The Should I Become a Mathematician? Thread


by mathwonk
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Hurkyl
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Jun7-06, 08:59 PM
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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.
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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
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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
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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
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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
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3b. Differential and Integral Calculus Volume 2
R. Courant
Bookseller: Pioneer Book
(Provo, UT, U.S.A.) Price: US$ 13.50
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Book Description: Interscience Publishers, 1947. rebound Hard Cover Good. Bookseller Inventory # 481571

4. ANALYSIS 1
Lang, Serge
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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]
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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
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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
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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
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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
(Flushing, NY, U.S.A.) Price: US$ 67.98
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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
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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
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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
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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
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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
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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
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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
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Book Description: Academic 1960, 1960. 361 pp., hardback, ex library, else text and binding clean, tight and bright. Bookseller Inventory # ZB472982
mathwonk
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Jun7-06, 09:08 PM
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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 any one 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
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Jun7-06, 09:16 PM
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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
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Jun7-06, 09:18 PM
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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
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Jun7-06, 09:44 PM
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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
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Jun7-06, 10:26 PM
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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
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Jun7-06, 11:00 PM
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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
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Jun7-06, 11:16 PM
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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 elast 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 any one 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
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Jun7-06, 11:22 PM
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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
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Jun7-06, 11:24 PM
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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
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Jun7-06, 11:32 PM
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Well I really don't know squat 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
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Jun7-06, 11:36 PM
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My older son was a math major with numerical emphasis at Stanford, and now does web based internet stuff for Arriba. 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
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Jun7-06, 11:38 PM
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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
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Jun8-06, 01:34 AM
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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
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Jun8-06, 03:55 AM
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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 indispensible 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
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#53
Jun8-06, 09:47 AM
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Quote Quote by matt grime
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
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Jun8-06, 11:22 AM
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" 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.


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