Shimura-Taniyama-Weil (STW) Solved 186
timbo_red writes "The BBC report that an international team of scientists have solved the STW conjecture. I vaguely remember what this is from reading the Fermat book, I'll have to check it again. " This really has me interested in the conjecture. Anyone have any good links for background reading?
Warning (Score:1)
I read it, and now my head hurts.
can you say esoteric? (Score:1)
"there exists a modular form of weight two and level N which is an Eigenform under the Hecke series and has a Fourier series".
I'd get a lot more excited if I knew what that meant.
More info (Score:5)
Re: (Score:2)
Who wrote this? (Score:3)
Only front-line mathematicians will really understand the STW conjecture. But you could say "there exists a modular form of weight two and level N which is an Eigenform under the Hecke series and has a Fourier series".
Ahh... it's all becoming clear now.
Re:More info (Score:2)
If you have a casual interest in this area of mathematics, good places to start might be Ireland/Rosen, A classical introduction to modern number theory, or Silverman, The arithmetic of elliptic curves (both Springer GTM). See also my bibliography [mit.edu] of math textbooks.
Esoterism is good for you :) (Score:1)
As to those of you who understand both the STW conjecture AND coding.....TTHHHHPPTTT
More accessible reference to STW theorem (Score:4)
Hmmmm... (Score:1)
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20th Century Mathematics (Score:2)
I find it facinating that math is so deep, that even though I took several math classes after Diff Eq. I still can only barely understand some of the stuff they are talking about.
One professor of mine once remarked that 20th century math concepts aren't really touch upon unless you are pursuing a math degree. The 'newest' math concept for most students being dot/cross-product notation - and I think that was 19th century, if memory serves.
I guess people that think in 9 dimensions scare me.
Tom
Hmmn.... (Score:2)
Links to STW Info (Score:5)
Like the article says, Wiles solved a special case of STW to knock off Fermat's Last Theorem. I guess this is a proof of the general version (but the article is a little vague--any number theorists around who are in the loop?)
Re:SUICIDE! (Score:3)
Disclaimer: IANAL, IANAM.
Re:Hmmmm... (Score:2)
Yes, STW is intuitively obvious to the casual observer once you have the Wiley proof of Fermat's Last Theorem.
I will leave the details as an exercise for the student.
Leaving aside the question of does the definition of "scientist" include "mathematician"
No. All scientists need to be mathematicians to some extent, but the reverse is not true. Science includes the formal consideration of experimental evidence as part of a model building process, but mathematics can be a purely abstract endeavor without an empirical component.
Don't worry, mathematicians (Score:2)
but we can always resort to lying and obfuscation.
Chuck
Re:Hmmn.... (Score:1)
an atrociously written article (Score:1)
NPR real-audio link (Score:5)
Before everybody starts screaming "this is old news" remember,
Re:Hmmmm... (Score:1)
Other way around, actually. A while after STW was published, somebody published a paper showing how to prove FLT given that someone had proven STW. (forget exactly who did it, since it's been a while since I've read Fermat's Enigma)
Re:20th Century Mathematics (Score:1)
Why all this is really, really important (Score:4)
I used to think that math wasn't much of a direct use, but this is incorrect and a lot of it has to do with how mathematics is taught in western culture (something we should be ashamed of; Most people don't do any calculus until senior high school if at all!).
What math does is provide a (perhaps the) universal language with which to describe the universe, science, language, everything. Everything can be represented and manipulated in some form with math - this is what computers do! (discreetly >:).
Discovering relationships between unrelated fields of math allows the scientists and engineers of tomorrow to use these descriptive tools to develop new cool gadgets. ;)
Kudos,
Re:20th Century Mathematics (Score:1)
Re:More accessible reference to STW theorem (Score:3)
Clark
--
Finding a job shouldn't be work.
Re:More accessible reference to STW theorem (Score:1)
Re:Esoterism is good for you :) (Score:1)
Andrew Wiles information, resources (Score:3)
Re:an atrociously written article (Score:4)
It is rather unfortunate that the BBC correspondent has very little idea about the subject he is writing about
It's always the same from the Beeb. "Real" intellectuals have arts or humanities degrees, mathematicians are just geeks and beneath contempt. Did you notice the "related links" they placed on the page ? The top feature was last year's "mathematics of biscuit dunking" story. This just shows what little significance the increasingly dumbed-down BBC now places on science or technology stories.
ObRant: Why is it that at the hypothetical mixed-background middle class dinner party, the scientists are expected to be literate, but the literati still revel in their innumeracy ?
It is true that the subject is too esoteric to be accessible to non-mathemticians,
There's probably at least a dozen people in this room as me who work on elliptic curves on a daily basis. OK, so I work in an unusual environment, but these things do actually have real world applications (crypto, natch) and not just for the NSA.
Re:Hmmn.... (Score:2)
1. There are 3 "main" types of geometry: euclidean, hyberbolic, elliptical.
2. In each of these the difference is that the sum of the angles in a triangle are respectively equal to, less than, and greater than 180 degrees. Note also that in euclidean geometry going there is exactly one line parallel to a first line passing through a given point not on the first line. In hyperbolic geometry there can be an infinite number and in elliptical there are none (for the last think of drawing lines on a globe - any two lines must intersect).
3. There is no problem with Euclidean geometry in 4 (or more) dimensions. Just picture 4 (or more) lines intersecting perpendicular to each other. Try not to induce a headache.
Re:More accessible reference to STW theorem (Score:1)
How to confuse Babelfish (Score:2)
In an effort to convert the British Broadcasting Company's text to English with Babelfish, I discovered a shortcomming in Babelfish's software. It could not convert this short article into English. DEC (or Compaq or whoever owns Alta Vista this week) really needs to improve their Math-speak to English converter. Maybe, they should OpenSource (tm) it so we could all help?
Perhaps we need a *math* section... (Score:3)
Rob, is there a way to get a math section? I know that crypto is a popular subject on Slashdot, and it's very closely tied with math. I know that a lot of geeks also like to hear about the STW conjecture being solved. It's all very reasonable-- fields as diverse as biology and physics have strong ties with mathematics.
I'm not complaining-- I love Slashdot, and I'm glad that this story was posted. I really think that, while the math is beyond my abilities, it's cool to at least know that the conjecture was proven. It's also pretty neat that I can find out why this is important to the rest of mathematics.
But when I see it posted under the "science" heading, I can't help but cringe a little. It's not likely that this is going to revolutionize science. And there are a lot of geeks who wouldn't care about it because of that fact-- no applications? Why the hell would you bother with it? Giving mathematics articles their own topic heading would most likely be useful to these folks.
I'm also seeing a lot of people joke and complain about the horrible headache that they received just viewing the article. If articles were placed under the mathematics heading, a lot of this can be prevented. This is partially due to the fact that users can filter out the stories, and partially due to the fact that anything under the "Mathematics" section would sort of carry an implied warning-- "Don't read this unless you are *really* in to high-level shit".
So perhaps it's best that a new section, "Mathematics" be created. It would be very much appreciated. I know you're a busy man and all, but it would please a whole lot of us anal retentive blowhards.
Background reading? Doesn't exist as such... (Score:1)
To really understand this sort of thing you have to do it for years and really be interested in it. Personally, I think it all sort of looks interesting until you sit through a lecture in algebra...
Re:Esoterism is good for you :) (Score:2)
As a practical example, I remember once compressing sparse matrices (parser tables) by mapping them to a graph (one line = a node, with node connectivity defined by line "overlap"), then using a minimal graph coloring heuristic.
Re:Who wrote this? (Score:1)
Apparently with a math minor I am able to understand:
What a fourier series does
What an Eigenvalue/vector is (which is prolly different than an Eigenform)
I do not know...
What a Modular form is
What a Hecke serires is
What an Eigenform is...
"under the"
Any person with a greater understanding care to explain what some of those things are?
Re:Hmmmm... (Score:1)
Personally I shudder to think what having to take all that math could do to a person if they had to get a job in something like computers! Basically it's nice to do something with math; however the approach to the subject needs a little work. It seems that the higher you go in math the more bland and unapproachable the subject becoms and the more difficult (difficulty!) it becomes. Why can't people produce a nice graphical textbook about complex math subjects with plents of examples and problems to work on? I even see calculus books that fall victim to the same type of thing. If anyone really cares I think that stories about random complex mathmatical subjects should not be covered if we don't have some of the more interesting political subjects. I have a bias towards things that have a lasting importance versus things that have a limited appeal. How can you tell little Billy about STW? You can't. Can you tell Billy about the current political situation in the Balkans (at least easier than some topic where most of the areas in math you can't get to with graduate levels of math education).
And for the mentality that Americans being brain dead and the rest of the world having a natural brilliance towards math I would beg to differ. I think it's only in places in the US where people have the free time and material wealth to do research and be able to have means to feed themselves makes things like this possible?
Are there any good textbooks (graphical, examples galore, problems) that would make someone an Einstein a little easier? Real world examples?
Attempted Math to Slashdot Translation (Score:5)
First, let's start somewhere seemingly unrelated that may be easier to deal with: Physics; specifically, gravity. I'm working under the assumption that everyone knows what gravity is, so, good. There are other forces that do similar things in Physics, however. The most common are the "Strong" and "Weak" electromagnetic forces. The force that holds electrons close to an atom, and that bonds atoms to each other in molecules are examples of these forces.
Now in Physics, there is a holy grail of theory called the 'Grand Unification Theory'. This is big important stuff. In an amazing oversimplification, it suggests that there is a single formula that relates all of these forces together. We _expect_ this from intuition, we currently just don't have any idea how to prove it, although progress is being made all over the place.
Now, skip back to mathematics. Mathematics is split into tons of different areas. Statistics, Number Theory (the stuff normally used in cryptography), Calculus, and so on. Robert Langlands proposed that there is a Grand Unification Theory (GUT) of sorts for mathematics. This is commonly referred to as the Langlands Proposition (or Program, according to the BBC article).
Some years ago, Yukata Taniyama (The 'T' in STW) asserted a conjecture that did two things. First, if proved, it would bring elliptic curves and modular forms together in the spirit of the GUT, thus giving the Langlands program a big push. Secondly and, while not really more important, at least more interesting to the public, he showed that if his conjecture was proved, the most famous unproved theorem at the time would follow. I speak, of course, of Fermat's last theorem (FLT). This was the holy grail of math at the time.
A few years ago, Andrew Wiles proved enough of Taniyama's conjecture to prove FLT. This was what made STW mainstream; had this not happened, noone would care and the BBC story would probably be overlooked. But it did happen, made lots of papers, was flawed, fixed, flawed again, and currently is believed to be correct.
What recently happened, in the BBC story, is that the _rest_ of the STW conjecture was proved. Not just the part that Wiles used to show FLT, but all of it. In math this elevates STW from a conjecture to a theorem and makes mathematicians everywhere giddy with joy since the Langlands Program is slightly closer to being proved.
And of course, giddy mathematicians are the types who post stuff to Slashdot, which is why this article is here at all.
Was that any better?
Re:Why all this is really, really important (Score:2)
Did Fermat really prove it? (Score:2)
Many people believe that Fermat had a flawed proof for his theorem. There are many reasons for this belief, most of which I am entirely unqualified to judge and the rest of which I probably shouldn't judge, but I think it rather likely that his proof was flawed. The sheer number of brilliant minds who attempted to prove it, and the fact that the final proof used such modern techniques, suggests to me that it is unlikely that Fermat had a valid proof.
I know this isn't really related to what you had to say, but I thought it was interesting enough to mention... and maybe someone who knows more about it will have something useful to mention.
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Re:Hmmmm... (Score:2)
Andrew Wiles did this. now, Conrad, Taylor, Diamond, and Bruile (?) have proved STW for all elliptic curves. that's the breakthrough, and the announcement was made earlier this year. it isn't "news" in the popular media sense.
by the way, the BBC has something wrong, the proof is not printed in the Notices Dec issue (ha!), an announcement is. it's been on my desk for about a week now.
- pal
interesting stuff (Score:1)
the proof of this conjecture is pretty cool, and i am looking forward to what it means for cryptanalysis and cryptography in general.
As a math major... (Score:1)
Just as a little note of interest, while it is amazing that these guys have proved this theorem, it has long been suspected to be true, since back in the '60's when it was proposed.
In fact, Wiles used a theorem that was proven years ago that IF the STW conjecture was true, then Fermat's Last Theorem would be true (Wiles proved a smaller subset of this problem).
So while the result is interesting and useful (and certainly needed), the consequences have already been explored.
LL
interesting stuff (Score:1)
the proof of this conjecture is pretty cool, and i am looking forward to what it means for cryptanalysis and cryptography in general.
Re:More accessible reference to STW theorem (Score:3)
On the flipside, I would like to warn people who do know some mathematics that they probably won't like this book. As a student of Mathematics (if very much a beginner) I found this book mostly frustrating, with long passages on the obvious stuff and no explanations where I got curious.
Which I guess goes for any reading of pop-science within one's own field. I'll just have to study for a few more years until I can understand tackle the true texts on the subject. No shortcuts in life...
-
We cannot reason ourselves out of our basic irrationality. All we can do is learn the art of being irrational in a reasonable way.
Re:Did Fermat really prove it? (Score:2)
a cyclotomic extension of the integers is what you get if you take the integers, throw in some new element x so that x^p=1 and x^q1 if qp, and close it under addition and multiplication.
kummer proved fermat in 1840ish using this technique, but it only works when the extension at hand has unique factorization (i.e., for "regular primes"). and fermat probably thought this was reasonable in 1637.
by the way, the fact that fermat mentions his proof for n=3 and n=4 repeatedly, but never (besides the one note in that one letter) again mentions the general case leads us to believe he may have realized his error, or at the least thought it was true and was not able to prove it.
- pal
Re:Elliptic Cryptography (Score:1)
The cryptogram newsletter last week had an article about products that advertize strong encryption with just such algorithms that currently use much smaller encryption keys than other systems like triple DES.
Does anyone out there know more about this?
Re:interesting stuff (Score:1)
modular arithmetic != modular forms. the former you are familiar with, obviously, but the latter is a complex analytic structure ("complex" here in the technical sense, this is not a judgement on my part).
- pal
Re:interesting stuff (Score:1)
I'm not aware of any crpyto schemes that use these curves as part of the encryption scheme, but I'm not that versed in those areas.
I agree that it would be interesting to see if modular forms can be used to aid ECM factoring methods, now that the two fields are so closely related. It'll take a while for people to hunker through the math to figure it out, however.
Re:Did Fermat really prove it? (Score:1)
a cyclotomic extension of the integers is what you get if you take the integers, throw in some new element x such that
a) x is not an integer
b) x^p=1 for some p
(ie, x is a root of unity). and then close this set under addition and multiplication.
for the math majors: Z[x]/.
- pal
Mathematical References for the really interested (Score:3)
Re:interesting stuff (Score:1)
Re:Hmmn.... (Score:1)
You are right about subdividing geometries like that but that's not the way how you construct a space.
First you take a set of elements and from there you can go in a couple direction:
-you can define on your set an inproduct and from there on you can define your angles. for example you can take a sphere and the triangles on the sphere doesn't necessarily add up to 180 deg.
-You take a field (field of real numbers, complex numbers,...) and define the action of those fields and creating a vectorspace (dimension is invariant of the chosen base)
-You take a ring R (R,+ is an abelian group, R,. is not necessarily communitative) and do the same as above. You'll get a module. The cool thing about this sort of stuff you can't even speak about a dimension because it could change when you chose another base.
-From modules/vectorspace you can construct a projective space. The most famous example is the Fano-configuration. Draw a triangle in the center of the lines and in the center are the other points (7 points total). In that projective plane the center points of the lines of the triangles can be connected by another straight line...
Another funny result is a square whos diagonals are parallel.
-...
When you study relativitytheory, you'll notice they aren't using euclidic space but Minkowsky space. The difference between those spaces is the inproduct. The bilineair form which defines the inproduct can be represented by a matrix with on the diagonal 1,1,1,1 and the rest zero (the last row/column is time)
A Minkowsky has 1,1,1,-1 as inproduct matrix. Which means if you calculate the inproduct of (1,0,0,1) with itself you'll get 0. This means that vector forms a right angle with itself. (I believe it's called isotrophic vectors)
One can always get the shortest path by calculating the geodesic lines (I only saw the 2-dim case but can probably be extended to n-dimensions)
Anyway, I just wanted to point out the theory can be as counter intuitive as you'd like.
---
I'd like to apologise for any brain damage suffered when reading this
Are you crazy? (Score:1)
Re:interesting stuff (Score:1)
:)
A few remarks (Score:5)
I followed a one-semester graduate course (by Laurent Clozel) on the proof of the semistable case of the Shimura-Taniyama conjecture (the case proven originaly by Wiles and which concludes the proof of Fermat's theorem). So I can make a few comments on the subject.
The Shimura-Taniyama conjecture (Weil's name is attached to it for rather dubious reasons: essentially, he mentioned the conjecture — as an exercice for the interested reader! — in a book he published; Serge Lang is always ready to flame anyone calling the conjecture by Weil's name, so let us omit Weil) concerns a correspondance between certain modular forms and certain elliptic curves (actually with Galois representations in between the two). That is, it states that every elliptic curve is associated to a certain modular form (the association can be stated in many different ways: they have the same L function; the eigenvalues of the modular form for the Hecke operators can be deduced from the number of points of the elliptic curve on finite fields, and so on). This conjecture was known (i.e. formulated) long before any relation with Fermat's theorem was observed.
Gerhart Frey had noticed that if a counterexample (A,B,C) (with A+B+C=0, A, B and C being p-th powers) to Fermat's theorem were found it would yield an elliptic curve y=x(x-A)(x+B) having certain miraculous properties, including being ``semistable'' and possibly violating the Shimura-Taniyama conjecture. Using works of Jean-Pierre Serre, Ken Ribet was able to prove this remark of Frey, so that the Shimura-Taniyama conjecture, and in fact even only the Shimura-Taniyama conjecture for semistable elliptic curves, would imply Fermat's theorem.
At that point it became obvious that it would be only a matter of time before Fermat's theorem were proven. Andrew Wiles, was able to complete the task. His first proof contained a flaw (in trying to construct an Euler system), which was noticed by Luc Illusie, but with the help of Richard Taylor, Wiles was able to replace the technique of Euler systems and use Gorenstein rings instead (and some very fine points of commutative algebra) and correct the proof. The full proof (Wiles' ``Modular Elliptic Curves and Fermat's Last Theorem'' and Wiles and Taylor's ``Ring Theoretic Properties of Certain Hecke Algebras'') was published in Inventiones Mathematicæ. Thus, the case of Fermat's theorem was settled.
The general case of the ST conjecture was still unproven. However, soon after Wiles' result, Fred Diamond improvement over it. To understand it, you must know that semistability of an elliptic curve is a ``local'' property, i.e. it can be tested for each prime number. An E.C. is (globally) semistable iff it is semistable at every prime number. (It is always semistable at all but a finite number of primes.) Wiles' result required the E.C. to be semistable at all primes; Diamond refined that and proved the modularity of elliptic curves that are modular at 3 and 5. This was a considerable progress, and it was then pretty obvious that these last conditions would be eliminated. Now they have been (every elliptic curve is known to be modular), but this is more a question of technique than a fundamental improvement.
One might be tempted to think that the proof of the ST conjecture is fascinating. In fact, I found it (or at least the semistable case, which has, it would seem, the gist of the ideas) terribly boring. It is all a matter of controling the behavior of the ramified parts of the cohomology groups of some Galois representations, and it is done in a succession of lemmata, each one seeming exactly the same as the previous one. In fact, the experts' opinion is that the proof of the conjecture is technically difficult but fundamentally trivial in that it does not use any deep results from (algebraic) geometry.
The ST conjecture is part of a more general scheme called the ``Langlands programme''. The Langlands programme is a correspondance (which has not been formulated in a completely satisfactory way, as far as I know, let alone proven) between higher dimensional abelian varieties (elliptic curves are abelian varieties of dimension 1), Galois representations and modular forms (disclaimer: I don't know half of what I'm talking about here). ``Class field theory'', the climax of the number theory of the beginning of the century, is the case ``GL1'' of the Langlands programme (the abelian case). The Shimura-Taniyama conjecture was the case ``GL2'' of the same programme. Some other cases have been proven, such as ``Sp4'' (these funny acronyms refer to certain algebraic groups: GL is the General Linear group, and Sp is the Symplectic group).
The Langlands programme actually splits in two parts: the ``number field'' (or ``global'') Langlands programme, the hard number-theoretic part, of which the ST conjecture is a particular case, and the ``function field'' (or ``local'') Langlands programme, which is an easier analogue of more geometric content.
The major news recently is that the ``function field'' Langlands programme has been proven, by Laurent Lafforgue. This is much more important than the full proof of the ST conjecture. And it also means that Lafforgue will be getting the Fields medal in three years (mark my words).
background reading: FERMAT'S ENIGMA (Score:2)
Get it h ere [fatbrain.com] at Fatbrain (does
Re:More info (Score:5)
I have been told by an applied math geek friend of mine that STW is another one of those "it's all connected, maaaan... [hightimes.com]"-type theories along the line of "e^(pi * i) + 1 = 0 [aol.com]", although a good deal messier [geocities.com]. I've also been informed that STW was used heavily in Wiles' proof, not unlike a load-bearing block in Jenga [demon.co.uk].
(Never mind "First Post!" I hereby start the new tradition of "Most Links!" After all, it's more productive, and more importantly, it's all connected, maaaaaan....)
Re:Perhaps we need a *math* section... (Score:1)
smth like `Mathematics and Algorithms'.
As to me, this is really wanted. I don't care
too much about crypto and such, but news on
computation theory, algs, data processing would be
very usefull.
Or called it `Applied Math'
TIA
Re:Why all this is really, really important (Score:1)
Green's functions were known and studied in the abstract for over a hundred years by mathematicians before Richard Feynman drove home the point (in the 1950's) that physical particles (electrons, protons, etc) are Green's functions.
A few more decades and that theory was transported to phonons, holes, etc. and lead to the explosion of understanding of semiconductors and transistors. And, of course, where would we be without that?
Hi-tech is not possible without the foundations of high-math.
Re:Who wrote this? (Score:1)
f(z) = (cz+d)^(-k) * f((az+b)/(cz+d))
with a,b,c,d integers, ad-bc = 1. These form a
vector space. (k is called the weight).
"Level N" is a technical condition on a,b,c,d.
Hecke series: Really a Hecke transform, it's a
linear operator on the space of modular forms.
Eigenform=eigenvector.
The fourier series business just means that the
form is 'nice' at infinity...
see, it's all simple.
Until they start talking about this
Galois cohomology business. Ugh.
Re:Attempted Math to Slashdot Translation (Score:1)
Re:Exactly what do you do with a degree like that? (Score:2)
Real-world relevance of higher math:
And you know, I'd hold "furthering the bounds of human knowledge" to be an good thing unto itself, regardless of any real-world applications for, say, generalized statements about Ramsey theory and the Party Problem (to choose something which I've been doing a bit of amateur reading on lately that might well lead to real-world applications).
My favorite open conjecture .. (Score:3)
Does anybody know the status of this problem? I recall reading something a while back about how somebody determined that this problem is undecidable, though I could be wrong.
When I was in college taking a History of Mathematics class years back, I was fascinated by this one. I even spent a fair amount of time hammering away at it, and while I came up with a few interesting ideas, nothing substantial came out of it. I was working using Euclid's famous proof of the infinitude of the primes as an inspiration. Anybody who's seen that proof knows that in mathematics, sometimes a correct proof can be completely unexpected and yet incredibly elegant and simple at the same time.
Re:interesting stuff (Score:2)
'All of the fastest algorithms for calculating discrete logs -- the number field sieve and the quadratic sieve -- make use of something called index calculus and a property of the numbers mod n called smoothness. In the elliptic curve group, there is no definition of smoothness, and hence in order to break elliptic curve algorithms you have to use older methods: Pollard's rho, for example. So we only have to use keys long enough to be secure against these older, slower, methods. Therefor, our keys can be shorter.
Does the proving of this conjecture open the way for a 'smoothness' function to be defined? Crypto-gram can be found at: http://www.counterpane.com/crypto-gram. html [counterpane.com]
Brian Haskin
Re:Elliptic Cryptography (Score:1)
I'm no expert in this, but I don't believe that this proof has too much impact on the mathematics of today. STW has been known to mathematicians for some years, and they believed that it is true, but they didn't prove it. They even built new theories using the asumption that STW is true. These theories were (if they are useful in that particular case) certainly applied to elliptic cryptography.
But it is important that STW is proven, and that we know now that theses new theories are really true. Just imagine a cryptosystem based on the assumption that STW is true and in reality it wasn't...
Re:Hmmmm... (Score:4)
All that learnin hurts the brain as we all know.
It seems that the higher you go in math the more bland and unapproachable the subject
becoms and the more difficult (difficulty!) it becomes
Well yes it becomes more difficult...just like coding for X is alot more complicated than hello world. However, it actually becomes MUCH more interesting. Think about it...addition and subtraction are pretty fucking boring while higher mathematics gives you stuning results such as the Banach Tarski Paradox (A sphere may be cut up into finitely many pieces and by translating and rotating the pieces reassembled into a two spheres of the orignial size).
Why can't people produce a nice graphical textbook about complex math subjects with plents of examples and
problems to work on?
Math books with examples and problems to work on are fairly common. The reason the textbooks often aren't (and shouldn't be) graphical is because mathematics is not a graphical pursuit. It would be like explaining perl via venn diagrams. Yes, some parts of mathematics MODEL the real world (such as R^3) but all to often people taught via pictures are restricted by them. As soon as they run into a problem without an obvious visual component (say a problem in R^4 (yes it can be useful)) they are stuck.
I have a bias towards things that have a lasting importance versus
things that have a limited appeal
Question who is more famous? Archimdes or the political leaders of athens? It in fact appears mathematics is of much more lasting imprtance than whatever war is occuring at the moment.
How can you tell little Billy about STW?
As we all know little billy is the ultimate judge of these matters. I imagine huffman encodings shouldn't be studied either.
Ohh while not a textbook their is a book On relativity or something either written by einstein or from his notes which is exceptionally good.
Re:Why all this is really, really important (Score:1)
This was drilled into our heads during EE classes at Princeton; it's a shame your professors didn't make the distinction clear. Phasor manipulation is a *short-cut*... not the real thing.
Math is actually the most versatile degree you (Score:4)
>Personally I shudder to think what having to take all that math could do to a person if they had to get a job in something like computers
Let's see.. impeccable logic... a rock solid understanding of algorithms...Top notch problem *defining* and solving skills.
yeah, not too useful in computers.
>It seems that the higher you go in math the more bland and unapproachable the subject becoms and the more difficult (difficulty!) it become
While there is no arguing that higher mathematics is difficult to wrap your brain around, I would rephrase the first part of this sentence.
I got a BS in mathematics taking several Graduate classes in the process(Real Analysis (The Horror) and Differential Equations/Dynamical Systems) and I would say rather than "bland and unapproachable"
:
Incredibly beautiful, deep, elegant and powerful, but with a much higher price of admission than any other field.
>Why can't people produce a nice graphical textbook about complex math subjects with plents of examples and problems to work on?
Well even in relatively simple math courses i.e. past the basic calc/diffEQ/linear algebra/ 2 year series, you are dealing with n-dimensional spaces.
The fact is there is no way to draw this. That is where the full power of the abstract approach is needed.
For example, take as your space the set of all functions from the real numbers to the real numbers. How the hell do you even draw anything dealing with this? If I remember correctly, this space has a cardinality ("number" of members) greater than that of the real numbers which is strictly greater than the usual "infinity" which is the cardinality of the Natural numbers/integers/rationals
>I think it's only in places in the US where people have the free time and material wealth to do research and be able to have means to feed themselves makes things like this possible
Well given that math and physics were almost completely re written a few hundred years ago by
Newton(England) Leibnitz, 23(?) different Bernoullis(SP),Gauss, Cauchy, Cantor,Riemann (Germany), and a few French people whose names slip my mind
Oops regarding computer theory we can't forget the Russians especially Kolmogorov
Are there any good textbooks (graphical, examples galore, problems) that would make someone an Einstein a little easier? Real world examples?
Einstein was a genius who did very poorly in school. He was not even accepted to any grad schools until he completed his Nobel prize winning work (Not General or Special Relativity either).
There is no easy way to understand the advanced results of mathematics without struggling your way up. Some people will have an easier time than others, but I feel that it is worth it even if I never use the specific facts I learned.
Mathematics has many "real-world" uses that haven't been discovered yet. In general Mathematics is decades and often centuries ahead of the relevant scientific fields. Abstract Algebra (not like in high school) was considered the most esoteric useless field by non-mathematicians until it became indispensible in quantum guage theory.
Superstring theory is built upon Some-old-guy-or-other's Beta function and Symmetry group theory.
General Relativity is written in the language of differential geometry.
To understand some of these theories is a mind blowing experience I would highly recommend.
Seriously though eve if you don't decide to pursue it you will be prepared for anything else you do want to do. You can go to grad school in almost any discipline, and your problem solving skills will exceed those of almost anyone you interview against for a job.
---CONFLICT!!---
Rich Media Links (Score:2)
Re:an atrociously written article (Score:1)
but these things do actually have real world applications (crypto, natch) and not just for the NSA.
I agree. For example Hamming codes (error correction, not crypto) are useful for telecommunications and folks who do that use some fairly high powered algebraic geometry over finite fields (I am not an expert in that, but i know some people who are doing that stuff).
Re:My favorite open conjecture .. (Score:3)
Interesting sidenote until recently whenever an example was needed in philosophy papers about a statement whose truth was unknown but which was in principle implied by the information at hand (i.e. proving we don't know the logical consequences of all our factual data) fermat's last theorem was used. They have had to switch over to the Goldbach conjecture.
Another wonderful unsolved conjecture is the collatz or 3n+1 problem.
Given x run the following algorithm
if x is even divide x by two
if x is odd take 3x+1
repeat until we get 1.
Does this algorithm always terminate? (Erdos was said to have remarked that we [the matematics community] was not ready for such problems).
Excersice for you assembly buffs out there how fast can you write an algorithm to check out the conjecture (i.e. test it for all starting x below some number). I tried writing it in C and even my shitty assembly was orders of magnitude faster. I believe the conjecture has been verified up to an incredibly large number.
Unique Factorization Domains (Score:3)
This is a bit of summarizing and paraphrasing from Joseph A. Gallian's Contemporary Abstract Algebra.
"Most likely, he made the error that his successors made by assuming that the properties of integers, such as unique factorization, carry over to integral domains in general."
In 1839, Gabriel Lame announced a proof to FLT. It involves a fairly simple factorization of x^p+y^p into factors with complex coefficients.
The problem is that in this situation, factorization into irreducibles is not unique. This is a property of the integers (45=3*3*5 and no other primes). This property is only true of certain types of algebras--called unique factorization domains. The algebra (or ring, if you're literate) involved in the factorization used by Lame did not hold the property of unique factorization. The proof is much simpler than Wiles' if you assume the property of unique factorization, which was likely Fermat's mistake.
Anyone who's interested in these terms should pick up a college text on abstract algebra. You'll need to read most of an introductory text....
By the way: MATHEMATICIANS ARE NOT SCIENTISTS. We are theorists. I expected more from the slashdot community.
Re:Why all this is really, really important (Score:1)
"It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance." -Bertrand Russell
Re:More accessible reference to STW theorem (Score:1)
Re:My favorite open conjecture .. (Score:2)
One (trivial) correction: greater than or equal to 4, since 4 = 2+2 can be written as the sum of two primes as well.
A lot of famous mathematicians have tried their hand at this problem, with no success to date. The first one was Euler: in fact this problem was stated by Christian Golbach in a letter to Euler, who apparently believed the conjecture to be correct.
There have been some "close" results. A Russian mathematician circas 1930s proved that every even number can be written as the sum of not more than 300000 (three-hundred-thousand) primes. As William Dunham points in "Journey Through Genius" this proof falls short of the goal slightly, namely by 299998 primes.
Back to the subject of open problems and the STW, it is a much welcome development that the STW has been proved finally. This is because a lot of time has been spent developing algebraic results of the form "If STW then..." FLT is certainly more interesting from a philosophical standpoint but very few results depend on it.
STW on the other hand is very similar to that other great open question, the Riemann hypothesis which factors into many important results. Starting with the 19th century, people used the Riemann hypothesis (and various generalizations) to "prove" results including the density of primes and even efficient algorithms for checking primality. STW being proved false would have some major repercussions, just as the Riemann hypothesis refutation will cause serious trouble.
--bluespower
Re:background reading: FERMAT'S ENIGMA (Score:1)
I imagine your local library ought to have a copy. If they don't, tell them to order one. Then go to your local College or University library. They better have a copy!
Re:A few remarks (Score:2)
Appreciate the breadth of your mathematical knowledge. Thanks for posting.
One question for you: if this is dull math, what do you consider interesting?
Salut.
Ken Ribet (Score:1)
The fact that FLT is true in itself doesn't however say anything about STW.
It seems that Ken Ribet is involved in the current full proof of STW.
Fermat's Last Theorem (Score:1)
Here's the address of the pbs page on the episode (Nova #2414)
http://www.pbs.org/wgbh/nova/proof/
It also has some links to some good math resources including Wiles' page.
Re:Hmmn.... (Score:1)
One of the largests gains that the proof of FLT has brought to math is that in the long run it will likely diminish the number of not-quite-sane people bothering mathematicians with their 'proofs' of FLT. I've had to deal with a number of those myself. You may think that a false proof is just that, and that pointing at the first error in the logic is sufficient, but it doesn't really work that way, because these proof just don't have any logic in them.
Re:Exactly what do you do with a degree like that? (Score:1)
Really, really, really bad writing. (Score:1)
I'll hunt the web to see what this is really about. As for the BBC, they should transfer Dr. Whitehouse to a job that doesn't require him to actually attempt to communicate with anyone else by any means whatsoever.
Re:Why all this is really, really important (Score:1)
First, I agree that math is a great tool for all of science, but: It is much much more than a language, even a universal one. Math is soooooo much more than a descriptive tool. Math is full of very deep and beautiful connections. Those can be described in mathematical language, but the description in itself isn't very interesting. In this STW case, the connection between elliptic curves and modular functions has always been there, and we as mankind have finally uncovered it and understood enough of its secrets to see why this connection is there.
So maybe it's not of immediate use. Maybe it will never be of any practical use, or maybe it will. I don't care all that much. I'm sure most research in psychology is of even much less value to society. And don't forget: most of these mathematicians teach classes too, and some of their students might indeed become scientists and engineers developing new cool gadgets.
Re:Attempted Math to Slashdot Translation (Score:1)
Re:My favorite open conjecture .. (Score:1)
Re:My favorite open conjecture .. (Score:1)
Re:My favorite open conjecture .. (Score:1)
As far as I know, the closest thing that has been proved is that any sufficiently large number can be written as the sum of two primes or a prime and a number that is the product of exactly two primes. This was proved by a chinese mathematician. I seem to remember that this mathematician died fairly recently.
Re:an atrociously written article (Score:1)
I don't know that is, but it is true, and in fact most scientists are very literate. I don't know why literati revel in their innumeracy, but I usually blame it on their education :)
Re:A few remarks (Score:1)
I however do have an interest in the history of mathematics and you are correct in calling the attachment of Weil's name dubious. In fact, I have seen it placed first. It is my understanding that in the sometimes political world of acedemia egos play a small role;) I believe others have mistakenly given more credit to Weil than he deserves,and he has done little to dissuade this.
Boy am I glad to be dumb enough not to have these worries.
All Conjecture aside... (Score:2)
I'd like to hear some examples of how this new technology is going to enable us. Will it allow visualization of data? Will it allow additional methods to be applied to the solution of formerly unsolvable problems?
I'd also like to say that I disagree with a previous poster's assertion that Mathematics and advanced number theory isn't science. A mathemetician see's patterns, theorizes, proves; how is that different from working with physical phenomena? Mathematics MODELS the physical - I believe that there isn't ANYTHING that exists that cannot eventually be modeled using mathematics. There is NO SCIENCE without numbers; ask Lord Kelvin.
The Greeks were right... working with numbers is the closest thing to being a magician; there is magic in it undeniably!
Re:a scientist who cant explain self to 7yr old... (Score:2)
The most advanced mathematics courses geek types typically take is differential equations, which usually consists of fairly mindless equation manipulation is hence is quite literally nothing like what a typical mathematician does. This is really unfortunate, as much of mathematics is quite beautiful. Great mathematicians are great artists, but appreciating the art has an extremely steep curve.
As for applications, people need to accept that going from understanding something to using an indirect consequence to build a sturdier lunch box could takes hundreds of years. It's a long chain, after all; math to physics to engineering to corporations to consumers.
There are deep and extremely important connections between number theory and physics, e.g. vertex operator algebras, string theories, zeroes of zeta functions, eigenvalues of large random matrices. Understanding these connections, in math as well as physics, is thus key to future progress.
--
Chris Long, Departments of Mathematics & Statistics, Rutgers University
well, maybe, but... (Score:2)
routinely used "unproven results" before the mathematical machinery is totally developed...
For example, Heaviside algebraic operator theory was used for solving linear differential equations
before the mathematicians finished proving the domain of applicability (Laplace et al)...
Newton's fluxions were used long before integral calculus formalized the operation of integration.
Not to mention infinite series, asymptotic analysis and the list goes on and on...
The quest for "truth" in mathematics has been a long, unexpected journey... If you haven't studied
up on it, read about Hilbert and his program to formalize math... then read about Godel and how
he showed that sometimes this mathematical foundation is really a mirage.
Sometimes practical use is more satisfying than theoretical comfort... So think about how the
"truth" of the FLT really affects things. I imagine it's a lot less effect than you might think...
clarification (Score:3)
It's a bit bold to regard Langland's program (not proposition) as a GUT.
- Shimura-Taniyama originated the idea about a deep connection between modular forms are related to elliptic curves.
- Weil made it plausible and precise but no one likes Weil (PBS) so sometimes his name is not added to the STW conjecture.
- Frey thought that STW-->FLT by using a solution to FLT to create an elliptic curve that probably wasn't modular.
- Serre made the framework of Frey's idea precise in his epsilon conjecture
- Ken Ribet proved Serre's epsilon conjecture establishing that STW-->FLT
- Wiles almost proved STW
- Wiles former student Taylor was brought in to help fill in an essentially small gap (something about deformations of Galois representations, wasn't it?)
- Long refereeing by people like Nick Katz...
Wiles work is a real tour de force. But, I cannot say that all mathematicians care mostly about Langlands program. There are tons and tons of mathematics just as interesting as this topic.Math is NOT a science (Score:2)
A science is a field of study which has a number of characteristics, the main one being that it is based on inductive reasoning from experiment and observation. Mathematics is based on deductive reasoning, not inductive, and therefore is not a science. The entire way we study mathematics is different than how we approach a real science.
Similarly "Computer Science" isn't. A science that is...
Cheers,
Ben
PS Disclaimer: I am not an unbiased observer in this. I am all but dissertation a PhD in mathematics.
MAA link (Score:2)
http://www.maa.org/mathland/math trek_11_22_99.html [maa.org]
Re:an atrociously written article (Score:2)
Please moderate this post down for your protection.
--
Re:My favorite open conjecture .. (Score:2)
Re:Simon Singh's book on Fermat and Wiles (Score:2)
There are many levels of mathematical knowledge. I am sure that this book was great for the complete layman (I did think it did a good job explaining what drives mathematicians to the unenlightened), which is whom it was written for.
However, as a third year mathematics major, I found that it was not suited for me at all (although I am far from good enough to actually pick up the proof and start reading). A lot of Slashdot readers have CS or technical degrees that include quite a lot of mathematics, so I think there are others here who would feel the same.
That was all I was trying to say. Not critisism of Singh, its just the nature of pop-science. Most physisists seem to find _A brief history of time_ appalling...
-
We cannot reason ourselves out of our basic irrationality. All we can do is learn the art of being irrational in a reasonable way.
...But it doesn't _have_ to be that hard to learn (Score:2)
But I differ with you in your answers about why the explanations/texts/etc can't be easier. There is some truth in what you and (moreso) others in the thread are saying, but there is also a heavy undercurrent of math groupthink. Just because no one takes the time to explain these higher concepts clearly doesn't mean there is not a way to do it. It's a hell of a lot harder to explain this stuff simply, but it could be done.
The fact that it generally isn't done is partly due to how hard it is to explain complex or highly abstract concepts clearly. But it's also partly due to the fraternity/hazing attitude in academia that "they should have to work as hard at it to get it as I did".
I have found that I can, with enough effort, find clear and simple (not necessarily short, though!) ways to explain even highly "esoteric" concepts. This involves the very difficult process of attempting to figure out how a newbie will be thinking about what I am saying, and trying to come up with accurate analogies to things that will already be familiar to them. Inevitably, after a lot of effort in this direction, I end up understanding the subject matter on a much deeper level.
This leads me to think that part of the reason that there are not clearer explanations out there is that you just have to understand it better than most people do before you can explain it that well, and at the same time you have to be thinking about how people outside of your field think.
The union of these two sets (one set being "those with a deep understanding of postgrad mathematics", and the other set being "those who spend a lot of time thinking about how to explain things clearly to newbies) may be vast, but the intersection is damn near the empty set (- that wisecrack is borrowed).
Intersect that with "those who have written math textbooks", and you'll get the picture.
It's not impossible, it's just hard, and, often, our cultural blinders don't let us see the payoff (if you want evidence of that, notice how quickly people reject the notion that more visualization would help--"if you learn with graphics, you'll suddenly quit understanding things when you get to 4-d or infini-d". It's baloney, but it's deeply ingrained baloney.)
Yet another barrier is that mathematicians make excellent use of the economy of notation. You can say a hell of a lot with a few symbols, and the very thought, once you've learned to use these symbols, of actually going back and writing out in english what you just expressed in symbols is anathema.
An analogy, for those who have messed with Perl, is regular expressions. How many people really comment their regular expressions? Once you've said it in such a nice, tight format, it just hurts to think about having to explain it in text.
For example, one of the first RegExp's in the perlre manpage ("man perlre" if you're on unix) looks like this:
s/^([^ ]*) *([^ ]*)/$2 $1/; # swap first two words
okay, that's commented--well, the "effect" is described. But imagine if you were trying to state what that expression does:
"Starting at the beginning of $_ (the default variable for matching), find the longest contiguous block (even if it's a block of length zero) of non-space characters (and store that in a variable called $1), then go past all the contiguous spaces after that, and group together the next contiguous block of non-spaces. Put this block of non-spaces into a variable called $2 [the "store that in $1 and $2" is implied by the presence of the parentheses, by the way]. Replace all of the matched text with a string consisting of the second block, a single space, and the first block."
Now that I've explained it in more excruciating detail, I understand it better. I can see that it won't work as advertised, for example. (try it on
foo bar baz
or even
foo bar
Maybe something like
s/(\S+)\s+(\S+)/$2 $1/;
would be better. Got to be careful with them *'s!
)
But look at the sheer number of characters in the text explanation! To another perlvert, the regular expression says the exact same thing. This is very similar to the situation in math--it's sooooo much easier to get the point across with a few terse symbols and references to theorems that it's really hard to get yourself to go through the effort required to explain it to the uninitiated (oops--pun inintentional).
Again, I'm not meaning to flame you, Darby--you hardly exhibited the problem compared to what other posts did. I'm talking about the general trend.
mike
Shimura-Taniyama-Weil (STW) Solved (Score:2)
Re:Science is not based on induction (Score:2)
This is kind of a paradoxical position, since the statement itself (the negation of an existential) is universal, and so itself isn't knowledge. Philosophers tend not to like to built upon such self-defeating foundations...
I rather like Popper, and I think his argument (adapted from Hume) about the invalidity of induction is sound. But his alternative I don't think should be adopted uncritically.