Should I Become a Mathematician?

[balletomane]

My school has pure, applied, and general math options, with the last giving almost complete control over which courses one takes. I would like to combine pure and applied math, but I can't really complete all of the requirements for both programs. What are the most necessary courses, esp. if I am planning on going to grad. school?

 

[mathwonk]

it would help some to see your schools website.

the most widely used and powerful subjects in math are linear algebra and calculus. that tends to mean advanced calculus, in which linear algebra is used to draw conclusions about non linear functions by means of calculus.

spivaks little book calculus on manifolds epitomizes what everyone should know about it, 1) inverse function theorem( if the linear approximation to a smooth map ,i.e. its derivative, is invertible, then so is the original maop at elast locally),
2) fubini theorem (a repeated integral may be comouted as an iterated integral, in any desired order),
3) stokes theorem (combines fubini and fundamental theorem of calculus to prove that the multi variable integral of a form which is an exterior derivative dw, can be calculated as the integral of w over the boundary of the original region).

linear algebra means not just the concepot of linearity, but a deeper study of the structure of matrices and linear maps, to include the theory of canonical forms (natrual representatives of conjugacy classes) such as rational form over any field, and (more useful) jordan form over a field in which the chracteristic polynomial splits. in particular the concept of characteristic polynomials and minimalpolynomials is crucial in finite dimensions. jordan forms should be used in a good treatment of linear ode's as well.

if this stuff sounds elementary to you, i can mentioin more advanced topics. of course one should also know about groups, including linear groups, and galois theory.

mike artin's book, algebra, is the best algebra book out there for most of this stuff.

you should also talk to your local math advisor, as she/he can tailor your needs with your department's offerings. they will also almost certainly be researchers themselves and have been to grad school.

it will be their recommendation that gets you in too. so one thing you want to do is meet them and learn what they expect from you.

best wishes.

 

[pivoxa15]

the riemann hypothesis is clearly an application of compelx analysis to number theory. put very simply, riemanns point of view was that a complex functon is best understood by studying its zeros and poles.

the zeta function is determined by the distribution of primes among the integers, since its definition is f(s) = the sum of (forgive me if tjhis is entirely wrong, but someone will soon fix it) the terms 1/n^s, for all n, which by eulers product formula equals the product ofn the sum of the powers of 1/p^s, which equals by the geometric d]series, the product of the factors 1/[ 1 - p^(-s)].

now this function, determined by the sequence primes , is by riemanns philosophy best understood by its zeros and poles.

hence riemanns point of view requires an understanding of its zeroes, which he believed to lie entirely on the critical line.

this hypothesis the allowed him to estimate the number of primes less than a given value, to an accuracy closer than gauss' integral estimate.

even with its flaws, this brief discussion allows you to see what areas of math you might need to know at a minimum. complex analysis, number theory, integral estimates, and (excuse me that it was not visible) mobius inversion.

Thanks for the introduction. It seems analysis and complex analysis is the key area.

You mention number theory. That is obviously a very broad field. One tend to associate algebra with it (i.e. http://www.math.niu.edu/~rusin/known-math/index/11-XX.html shows it is nearly all algebra) but the Riemann hypothesis is (at the heart of?) number theory but you didn't mention any algebra. Isn't algebra useful for this problem?

 

[mathwonk]

well i don't see much algebra in the riemann hypothesis, but the mobious inversion formula is algebra. what riemann shows as i recall is that the integral estimate of gauss estimates not just the number of primes less than a given quantity, but also the number of squares of primes, and the number of cubes of primes etc... so he inverts this to get just the number of primes. the inversion process is algebra.

but the proof is still open to completion. you should read riemann's own account. the paper is not that hard to read. or read the book on the topic by harold edwards.

 

[pivoxa15]

How would you classify foundations of maths?

What do you see in that discipline? What is your opinion about it?

What are some of leading departments research this topic?

 

[mathwonk]

well i know very little about it. when i hear that term i think of goedel's work, and the work of paul cohen in the 60's completing the proof of the independence of the continuum hypothesis. i saw cohen lecture at harvard in about 1965, but have heard very little about this field since then.

as to my opinion of it, what little i know is a fascinating but small body of results, that especially interested me as a young student. I recall also that my algebra teacher Maurice Auslander described Paul Cohen as perhaps the smartest man he knew, the only person he knew who could read and understand a math text without writing out many pages for each page read.

Goedel's work of course dealt with the theory of provability of statements, so is a branch of logic. If the theory of algorithms is included, i.e. the theory of existence of methods of deciding whether solutions of problems exist, and finding solutions, then there is deep current work on the topic.

I would include Rumely's work extending Hilbert's problem, which had a negative solution for rational integers (Hillary Putnam), to the case of algebraic integers, where he showed it has a positive solution.

Rumely is at Univ of Georgia.

let me do some google research. and maybe others, e.g hurkyl, or matt grime, will pitch in.

 

[mathwonk]

look here http://sakharov.net/foundation.html

in the usa, this page lists stanford, michigan, ucla, berkeley, irvine, notre dame, rutgers, penn, penn state, forida, etc...

abroad they mention oxford, leeds, barcelona, steklov institute, bonn, vienna,...many very famous places are represented in this field which implies it is a thriving subject. i would think stanford is an outstanding department, but i did not read the activity lists at each place. i am encouraged by the existence and richness of the link above, as to the activity at bonn and vienna. also there is an international meeting next year in china on a related topic. all this is on that link.

 

[mathwonk]

here is part of a description of "what is foundations of math?" from a practitioner at Penn State:

"If X is any field of study, "foundations of X" refers to a more-or-less systematic analysis of the most basic or fundamental concepts of field X. The term "basic" or "fundamental" here refers to the natural ordering or hierarchy of concepts (see point 1 above). For instance, "foundations of electrical circuit theory" would be a study whose purpose is to clarify the nature of the most basic circuit elements and the rules of how they may be combined. The study of complicated types of circuits (e.g. radio receivers) is to be formulated as an application of the basic concepts and therefore would not be called "foundations" in this context.
In the history of particular fields of study, the foundations often take time to develop. At first the concepts and their relationships may not be very clear, and the foundations are not very systematic. As time goes on, certain concepts may emerge as more fundamental, and certain principles may become apparent, so that a more systematic approach becomes appropriate. An example is the gradual clarification of the concept of real number through the centuries, culminating in axioms for the real number system.

The foundations of X are not necessarily the most interesting part of field X. But foundations help us to focus on the conceptual unity of the field, and provide the links which are essential for applications and for integration into the context of the rest of human knowledge."

unfortunately i am not of a very philosophical bent, so i find this a bit off putting myself.

the positive impression i have stems from hearing people like Paul Cohen speak, and knowing Robert Rumely and hurkyl, and another friend of mine, Dave Anderson of cwsu, and seeing how bright they are. so the experts are much better sources for impressions of this area than onlookers like me. best to seek input from someone who really sees the beauty of the area. so hopefully some of these will chip in.

 

[mathwonk]

ok here is a survey by experts on the current state of "proof theory" solicited by an absolutekly brilliant expert, soloman feferman:

it's technical, but they know what they are talking about.

http://www-logic.stanford.edu/proofsurvey.html

 

[mathwonk]

my latest post got lost. i said something about this subject appealing to very smart people who have not lost their healthy sense of naivete and wonder about the existence of actual concrete solutions to problems.

Many of us lose this gradually as we absorb abstract "existence proofs". the quote by Russell is perhaps relevant:" the axiomatic method has much to recommend it over honest work".

 

[pivoxa15]

What does Russel mean when he said "the axiomatic method has much to recommend it over honest work".

Would you say this subject is a branch of logic rather than maths?

 

[pivoxa15]

"...who could read and understand a math text withut writing out many pages for each page read."

That is amazing, I have to do lots of writing even when reading undergrad texts.

 

[mathwonk]

maybe Russell meant something like what Weyl said more clearly:

"Important though the general concepts and propositions may be with which the modern and industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics, and that to master their difficulties requires on the whole the harder labor."

Herman Weyl, quoted by Michael Artin, in his Algebra.

 

[jammidactyl]

I slipped through undergrad with no research experience... how do I fix this now?

I'll get out of the military July 2007 and want to apply to school to start fall 2008. Should I get a master's degree first? What kind of work should I look for in that year between?

 

[mathwonk]

if you want to start grad school in math in fall 2008 trying for a PhD, it might make more sense to start first for a masters and get a refresher course in stuff that has got rusty.as to the year before, what i did was teach in a college and learn as much as i could by teaching it. that is probably hard nowadays. maybe teaching math in a private school would help, or maybe just making some money would help so you will have more time to study when you do begin school.

good luck.

 

[jammidactyl]

In planning things out I'm trying to figure out the "best" path to go down, but it's hard to even come up with criteria to measure "best"! It's hard finding a balance between what I'm qualified for, what I'm interested in and what will put me in the best position. Ultimately I'd like to get into a top 30 or so PhD program. I know I have that potential, but my record doesn't currently reflect that.

I agree that master's program might be the best bet initially, but it doesn't seem common. I feel I could get research experience and solidify my math knowledge with the side benefit of polishing my application for a PhD program. Is this realistic?

My interests are in geometry/topology, but finding a place with a good logic program would be nice as well!

Right now I'm looking at a master's degree at the University of Arizona. This interests me for a number of reasons. I can start in the spring (not really that important, but nice). They claim to have strong ties to LANL (research experience). It also seems well within my reach to get accepted, as well as being a solid middle-tier place I wouldn't mind staying.

Moving up the ranking chain, I see Carnegie Mellon and Indiana University that would interest me and appear to offer funding with their master's degrees. They also have good logic programs, as well. Not as likely for me to get in, but doable.

Can anybody suggest any other programs? Also, how often do people in mathematics get an MS as a stepping stone to a PhD? Anything I should be aware of vs going directly into a PhD program? Assuming I study hard and get some research experience will such a plan have the desired effect of making me more attractive to a "better ranked" PhD program? (I realize rankings aren't everything, but the pragmatist in me says it's often not what you know, but who you know.)

All this could change over a few hours in April when I take the GRE!

 

[mathwonk]

for some reason i want to discourage what i see as your somewhat classist approach to grad school in math. Instead of aiming for a "top 30" program, if i were you, i would just be glad to get into any program that helps you learn some math and polish your research skills.

If as you say your record does not show much visible evidence of promise, there is no real reason to expect that you will be a great researcher. No one knows this until they begin to work at it, Even David Hilbert took a teaching certificate as a fall back.

If your research pans out, it will be very visible to people at any decent place, certainly at the univ of arizona. in research academics I do not believe at all it is "who you know: but in fact "what you know", and perhaps more what you can do.

Of course it does help to have someone famous find out who you are, but if you do some good work, you only have to publish it and it will be noticed.

The rewards in academics are not great enough to make it worth the while of an ambitious person, and to me you sound a little on the ambitious side.

research academics is more about doing it for love of intellectual excitement. Of course I could well be wrong, and I do not mean to judge you, just advise you.

it is true that self confidence is quite helpful in academics and you do seem to have that, so good luck to you.

 

[Ki Man]

Most online textbooks that i find on math are college level and beyond

why arent there as many high school level sources

 

[jammidactyl]

I understand your reaction to my post, allow me to clarify. I know I have it in me to put together an application competitive at that level. I have worked a few years and made a little money, and I'm sure I could land a sweet contracting gig or something, but that's really not for me. I'd like to learn mathematics and teach others to do the same. I understand the PhD->tenure system is but one path to do that, and I just want to set myself up on the best possible path within that system. As I said, measuring "best" is very hard, so please don't confuse focus with ambition! (I attach sort of a negative connotation to the second term.)

I spent time as an undergrad with neither ambition nor focus. I returned to school for meteorology while in the military and had much ambition, but still no true focus. That department didn't match my personality and I opted not to stay for a master's degree. So I'm really just done wasting time and want to get down to some math!

Let my last comment be that I think Richard Hamming outlined the situation best in his speech, "You and Your Research". (Great read for anybody who hasn't already.)

 

[jammidactyl]

Just kidding, one more comment on the subject.

I equate ambition with going through the motions because you feel that alone will bring success.

I equate focus with wanting to deal with the system in the most efficient manner possible in order to free time for other activities.

 

[pivoxa15]

My university dosen't teach classical eucliden geometry to undergradates. Is this true for most universities? If so why?

I heard there were some flaws in this kind of geometry (at least in Euclid's orginal book) is this true?

Could you recommend some good modern books that offer a rigourous introduction on this subject? Or is Euclid's book still the best.

 

[mathwonk]

i like the modern book by milman and parker. a metric approach with models.

classical euclidean geometry was traditionally taught in high school, not as much now.

some colleges with math ed programs still teach euclidean and other geometries aimed at future high school teachers.

the best modern high school geometry book in my opinion is the one by harold jacobs, the most fun and the best coverage of classical geometry including logic. it is a modern pedagogical classic. it would make a good book for college courses in proofs and geometry for future teachers, especially those whose geometry is rusty at best.

 

[mathwonk]

the so called logical flaws in euclid are probably visible only to professionals. the book is worth looking at, but it may be easier to study from modern books. the old smsg series for high school from the 1960's is recommended but out of print. it is found in some math or math ed libraries.

edit: (much later) Apparently shortly after posting this, I taught from the original Euclid and quickly became thoroughly convinced it is by far the best book!

 

[symbolipoint]

My university dosen't teach classical eucliden geometry to undergradates. Is this true for most universities? If so why?

I heard there were some flaws in this kind of geometry (at least in Euclid's orginal book) is this true?

Could you recommend some good modern books that offer a rigourous introduction on this subject? Or is Euclid's book still the best.

That particular Geometry course is remedial and so is not considered undergraduate level; that is why your school does not offer this Geometry course. Community colleges should still offer the course if the university does not.

One very excellent Geometry book for "remedial" or college preparatory purposes is the Prentiss-Hall book, Geometry (the one with the purplish coloring). Do not be mislead by the jazzy features shown on many of the pages! The main topics content is very well developed.

symbolipoint

 

[pivoxa15]

That particular Geometry course is remedial and so is not considered undergraduate level; that is why your school does not offer this Geometry course.

symbolipoint

So you think it is too easy for undergraduates? Or is it that they expect pure maths majors to know these things while being unnessary for applied maths majors.

It is surprising that my univeristy offers no geometry subjects for undergraduates but a differential geometry at honours or 4th year level.

 

[acm]

How would you rate H.S.M Coexter's : Introduction to Geometry?
In terms of relevance to an undergraduate mathematics course.

 

[mathwonk]

from what i know, anything by Coxeter would be outstanding.

 

[pivoxa15]

Mathwonk, What do you think about the use of computers in pure mathematics? Do you see pure mathematicians become redunant one day since some computers are able to do non trivial theorem proving and they can only get more sophisticated.

Do you use computers in your work and how do you use them?

 

[J77]

Mathwonk, What do you think about the use of computers in pure mathematics? Do you see pure mathematicians become redunant one day since some computers are able to do non trivial theorem proving and they can only get more sophisticated.

Do you use computers in your work and how do you use them?

For me, applied math is quite simply the application of pure math.

This application can be done using pen and paper, or by computer.

It's necessary to know the "pure" maths behind the particular applied math technique you're using - if not, you don't really know what you're doing.

This is not always clear at undergrad level where you're taught techniques and can apply them blindly - but when you get into mathematical research the two "separated" parts of maths pretty much merge to one.

In other words, mathematicians don't use computers to prove results - but, more likely, to verify "pure" ideas in "applied" situations

 

[mathwonk]

As to whether computers are used to prove pure math theorems, it depends on what you are proving, and what you call a "proof". E.g. suppose you want to prove that the decimal expansion of e begins 2.718281828459...

What would you consider a valid proof of that? I have checked it myself, (except for the last 9) by hand. Many people would consider it a proof to plug it into a program like mathematica, and read off the result. Or perhaps to give the taylor series, and say how many terms needed to be added up using the error estimate, to get that acuracy. but then someone has toa dd up the terms. often that someone is a calculator, and in fact the error rate of calculators at this work is less than that of humans.

In a pure math proof, one may e.g. reduce the proof down to showing say that a certain diophantine equaton has no integer solutions les than 100,000. then how to complete the proof? one way is to plug in all 100,000 candidates and see if they work. this is reasonably a job for a computer.

I myself do not use them because i do not know how to use them technically, and cannot think of anything they could do for me in my work. computers are basically for calculating, and the mathematician's work is to decide what to calculate and how to calculate it.

but more advanced and savvier people then me, have used computers in their pure mathematical work. e.g. David Mumford and Joe Harris used a computer program, whcih they included in their paper, to check all outstanding cases in a proof that the moduli space of curves of odd genus at least 25 or so was not unirational. they needed to verify a bound on some slopes of divisors, and there were too many to check by hand.

David once tried to think of a way to prove the Schottky problem by computer when we were having lunch, but did not finish his thought. his fascination with computers led him later to give up algebraic geometry (with a fields medal) and become a researcher in AI.

also people like Jon Carlson and Dave Benson, use computers in their work on group algebra representations. matt would know.

computers are wonderful at calculating and this can play a role in completing a proof. they are also good at displaying things which can help inspire a proof idea. they are also good at compiling and listing data which can help inspire both ideas and confidence in ones ideas.

any time you can reduce a proof to a finite amount of computation, a computer can conceivably carry it out.

 

[mathwonk]

as to whether computers will render mathematicians redundant, i cannot say. take the question of whether they will render teachers redundant, and instruction will all be done online, possibly by instructional programs.

First of all computers certainly cannot do what people can do, so in that sense they will not render people redundant. But some institutions seem to prefer cheap instruction to good instruction, so it is conceivable that computers will actually replace human teachers even though they cannot do anywhere near as good a job. I.e. bureaucrats may try to use cheaper computers over costlier people and that may render them non existent even though still highly needed.

this is up to the public. As to research mathematicians, they are mostly a group of people who do what they do for love anyway, so they will continue to do it. but support for them may get even less than now. already most grant support is not for pure mathematicians but for computer scientists, educators, and statisticians.

We are part of a brotherhood and sisterhood of like minded creative people, moving down through the centuries mentally arm in arm. It is wonderful activity. As long as there are people like you asking us about our work, we will enjoy discussing it.

 

[pivoxa15]

It seems like the computer is still a tool for the pure mathematician just like it is for the scientists.

Have you read Roger Penrose's "Emporer's New Mind" and "Shadows of the Mind" where he argues that computers will eventually do much more complicated maths but cannot do some basic things because of limitations imposed by Godel's theorem and so will not replace human mathematians no matter how vast their mathematical capability. If you have what is your opinion on it?

Maybe automizing more of the teaching is effective and good because humans make mistakes and they have to do it every year which can become boring. It will also give the acadmics more time on their research.

 

[mathwonk]

well maybe. it seems to me that key aspects of good teaching include conveying enthusiasm, sensing difficulties that the student has, and encouraging them, also providing role models. i do not see how a computer can do very well at those things.

when i was a student, and even a professional, an essential part of doing my work was having someone to tell it to. i think heard someone say about the great teacher r.l.moore, that students worked so hard for him because he was just so pleased by good work.

 

[mathwonk]

i have not read penrose's books but have had the delightful pleasure of meeting and chatting with him for a few hours. i think anything he writes should be interesting.

 

[robphy]

i have not read penrose's books but have had the delighful pleasure of meeting and chatting with him for few hours. i think anything he writes should be interesting.

Did you get a chance to discuss your criticism of his motivation of the "abstract index notation" (quoted in blue in post #11 https://www.physicsforums.com/showthread.php?t=107389 )?