Book Cellar, LLC
(Nashua, NH, U.S.A.) Price: US$ 39.99
Shipping within U.S.A.:US$ 4.00
Book Description: Addison-Wesley 1968., 1968. Fine in Good dust jacket; Light shelf wear to book. Heavy wear to DJ. 460 pages. Binding is Hardcover. Bookseller Inventory # 374309

5. Calculus of One Variable
Joseph W. Kitchen, Jr. Bookseller: Antiquarian Books of Boston
(Winthrop, MA, U.S.A.) Price: US$ 150.00 [sorry]
Shipping within U.S.A.:US$ 3.50
Book Description: Addison-Wesley Publishing, Reading, Mass., 1968. Hard Cover. Book Condition: Very Good. No Jacket. 8vo. xiii, 785 pages. Tightly bound and clean. No writing in book. The book also deals with plane analytic geometry and infinite series. Bookseller Inventory # 7620

also Honours Calculus*(ISBN: 0965521117) $24. from the author.
Helson, Henry
http://members.aol.com/hhelson/


Calculus of several variables.
6. CALCULUS ON MANIFOLDS: A MODERN APPROACH TO CLASSICAL THEOREMS OF ADVANCED CALCULUS
Spivak, Michael Bookseller: BRIDGEWAY ACADEMIC BOOKSTORE, ABA
(TAOS, NM, U.S.A.) Price: US$ 25.00
Shipping within U.S.A.:US$ 6.50
Book Description: W. A. Benjamin, NY, 1965. PAPERBACK COPY. Book Condition: Very Good. VERY GOOD CONDITION, PAPERBACK, 146pp. Bookseller Inventory # 001874

7. Mathematical Analysis: A Modern Approach to Advanced Calculus
Apostol, T. M. Bookseller: Textsellers.com
(Hampton, NH, U.S.A.) Price: US$ 12.50
Shipping within U.S.A.:US$ 3.50
Book Description: Addison Wesley, 1957. Book Condition: Good. Dust Jacket Condition: Fair. 8vo - over 7¾" - 9¾" tall. Hardcover, 559 pp. Notes, jacket has edge chips. Bookseller Inventory # 011916

8. Functions of Several Variables.
Fleming, Wendell H. Bookseller: Significant Books
(Cincinnati, OH, U.S.A.) Price: US$ 12.00
Shipping within U.S.A.:US$ 3.50
Book Description: 337 pp. Addison Wesley (1965) (Hardback) Good condition, ExLib. Glue Spot on cover. Bookseller Inventory # MATH10273

9. Advanced Calculus
Loomis and Sternberg
free download from Sternberg’s website.

Linear Algebra:
10. Linear Algebra : A Geometric Approach (ISBN: 071674337X)
Malcolm Adams, Ted Shifrin Bookseller: www.EMbookstore.com[/URL]
(Flushing, NY, U.S.A.) Price: US$ 67.98
Shipping within U.S.A.:US$ 3.25
Book Description: W. H. Freeman; (August 24, 2001), 2001. Book Condition: New. Free Delivery Confirmation! Brand New Hardcover, US Edition, Quality Paper Printed in USA. Bookseller Inventory # 071674337X-2

11. Linear Algebra.
Hoffman, Kenneth, & Ray Kunze Bookseller: Zubal Books
(Cleveland, OH, U.S.A.) Price: US$ 11.46
Shipping within U.S.A.:US$ 6.50
Book Description: Englewood Cliffs: Prentice-Hall 1965, 1965. 1st edition, fourth printing (1965) 332 pp., hardback, wear to spine & covers, previous owner's name to front free endpaper else textually clean & tight. Bookseller Inventory # ZB471098

Ordinary Differential Equations
12. Ordinary Differential Equations (ISBN: 0262510189)
V. I. Arnold Bookseller: A1Books
(Netcong, NJ, U.S.A.) Price: US$ 28.77
Shipping within U.S.A.:US$ 4.95
Book Description: Brand new item. Over 3.5 million customers served. Order now. Selling online since 1995. Few left in stock - order soon. Code: M20060602184422T0262510189. SKU: 0262510189-11-MIT. Bookseller Inventory # 0262510189-11-MIT

13. Lectures on Ordinary Differential Equations.
Hurewicz, Witold. Bookseller: Significant Books
(Cincinnati, OH, U.S.A.) Price: US$ 7.00
Shipping within U.S.A.:US$ 3.50
Book Description: Book Condition: Good condition, no dj. 122 pp. Wiley (1958 ) Hardback. Bookseller Inventory # MATH12978


Topology
14. First Concepts of Topology
Chinn, W. G. & Steenrod, N.e. Bookseller: aridium internet bookstore
(Cranbrook, BC, Canada) Price: US$ 8.32
Shipping within Canada:US$ 8.95
Book Description: SInger, 1966. Trade Paperback. Book Condition: Very Good. First Printing. Usual library markings in and out. non-circulating. very light use, clean crisp pages. edge rub/wear. A solid copy. Ex-Library. Bookseller Inventory # 010917

15. Differential Topology: First Steps
Wallace, Andrew Bookseller: Books on the Web
(Winnipeg, MB, Canada) Price: US$ 30.25
Shipping within Canada:US$ 5.50
Book Description: NY: W.A. Benjamin, 1968, 1968. paper bound, 1st edition, illustrated in colour, 130pp including bibliography and index. As new. Bookseller Inventory # 16779

16. An Introduction to Algebraic Topology
Wallace, Andrew H. Bookseller: BOOKS - D & B Russell
(Shreveport, LA, U.S.A.) Price: US$ 12.00
Shipping within U.S.A.:US$ 4.00
Book Description: Pergamon Press, New York, 1963. Book Condition: Very Good hard cover/ no dust. Octavo, 198 pp., Last name of prior owner inside front cover. One of a series of the International Series of Momographs in Pure and Applied Mathematics. Bookseller Inventory # 013208


Abstract Algebra.

17. Algebra (ISBN: 0130047635)
Artin, Michael Bookseller: DotCom Liquidators / DC 1
(Fort Worth, TX, U.S.A.) Price: US$ 44.50
Shipping within U.S.A.: US$ 3.50
Book Description: Bookseller Inventory # NA/DC8/T999/*114552

Abstract Analysis

18. Foundations of Modern Analysis. Pure and Applied Math., Vol. 10
Dieudonne, J. Bookseller: Zubal Books
(Cleveland, OH, U.S.A.) Price: US$ 9.49
Shipping within U.S.A.:US$ 6.50
Book Description: Academic 1960, 1960. 361 pp., hardback, ex library, else text and binding clean, tight and bright. Bookseller Inventory # ZB472982

 

[mathwonk]

notice there is a dearth of books listed for elementary diff eq since few of them inspire much admiration among people. on the other hand i have found some amazing bargains for you, including courant, apostol, hurewicz, hoffman/kunze, and dieudonne, at prices about 1/5 to 1/10 those often seen. sorry about kitchen. its a nice book but at that price it is absurd to buy it, given that copies of fleming, dieudonne, courant, etc... exist for so much less. almost anyone of these books will give you an enormous amount of education. i have also shortchanged complex analysis, but you will find another example on henry helson's website. he is a student of loomis i believe, and former berkeley professor who writes excellent books and publishes and sells them himself at reasonable rates, with some written by others. he has a linear algebra book too but i have not seen it.

 

[mathwonk]

here are two more really good, really cheap books:

Elementary Theory of Analytic Functions of One or Several Complex Variables
Henri Cartan
Format: Paperback
Pub. Date: July*1995
B&N Price: $13.95
Member Price: $12.55
Usually ships within 2-3 days

also: Differential Forms
Henri Cartan
Format: Paperback
Pub. Date: July*2006
NEW FROM B&N
List Price: $12.95
B&N Price: $11.65*(Save*10%)
Member Price: $10.48

 

[courtrigrad]

For any mathematicians (pure or applied) did you guys intern anywhere during your summers? I am trying to find places where an applied mathematics major could go and intern during the summer (freshman). Maybe I could go abroad? Typically, does a math major do research over the summer or intern if he opting for a pHd? Does it have to be necessarily math related? Also, for an applied mathematician, what would you say is the most important area to know? Would it be ODE's / PDE's. I might be interested in going into quantitative finance, or something like biological math. This summer, I want to try to focus on learning a range of math rather than a depth of math (i.e. only studying Apostol, but not studying other areas of math like probability theory). Sure, I may not be a scholar in the end in any of the particular fields, but I can always go ahead and brush up later when the time calls for it (i.e. if I do a pHd). I find that the internet offers me the most versatility in learning different fields of math.

 

[mathwonk]

I did not intern myself. Today there are several programs for math types in summer funded by VIGRE grants from NSF. Some schools also offer sumemr research opportunities but these are often voluntary activites by faculty, hence may fall short of volunteers. I.e. we are asked to do it for free, and that is something hard to sustain for long.

 

[mathwonk]

hurkyl, i do not see how you can resist getting paid plus free tuition to study something interesting. how can you lose? it also adds to your resume for pay increases, new job opportunities, etc. i say grab it. you will do it easily. you are really strong mathematically. I am sure of this.

 

[mathwonk]

Becoming a mathematician part 6) basic graduate books

Here are some foundational graduate books for future professionals. * means an especially high level recommended book.

Grad math books:

Algebra:
1. *Lang, Algebra,

2. Jacobson, Basic algebra 1,2.

3. Dummit - Foote, Algebra

4. Hungerford, Algebra

Reals

5. Measure and Integral: An Introduction to Real Analysis
Richard L. Wheeden, Antoni Zygmund

6. Royden, Real Analysis

7. Rudin, Real and complex analysis

8. * Functional Analysis, Riesz - Nagy

Complex

9. Ahlfors, Complex analyhsis

10. Conway, complex analysis

11. *Hille, Complex Analysis

12. Complex Analysis in One Variable, R. Narasimhan,

Topology
13. Fulton, Algebraic topology

14. *Spanier, algebraic topology

15. Hatcher, algebraic topology.

16. Vick, Homology theory.

 

[mathwonk]

A remark for graduate students that they do not always seem to understand: Your instructor in a basic graduate course is often an expert in the field, at least on a level with many authors, although perhaps not all, of basic books. Hence it is not to be expected that the instructor will slavishly plow through a standard book on the topic, but may well merely present the material as best suits him or her. Do not be automatically disappointed if your instructor lectures from his/her own notes as they are often actually superior to what is found in many books. At the least the lecturer will probably select from the best presentations available for each topic.

This is a plus for the student. I am having difficulty citing here standard books for each subject, since at this level the presentation given in class is normally better than that found in anyone book, for one thing as it is more up to date, being given by a practicing professional. I.e. at this level the best instruction is often obtained in person rather than from books.

 

[mathwonk]

Another remark: You will notice that all the books I have cited are theoretical ones, on specific bodies of theory, rather than being say problem books. This is the way I was taught, proving theorems. We were expected to find and work problems on our own.

But in Russia e.g., there is a wonderful tradition of problem solving and problem teaching. This type of activity was what brought me to math in high school but was slighted in my college instruction. Nonetheless it is gresat fun, and leads well toward the experience of doing a PhD and solving open problems.

Thus it would be good to list some books of problems, but I will have to do some research to find them.

 

[Sisyphus]

Hey mathwonk, would you mind doing a little comparison between pure math and applied math? As in the types of classes you'll take in each major, their differences, what you can do with each degree, etc

I am starting my undergraduate studies in September. While I don't have to decide on my major until I am done by first year, I'm still kind of curious as to how it all works.

Awesome thread, by the way. I've been reading it since you've started it.

 

[mathwonk]

Well I really don't know anything about applied math, but i gather you should go heavy on the ode, partial de, and numerical analysis courses.

Thanks for the feedback.

I have been dominating the discussion but I want to explicitly solicit reports from other math people on their experiences in school, getting ready, what helped, what was a problem, what led to productive results at work, etc,...

Perhaps Matt could shed some light on his journey to a math PhD, and Hurkyl on his path to gainful employment, and J77 on his life as an applied math guy. Also physics guys like Zapper and others could help us with input on what math you really need if you might want to get into physics, or mathematical physics.

My friends in physics have emphasized group representations, but that was a long time ago. More recently it has been Riemann surfaces and algebraic geometry.

 

[mathwonk]

My older son was a math major with numerical emphasis at Stanford, and now does web based internet stuff. He likes it. He also needs some business skill, as in a company you have to manage people who work for you, motivate, sell, service, hire and fire, and educate customers and clients.

 

[mathwonk]

My wife was also a math major and is now a pediatrician. Math is not her main resource but all dosages require mathematics to scale them to suit each child by weight. I may be trivializing her math usage, but math majors can do a lot of things because they can reason and calculate well. She also needs to manage people and service customers.

Besides her ability to deal with all people she meets, her main skill that impresses me is her terrific diagnostic ability. She actually saves lives when she detects a serious infection by its outward signs. This is deductive ability appield to real life emergencies.

 

[mathwonk]

One thing i can guarantee, everyone needs to take linear algebra, pure applied, whatever. The thing that is so frustrating about the AP courses in high school is their focus on calculus instead of linear algebra. I.e. linear algebra is easier than calculus, more important for more people than calculus, and even a prererquisite for understanding calculus.

So it sems odd to make calculus the focus of high school AP courses instead of linear algebra. Unfortunately no one listens to math professors when planning math education curricula.

 

[matt grime]

Some catch up points:

1. I didn't intern, some do, though, at investment banks and so on

2. I wouldn't bother with a research program with the aim of getting research under your belt if your intention is to do pure maths; it seems highly unlikely that anything you do will be representative of a pure PhD. However it can be a good experience of how mathematicians work, and you might get a glimpse of the future.

3. Books: I'd like to weigh in with some none analysis stuff, at the graduate level,

a) Fulton and Harris 'Representation Theory' for anyone considering doing algebra or theoretical physics. (contains all you need to know about semi simple lie algebras)b) James and Liebeck 'Representations and Characters of Groups' (brilliantly written intro to complex reps of groups)

c) LeVeque 'fundamentals of number theory' (all the basics)

d) Cox 'primes of the form x+ny' (very good intro to things like class field theory, must know number theory first, eg quadratic reciprocity first)

e) Weibel 'Introduction to Homological algebra' (all you wanted to know about homology theory but were afraid to ask, even introduces derived categories which are indispensable these days)

f) Alperin 'Local representation theory' (this is very specialized, but very accessible, worth a look for the out and out group theorist)My reasons for holding back are that I have a background in the UK and it is completely unrelated to the story unfolding here: there is no such thing as major and minor for a degree, you pick the subject whilst you're in high school that you'll do in university, and do it from day one when there. Doing a PhD in the UK is also vastly different: all those things that are taught in a US program are either things you're expected to know before you start or things you're expected to teach yourself if they're relevant to your area.

If you want to do a PhD in the UK, I'd very strongly recommend going to Cambridge to do Part III first (and this applies to international students too; I know plenty of Americans who did that year before going back to the states for their PhDs). It is the mathematical equivalent of basic training in the marines.

Don't let your mind go fallow either (one reason I've been posting here frequently in the last week is because I've got mathematicians block, and I'm trying to keep my mind active until it pops back into doing my research) and don't be afraid to look outside your area of interest. I see too many people dismiss something as being 'rubbish' just because it marginally falls outside their narrow ideas of what maths ought to be. It is to the UK's discredit that right now people are graduating with PhDs in this country in maths yet they don't know what a Riemann surface is, they've never seen any category theory, don't know a single cohomology theory. I can forgive any mathematician for not knowing what a sheaf is, but not for being ignorant of Galois theory, yet even that is missing from many of their memory banks.

 

[J77]

If you want to do a PhD in the UK, I'd very strongly recommend going to Cambridge to do Part III first (and this applies to international students too; I know plenty of Americans who did that year before going back to the states for their PhDs). It is the mathematical equivalent of basic training in the marines.

I would say that other masters courses - particulary those by advanced study and research - are equally beneficial as preparation for a PhD in the UK.

And with the Part III, you can also specilise in Applied or Pure, right? So what's so special about the Part III?

I like how this thread's going, and that question's not a swipe at Cambridge, I'm interested...

 

[matt grime]

" And with the Part III, you can also specilise in Applied or Pure, right?"

No, you get to do whatever the hell you like.

In Part III the lectures are intense, far reaching, there are many different courses, far more than the average university is capable of handling, and widely recognised at international level to be outstanding. None of that applies to other taught masters courses in the UK, which then to be very narrowly focused on one particular area. You want to do graduate level courses in QFT, Lie Algebras, Differential Geometry, Non-linear dynamics and Galois Cohomology of number fields? Could be arranged, depending on the year (that was a selection of courses available when I did it). Where else would you be able to do that?

Feel like finding out about modular representation theory, combinatorics, functional analysis, fluid mechanics, and numerical analysis? Again, quite likely you can do that.

Of course, why you would want to do that is a something else entirely, but in terms of scope of work and expectations placed upon you it is the best preparation out there, far more so than most (ifnot any, but I can't bring myself to make such sweeping statements) MSc's by research, and certainly more so than any MMath course.

If you even want to do a PhD in maths at Cambridge, they will demand part III, and many other places use it as a training ground and ask their students to go there.

The reason it is the best is because in some sense it is 'the only': there is no other university with the resources to be able to offer a program like it. Even Oxford can't compete, and most UK maths departments are just too small to offer anything comparable.

 

[mathwonk]

Matt's remarks on differences in expectations in US, UK remind me of a talk I heard at a conference. The speaker said something like, "this proof uses only mathematics that any sophomore undergraduate would know", then paused and added, "or here in the US, maybe any graduate student". This is true and getting worse.

Not only do we need to teach incoming grad student essentially beginning abstract algebra and analysis, but increasingly today also advanced calculus, and even basic proof writing in some cases.

This all goes back to the same problem - almost non existent training in basic math in high school, because of the ill conceived AP program. In the 1960';s there was a very ambitious and excellent set of hiogh school level books put out by the SMSG (school mathematicsa study group) via Yale University Press.

These constitute an excellent high school math preparation, including linear algebra, geometry, and calculus, but they are very hard to find now, being long out of print. One possible place to find them is math ed libraries in colleges of education.

 

[mathwonk]

Thanks for the book list Matt. That would be my next category, specialized books, as opposed to the ones so far covering what "everyone" should know. It is harder to choose them though, so I appreciate the help from various perspectives. Many will disagree already with some of my choices for "everyone". A future algebraic geometer should ideally know at least abstract algebra, commutative algebra, homological algebra and sheaves, complex analysis (one and several variables), algebraic topology, and differentiable manifolds, hopefully differential geometry. Also something about projective geometry and plane curves. Not to know cohomology and some Hodge theory also seems unthinkable. I myself have alkways felt handicapped by a lack of knowledge of group representations.

Real analysis is used less, but it is used to prove Serre's duality theorem in cohomology. It is hard to prove finite dimensionality of a priori infinite dimensional cohomology spaces without some functional analysis.

But I want to be careful here of blowing away young students by overwhelming them with "what one should know" as the list is potentially unbounded. Getting a PhD means having basic training, then acquiring a lot of knowledge about something very restricted, and verifying some little new insight about it. All the rest of this information just builds up over time. I found it very important after my first post PhD job, to have a learning seminar every week, reading some good paper, or even teaching basic information to each other. You would be amazed how many PhD professors will come up quietly and say, I have always wanted to know what a differential form is, or a manifold, or a sheaf, or a toric variety, or algebraic variety, or chern class,... I myself taught galois theory and wrote my "graduate" algebra notes the way i did because i had never understood that subject. Just keep pecking away, learning, teaching, and doing mathematics, and eventually you will know a lot.

 

[mathwonk]

As to my own history of things I did and didn't know, I entered college knowing what a group is, at least the definition, but upon entering grad school still did not know what an ideal was. Hence the note on the bulletin board with the "pre class" reading assignment for first yr algebgra, chapter 4 of Zariski - Samuel, was energizing. This chapter covered noetherian rings, including Noether Lasker decomposition theory of ideals, prime, primary, principal ideals, unique factorization, localization, etc etc..

Nontheless, after one year I was among the subset who passed algebra quals, but not my friend who had written his senior undergrad thesis on Zariski's main theorem in algebraic geometry. Tests are so odd, but I did not complain, having usually been treated better than I derserved by them.

There is a big difference between "knowing" a topic and understanding it too. Having read Lang's Analysis II, I "knew" the implicit function theorem to be a special case of the rank theorem, that a smooth function with locally constant rank near p, can be written locally, after a smooth change of coordinates, as a composition of a linear projection and a linear injection.
But when a professor remarked that the theorem meant you "can solve for some of the variables in terms of the others", I thought "huh?".

Another time at a high level research and instructional conference in algebraic geometry I sat next to Professor Swinnerton - Dyer and his student. The student was amazed that the Fields medalist speaker was hesitating over some elementary point in projective geometry, and his Professor was assuring him "they only teach that at Cambridge nowadays".

Perhaps it would help if Matt would describe a pre college and college preparation in the UK for maths. That could give us a more ideal pattern than ours here. British books are often the best to learn from as well, compared to American ones, since they seem to be written more often by people who know the language.

 

[mathwonk]

I might mention, in case it is relevant for someone, I have not tried to talk to the upper upper percentile of math persons. I have tried to keep it reasonable for a bright math loving student. There are a few people who can handle anything thrown at them, just not most of us.

From 1960-1964 there were undergrads I knew at Harvard, maybe even the typical very good math major, who took the following type of preparation: 1st year: Spivak calculus course, plus more; second year: Loomis and Sternberg Advanced calculus, Birkhoff and Maclane, or Artin Algebra; 3rd yr: Ahlfors and maybe Rudin Reals and Complex; 4th year: Lang Algebra, and Spanier Algebraic Topology.

(Actually most of those courses did not even use books, just the professors notes, but those books are an approximation.)

Others actually began as freshmen in graduate courses. (I myself was briefly placed in an advanced graduate course in mathematical logic by Willard Van Orman Quine, but I did not care for the pace of it.) These were students who went to Princeton or Berkeley or Harvard afterwards for PhD.

Many added more advanced topics courses to these and wrote a thesis. One kid provided a small but key step for the classification of finite groups.

Indeed I might have even survived this regimen with much better study skills. But there were also people like Spencer Bloch, and John Mather there as undergrads who are famous figures in mathematics now. And the program there was ideal for them.

The point is not to compare ourselves too much with others, just to use them as inspiration, not discouragement, and keep on at our own pace, enjoying it as much as possible.

 

[Beeza]

Mathwonk, is there any chance that I could PM you my Email, and you send me a typical syllabus for what you teach in Calculus I and II and with what book? For me, Calculus I and II were taught out of Stewart's.

I'm currently in calculus II, and sometimes I think that my calculus I course was a bit "mickey mouse"-- as I never really had to study beyond doing the homework, and I found the exams relatively easy compared to what I was expecting. I honestly don't like the idea of having an easy course, and I want to excel when I eventually reach graduate school--not be blown away by people that had more rigorous courses during their undergrad.

Although I'm a physics major, I'll admit that my favorite part of physics is the math.

 

[George Jones]

My brother was Bill Monroe's fiddler in college.

Wow!

By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me.

Shmoe (post #20 in this thread) is a fairly regular poster.

Also physics guys like Zapper and others could help us with input on what math you really need if you might want to get into physics, or mathematical physics.

Today especially it is important to know some physics even if if you only plan to do math.

And vice versa, i.e., today especially it is important to know some math even if if you only plan to do physics. In this, I including some pure mathematics, otherwise you might end with nonsense like 0 = 1, as I pointed out in https://www.physicsforums.com/showthread.php?t=122063".

When I was a student I enjoyed mathematics courses both because I was interested in the applications of mathematics to modern theoretical physics, and because I enjoyed mathematics. Here are some of the math courses that I took.

Real Analysis: Analysis in Euclidean Space by Kenneth Hoffman
Measure Theory: The Elements of Integration by Bartle
Algebra: Basic Algebra I by Jacobson
Topology: Topology a First Course by Munkres
(Baby) Functional Analysis: Introductory Functional Analysis with Applications by Kreyszig
Representation Theory: Linear Representations of Groups by Vinberg

Below, I repeat a post that I wrote in response to a question about math references for physicists. The topics are pretty basic for mathematicians, but many physics students never see *any* of this stuff done in a "mathematical" style, and all the books listed below, except maybe Nakahara, are written in this style.

It is probably impossible for anyone person to learn all the mathematics useful for physics, so you have choose what mathematics you want to study and how much time you want to spend on it.

Typically, mathematical physics courses emphasize techniques for solving differential equations, e.g., special functions, series solutions, Green's functions, etc. These techniques are still very important, but, over the last several decades, abstract mathematical structures have come to play an increasingly important role in fundamental theoretical physics. Consequentlly, useful courses include real/functional analysis, topology, differential geometry (from a modern perspective), abstract algebra, representation theory, etc., and, usually, should be taken from a math department, not a physics department.

These courses, supply vital background mathematics, and, just as importantly, facilitate a new way of thinking about mathematics that complements (but does not replace) the way one thinks about mathematics in traditional mathematical physics courses.

A number of good books on "modern" mathematics exist. Among these, my favourite is https://www.amazon.com/gp/sitbv3/reader/104-8106425-5831130?%5Fencoding=UTF8&asin=0226288625#reader-link"&tag=pfamazon01-20 by Robert Geroch. Geroch purposely and provocatively chose his title to indicate that, these days, mathematical physics includes topics other than those covered in more traditional mathematical physics courses. He starts with a few pages on category theory!

Geroch's book contains a broad survey of abstract algebra, topology, and functional analysis, and it does a wonderful job at motivating (mathematically) mathematical definitions and constructions. Surprisingly, since Geroch is an expert, it contains no differential geometry. Also, its layout is abominable.

At slightly lower levels are https://www.amazon.com/gp/sitbv3/reader/104-8106425-5831130?%5Fencoding=UTF8&asin=0521829607#reader-link"&tag=pfamazon01-20 by Chris Isham.

Geroch's book should be supplemented by more in-depth treatments of topics. For example, a good mathematical introduction to group theory is https://www.amazon.com/gp/sitbv3/reader/104-8106425-5831130?%5Fencoding=UTF8&asin=0521248701#reader-link"&tag=pfamazon01-20 by Shlomo Sternberg.

Also, none of the surveys that I listed treat fibre bundles, which are so important in modern gauge theories, and in other areas. Treatments include https://www.amazon.com/gp/product/9810220340/?tag=pfamazon01-20 by Chris Isham.

This is just the tip of the iceberg - there are many, many other good books including Nakahara, Choquet-Bruhat et al., Reed and Simon, Fulton and Harris, Naber, ...