Should I Become a Mathematician?

[mathwonk]

here is a site with free notes on a wide variety of topics:

http://us.geocities.com/alex_stef/mylist.html

 

[balletomane]

Awesome. Thanks.

 

[mathwonk]

Galois' theorem on solvability of polynomials by radicals
We will prove next that in characteristic zero, a polynomial whose
galois group is not a solvable group, is not "solvable by radicals", and
give an example of a polynomial that is not solvable by radicals.

Lemma: The galois group of a polynomial is isomorphic to a subgroup of
permutations of its distinct roots. If the polynomial, is irreducible, the
subgroup of permutations is transitive on the roots.

Cor: If an irreducible polynomial over Q has prime degree p, and exactly 2
non real roots, its Galois group is isomorphic to S(p).

Def: A primitive nth root of 1 (or of "unity"), is an element w of a field
such that w^n = 1, but no smaller power of w equals 1.

Lemma: If char(k) = 0, for every n > 0, there is an extension of k which
contains a primitive nth root of 1.

Theorem 1: If char(k) = 0, the Galois group G of X^n -1 over k is
isomorphic to a subgroup of the multiplicative group (Z/n)*, hence G is
abelian.
Rmk: If k = Q, then G ≈ (Z/n)*, as we will show later.

Theorem 2: If k is a field containing a primitive nth root of 1, and c is
an element of k, the galois group G of X^n-c is isomorphic to a subgroup of
the additive group Z/n, hence G is abelian.
Rmk: It is NOT always true that G equals Z/n, even if k is the splitting
field of X^n-1 over Q.

Theorem 3: If ch(k) = 0, and w is a primitive nth root of 1, and if k0 =
k(w), ki = k(w,a1,...,ai), where for all i = 1,...,m, ai^ri = bi is some
element of ki-1, and ri divides n, then the galois group of km =
k(w,a1,...,am) over k is solvable.

Def: A radical extension E of k, is one obtained by successively adjoining
radicals of elements already obtained. I.e. E = k(a1,...,am) where for
each i, some positive integral power of ai lies in the field
k(a1,...,ai-1).

Def: A polynomial f in k[X] is "solvable by radicals" if its splitting
field lies in some radical extension of k.

Theorem 4: If k has characteristic zero, and f in k[X] is solvable by
radicals, then the galois group of f is a solvable group.
Rmk: The converse is true as well, also in characteristic zero.

Cor 5: The polynomial f = X^5-80X + 2 in Q[X], is not solvable by radicals.
proof: The derivative 5X^4 - 80, has two real roots 2,-2, so the graph has
two critical points, (-2, 130), and (2, -126). Since f is monic of odd
degree, it has thus exactly 3 real roots, and 2 non real roots. The galois
group is therefore isomorphic to S(5), which has a non solvable subgroup
A(5) ≈ Icos. QED.

 

[Ki Man]

does fractal geometry have any practical uses

 

[mathwonk]

check this out;

http://classes.yale.edu/fractals/

 

[0rthodontist]

I read once that somebody made a fractal image compression algorithm for nature scenes. I don't think it was dramatically better than other algorithms for most subjects, but apparently it worked.

 

[pivoxa15]

In your very first post you mentioned

"basically 3 branches of math, or maybe 4, algebra, topology, and analysis, or also maybe geometry and complex analysis"

But what about branches like Statistics, Probability/
Stochastic Processes, Operations Research?
Do they fit into one of the 4 major fields you suggested? If so how would you put them in?

 

[jammidactyl]

The comment probably should have been, "three branches of PURE math".

Those topics you mentioned all fall under applied mathematics, with things like statistics and stochastic process borrowing heavily from analysis. The question is more, can I use some or invent some mathematical technique to solve this problem? So it doesn't really matter what branch that technique comes from.

http://www.math.niu.edu/~rusin/known-math/index/mathmap.html"

 

[mathwonk]

yes, i admitted later that i am incompetent in those other areas of applied math, and i appreciate any input on those topics anyone is willing to offer. i apologize if i gave the impresion my advice is comprehensive, as i am obviously limited by my own knowledge and experience.

i myself have studied only pure math, with courses in algebraic topology, algebraic geometry, functional analysis, riemann surfaces, homological algebra, complex manifolds, real anaylsis and representations.

then my research was entirely in riemann surfaces, singularities of theta divisors of jacobians and prym varieties, and their moduli.

so i am pretty ignorant of analysis, algebra, topology, finite math, probability, statistics, gosh almost everything.

I do know a little about theta divisors.

But I still feel free to offer advice!

i just meant to start this thread, not to dominate it. my apologies for its shortcomings.

 

[sherlockjones]

can one be both a pure and applied mathematician?

 

[pivoxa15]

Why don't you include complex analysis in analysis and geometry in topology so that there are (definitively) 3 branches of (pure) mathematics.

 

[mathwonk]

you could very reasonably do that. complex analysis , at least the homological kind I know most about, does have a rather different flavor from real analysis, but deep died analysts use a lot of real analysis to do complex analysis, via harmonic functions.

 

[pivoxa15]

you could very reasonably do that. complex analysis , at least the homological kind I know most about, does have a rather different flavor from real analysis, but deep died analysts use a lot of real analysis to do complex analysis, via harmonic functions.

That is interesting. So the complex analysis you do is more related to algebra than analysis?

Now I am going to ask, most likely a very stupid questions.

If someone wanted to solve the Riemann hypothesis, which branch of mathematics should they get into?

 

[radou]

In general, the more branches you get in, the better.

 

[JasonRox]

In general, the more branches you get in, the better.

Yeah, but you might want to do the relevant ones first.

I'd say Complex Analysis, Number Theory and Abstract Algebra would be the most relevant.

Not entirely sure if there are more important areas.

You might want to read "The Music of the Primes". That might give you an idea of what you're getting into.

 

[quasar987]

What is everyone's favorite book on the Riemann hypothesis? There's

Prime obsession
The Music of the Primes
The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
Riemann's Zeta Function

And probably a miriad of other ones. Which one's the most interesting to read?

 

[mathwonk]

i rather liked Riemann's own paper.

 

[mathwonk]

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.

 

[sherlockjones]

mathwonk, what do you think of the math department of this college:

http://www.rose-hulman.edu/math/home.php

 

[mathwonk]

i never heard of it before but it looks like a really good undergraduate college department, with a deep commitment to teaching and nurturing undergraduates. i like what i see on their website.

 

[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".