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Shimura-Taniyama-Weil (STW) Solved 186

Posted by justin++
from the proved? dept.
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?
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Shimura-Taniyama-Weil (STW) Solved

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  • Stories like this should come with a warning from the Surgeon General.

    I read it, and now my head hurts.

  • by Anonymous Coward
    STW conjecture:
    "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.


  • by rde (17364) on Monday November 22, 1999 @08:44AM (#1512436)
    I read this a couple of days ago, and understood several of the words involved. Further reading, with decent enough explanations, can be found here [treasure-troves.com].
  • by MrP- (45616)
    "The brilliant but ill-fated Japanese mathematician Yukata Taniyama was the first person to propose some of the ideas behind the STW conjecture in 1955. A few years later, at the age of 31, he committed suicide."

    hmm, after reading this article and trying to understand it I think ill do the same...geesh

    any math experts here on /. that wanna translate this article to english for us? (maybe a run through babelfish back to english will make it make sense =P )

    #----------------------------
    $mrp=~s/mrp/elite god/g;
  • by Danse (1026) on Monday November 22, 1999 @09:01AM (#1512438)

    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.

  • A research announcement, with enough details to be intelligible to a pure math grad student, is in Notices of the AMS 46:11 (December 1999), available in pdf form here [ams.org].

    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.

  • I think it's good for programmer/geeks to get a dose of stuff they don't understand. I imagine that I had much the same look on my face when I read "modular form of weight two and level N which is an Eigenform under the Hecke seriers and has a Fourier series" as my friends & relatives get when I talk about pointers, polymorphism and other programming stuff. A little confusion can be educational.

    As to those of you who understand both the STW conjecture AND coding.....TTHHHHPPTTT :)
  • by Scryer (60692) on Monday November 22, 1999 @09:06AM (#1512441)
    While I don't pretend to understand the math involved, Simon Singh's book Fermat's Enigma [amazon.com] gives a good explanation of why the Shimura-Taniyama-Weil conjecture is interesting and important, even beyond its application in proving Fermat's Last Theorem. It serves to unify two unexpectedly related fields of math. I recommend the book -- although nonmathematical, it gives a feeling and appreciation for the mathematical discovery process, and is a gripping read. It's a midway point between "popular math" and real math.
  • Leaving aside the question of does the definition of "scientist" include "mathematician", I thought STW was automatic once we had Fermat.
    ---
  • by Anonymous Coward
    I remember from documentary on solving Fermat's Last Theorem that Wiles spent ALOT of time grappling with the Shimura-Taniyama conjecture. Although I can't remember if he used a part of it, or managed to circumvent it. Anyone know if Wiles circumvented it? And if so, was Fermat's theorem used in this proof? Seems like a good tact if so.

    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
  • by c4 (92513)
    Maybe I'm delusional....but..... there are 3 different types of trigonometry, -euclidian (all angles = 180 deg.) --euclidian does not exist in more than 3 dimensions. - and two other kinds...both made with compasses, 3 circles intersecting eachother, one kind has more than 180 deg angles, the other has less - visualize! so heres my point... I was once told by a psycho math/physics professor that if they could actually figure out how to use that info then they could make lots , i mean LOTS of breakthroughs. Hmmn, i am thinking - like 30 years in future or more with the right research - being able to cross dimensions......from 3 to 4. ----- Whats the shortest distance between 2 points? -Straight line? Nope. None at all. Fold the paper from 2 dimensions into 3 and touch the dots. The shortest distance is no distance at all. Just bend 3 dimensions into 4. Sounds easy. Nearly impossible to comrehend with more than 4 dimensions...but...leave that to the people with brains! -What do ya think?
  • by gregbaker (22648) on Monday November 22, 1999 @09:16AM (#1512446) Homepage

    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?)

  • by SpinyNorman (33776) on Monday November 22, 1999 @09:19AM (#1512447)
    As I understand it, Taniyama-Shimura establishes a correspondence between elliptic curves and "modular forms" which are a set of functions that satisfy a certain set of critera, and are based in number theory. Before it was [just] proved, T-S was known to imply FLT, and Andrew Weil's key breakthrough was to prove T-S for the classes of elliptical curves required for FLT. He did this by a novel method of counting both sets (elliptic curves and modular forms), and showing they had the same number of members, hence implying the correspondence. The complete general case of T-S has now been proved. There was a great documentary on FLT a few days ago (PBS I think), which is a must see if it gets reshown.

    Disclaimer: IANAL, IANAM.
  • I thought STW was automatic once we had Fermat.

    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.

  • we McSE's are regularly baffled and defeated,
    but we can always resort to lying and obfuscation.
    Chuck

  • Sounds like "A Wrinkle in Time" by Madeleine L'Engle, eh?
  • It is rather unfortunate that the BBC correspondent has very little idea about the subject he is writing about. The bit about "front-line mathematicians" is horrific. It is true that the subject is too esoteric to be accessible to non-mathemticians, but that is no excuse for a poorly written article.
  • by FreeUser (11483) on Monday November 22, 1999 @09:29AM (#1512452)
    The existence of a proof of the full Taniyama-Shimura conjecture was announced at a conference by Kenneth Ribet on June, 21 1999 (Knapp 1999), and reported on National Public Radio's Weekend Edition on July 31, 1999. The proof was completed by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, building on the earlier work of Wiles and Taylor.

    Before everybody starts screaming "this is old news" remember, /. posts what we submit. Though, I think monitoring NPR would be a good source for stories -- they reported [npr.org] this one a while back. Perhaps links like this one to the real-audio recordings of their broadcasts might be a nice touch.
  • 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)

  • My understanding was that Wiles proved a limited subset of the Shimura-Taniyama conjecture- just enough to prove FLT.
  • by xtal (49134) on Monday November 22, 1999 @09:30AM (#1512455) Homepage

    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,

  • AC wonders whether Wiles used the STW conjecture or circumvented it. In fact he proved the part of it that he needed for FLT (semistable elliptic curves), and the full conjecture is what was proven last summer building on Wiles' work. STW doesn't follow from FLT -- FLT follows from part of the STW. That's all in the BBC article.

  • by ccf (116263) on Monday November 22, 1999 @09:32AM (#1512457) Homepage
    I agree. Fermat's Enigma is a beautifully written account of the history of Fermat's Last Theorem. He talks about who Taniyama and Shimura were and when and how they did their work, and in a general sense, how it relates to Fermat's Theorem. Without having to know that much math, you get a real sense of what the mathematical process is like. Singh covers Euler, Gauss, and even has a section about Alan Turing and the first code-cracking computers in WWII. A great read.
    Clark

    --
    Finding a job shouldn't be work.
  • Oddly, this book, according to Amazon.com, is popular in Clearwater, FL (world headquarters to and general stomping grounds for the Church of $cientology). I wonder what the connection is. Do they talk a lot about clams in this book, bychance?
  • Well actually I think confusion is a very bad thing. I really want to persue more of the interesting stuff in science. I am currently having a bear of a time with Calculus (most likely failure) and find that confusion causes people undue levels of negative energy. Statements such as these most likely have revelence to only about 10 people in the world. And of those 10 about 3 are actually qualified to do any of these things that they talk about. Then about 1 person is left who can do this without getting a very large migrane or stroke.
  • by Randym (25779) on Monday November 22, 1999 @09:43AM (#1512462)
    Look here [st-and.ac.uk] for biographical information about Andrew Wiles. Also look here [best.com] for some more resources including a pointer to Wiles' original article on solving it. And a good, fairly non-technical book on the subject is Simon Singh's Fermat's Enigma [amazon.com].

  • by dingbat_hp (98241) on Monday November 22, 1999 @09:47AM (#1512463) Homepage

    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.

  • by Anonymous Coward
    Couple clarifications:

    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.
  • While I'm far from a defender of the Scientologists, methinks you are seeing a bit too much correlation here...
  • by Anonymous Coward
    there exists a modular form of weight two and level N which is an Eigenform under the Hecke series and has a Fourier series

    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?

  • by Anonymous Coward on Monday November 22, 1999 @09:56AM (#1512467)
    I'm not trying to be offtopic, and I know I'll probably get moderated down for this, but:

    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.

  • Hmmm... Lets see. I don't exactly think background reading is the appropriate question for such an inquiry as to the nature of this material. Adequate background information would be more at issue. To gain adequate background in such an inquiry one would have to be well aquatinted with papers in algebra (Ph.D. level, not high school or undergrad), algebraic topology, topology, and throw in analysis (just for good measure - ha ha). That would put you about third year into a respectable Ph.D. program in Mathematics.

    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...
  • This is actually not as obscure as it may sound. Simply put it relates topology to number theory, thus allowing problems in one domain to be translated to the other. That FLT was able to be solved (albeit not in the same way that Fermat did it) using this technique is an indication of the power of being able to do this: suddenly the power of techniques developed in one domain become applicable to problems in another.

    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.
  • lol...

    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?
  • Well your comment about slashdot I think is a little biased. I think there is a big difference between a person who has math info and one who has information regarding computers and the like. The separation between those in the math world and those in the computer world ended with the use of the personal computer. Once computers didn't cost $9,999 (taken from a magazine in 1979) people could use a computer without needing a Phd degree in math.

    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?
  • by ZahrGnosis (66741) on Monday November 22, 1999 @10:12AM (#1512472) Homepage
    Alright, I'm no expert on this stuff, but I'm going to take a stab at explaining why anyone would care about the STW conjecture.

    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?
  • Just to accentuate the point - Imagninary number theory existed for around 100 years before Electrical Engineers found that it could be used to describe how AC circuitry works, and allows a consistant tie between DC and AC theory. Ohms simple ratio of I=E/R still holds true in both domains thanks to imaginary numbers on a polar coordinate system.
  • That FLT was able to be solved (albeit not in the same way that Fermat did it)

    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.
    ---

  • actually, a special case of STW gives you FLT. (if you can show STW for a subclass of elliptic curves, "semistable" ones, then you have proved FLT). this is due to Ribet, Serre, and perhaps others.

    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
  • so, after reading fermat's last theorem and how modular arithmetic and elliptical curves were related, i began to wonder if crypto, which relies heavily on modular arithmatic, could also be done using elliptical curves. in fact, not being a mathematician by any means, i was way behind and stumbled across what was obvious to people long before me.

    the proof of this conjecture is pretty cool, and i am looking forward to what it means for cryptanalysis and cryptography in general.
  • This stuff is still almost incomprehensible.

    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
  • so, after reading fermat's last theorem and how modular arithmetic and elliptical curves were related, i began to wonder if crypto, which relies heavily on modular arithmatic, could also be done using elliptical curves. in fact, not being a mathematician by any means, i was way behind and stumbled across what was obvious to people long before me.



    the proof of this conjecture is pretty cool, and i am looking forward to what it means for cryptanalysis and cryptography in general.
  • by Hobbex (41473) on Monday November 22, 1999 @10:25AM (#1512482)

    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.
  • most people that i know believe that fermat had a "proof" that relies on unique factorization in cyclotomic extensions of the integers.

    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
  • I know this thread had already been null-moderated, but I'm curious about what effect this solution may have on cryptography that uses elliptic curve algorithms.

    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?

  • actually,

    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
  • A lot of modern crypto relies on large numbers with big prime factors. Breaking encrpytion schemes such as PGP that rely on this can be boiled down to factoring these large numbers into their big prime factors. Currently, one of the fastest methods for finding factors of an arbitrary number is the ECM (Elliptical Curve Method, I believe) which, clearly, relies on elliptical curves.

    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. :-)
  • stupid html tags.

    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
  • by arri (95615) on Monday November 22, 1999 @10:43AM (#1512489) Homepage
    P. Ribenboim, "13 Lectures on Fermat's Last Theorem", Springer-Verlag, 1979, ISBN 3-540-90432-8 (assumes undergraduate maths). You might notice that this book's publication date is way before Wiles, it contains material on which Wiles then expands, e.g. elliptic curves (cf. cryptography too!) and modular forms. A simpler text, still by Ribenboim is "Fermat's Last Theorem for Amateurs", 1999, Springer again, ISBN 0-387-98508-5, which, as the title sort of implies is a tad easier. I wouldn't say it is exactly trivial but it is a very good self-contained book with a number of chapters explaining the number theory you need and a good attempt at explaning Wiles' proof. Borrow this one from your local library if you are really interested and have some mathematical background, the first one if you are into higher mathematics.
  • It's been done. Here [pipex.com] is a FAQ on cryptography using elliptic curves dated Dec. 1997. The FAQ indicates that keys can be shorter than RSA keys for the same level of cryptographic difficulty.
  • by Anonymous Coward
    I study pure maths (which gives me a right to tell you all that you are wrong :P Just kidding)

    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
  • Don't say it, man! The Scientologists will shut down Slashdot!

  • :) thanks for the clarification. it is appreciated. i haven't the foggiest about most of this, i just stand back and say, "cool."

    :)
  • by David A. Madore (30444) on Monday November 22, 1999 @10:54AM (#1512494) Homepage

    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).

  • Simon Singh, Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem. Based on the (excellent) BBC/PBS television show (it was a Nova episode). Highly recommended.

    Get it h ere [fatbrain.com] at Fatbrain (does /. get a cut that way?), or here [amazon.com] at Amazon.com.
  • by Master of Kode Fu (63421) on Monday November 22, 1999 @11:10AM (#1512496) Homepage
    Alas, until I read Paul Hoffman's The Man Who Loved Only Numbers [maa.org] , a great biography of prolific math-geek Paul Erdos, all I really knew about Fermat's Last Theorem [cs.unb.ca] came from a painfully bad Star Trek episode [startrekcontinuum.com]. In the Trek universe, the proof still eludes everyone in the 24th century, even Data and a room full of math geeks [demon.nl]. While not really a math guy, [upenn.edu] Picard likes trying to solve it as a hobby and the innumerate [amazon.com] Riker hasn't even heard of it, owing the the constant warp core breach [connect.ab.ca] in his pants). The book devotes a couple of pages to Andrew Wiles' presentation of his proof [pbs.org], in which he threw "the entire kitchen sink" of twentieth century mathematics [amazon.com] and how it's unlikely that Wiles' proof is similar to Fermat's (assuming it existed). Perhaps Fermat thought he had a proof when he really didn't [ndi.net], or maybe it was his way of pulling a fast one on future generations [isetan.com.my].

    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....)

  • I'll second this request. Probably, the title could be
    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
  • Or to be a tad more obscure:

    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.

  • Modular Form: complex functions satisfying
    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.
  • That Taniyama implies FLT was shown by Ken Ribet.
  • Oh, for moderation points right now. I'll make what is probably a drastic error, dust off my undergraduate degree in mathematics, and respond in due seriousness to this little piece of trollbait.

    Real-world relevance of higher math:
    • Number theory is what makes modern cryptosystems go. (As someone has noted elsewhere in the responses to this article, cryptosystems have been devised based on elliptical curves.)
    • Scheduling problems (as seen in, say, multitasking processors) require a surprising amount of deep math for optimized solutions.
    • To the best of my knowledge, both real and complex analysis are required for several branches of upper-level physics. Someone with a degree in physics may wish to correct me in this.


    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).

  • by cje (33931) on Monday November 22, 1999 @12:26PM (#1512512) Homepage
    While we're on the topic of open mathematical conjectures, my favorite still has to be Goldbach's Conjecture. It's tantalizingly simple; it states that any even integer greater than 4 can be expressed as the sum of two prime numbers. It seems intuitive, and it's certainly easy to verify "by hand", at least for relatively small numbers (i.e., 31 = 13 + 17). Indeed, computers have been unable to find a counterexample, regardless of how high they've gone.

    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.
  • But the reason that the keys could be shorter may now be invalidated with the proving of this conjecture, I don't know enough of the math but to quote from the latest Crypto-gram newletter:
    '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. ... Whether this recommendation makes sense depends on whether the faster algorithms can ever be made to work with elliptic curves. The question to ask is: "Is this lack of smoothness a fundamental property of elliptic curves, or is it a hole in our knowledge about elliptic curves?" Or, more generally: "Are elliptic curves inherently harder to calculate discrete logs in, or will we eventually figure out a way to do it as efficiently as we can in the numbers mod n?" '
    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
  • I know this thread had already been null-moderated, but I'm curious about what effect this solution may have on cryptography that uses elliptic curve algorithms.

    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...

  • by PG13 (3024) on Monday November 22, 1999 @12:45PM (#1512515)
    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!

    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.
  • Actually, the electrical engineering connection is a red herring. Phasor analysis (which is what I assume you're referring to) does use imaginary numbers, but there is *no* fundamental link to the mathematics. It just turns out that the easily-remembered rules to manipulate imaginary numbers correspond with worked answers to commonly-occuring problems of frequency analysis. The real connection occurs via the frequency domain and a fourier transform... it just so happens that you can easily remember the results of the double transformation using arithmetic on imaginary numbers.

    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.
  • by Darby (84953) on Monday November 22, 1999 @12:57PM (#1512518)
    can get bar none!

    >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 :-) I think that this is a very poor take on this situation.
    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!!---
  • from the front page would be a nice addition also. It seems quite often that some good links are provided in the discussion but (even with moderation) that requires some diggin'. Short form: More film and audio links from the front page. Let's *really* smash some servers. :-)
  • Yes, as a matter of fact I did notice the links. It is rather amusing to have the bit about dunking biscuits sitting next to a major breakthrough in math. But the one about the stock marked is even worse - either the implications of the model are trivial or the journalist entirely missed the point. Don't know which... I like BBC a lot and it makes me upset to see them make fools of themselves. A lof of other well-respected news sources are just as bad when it comes to science. In NY times for example one writer wrote (talking about the mirror symmetryin physics) that the reflection in the mirror has left and right, and up and down switched...

    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).
  • by PG13 (3024) on Monday November 22, 1999 @01:05PM (#1512521)
    The Goldbach conjecture is still open as far as I know (and no it hasn't been shown to be undecidable in standard number theory).

    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.
  • by Brecker (66870) on Monday November 22, 1999 @01:05PM (#1512522)
    Not much of anybody in the mathematical community thinks that Fermat had anything resembling a proof to this one. There is a fairly reasonable explanation for where Fermat went wrong.

    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. :)
  • You have reminded me of a quote that serves as a nice counterpoint to your argument:

    "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
  • Is that better or worse than Scientologists?
  • While we're on the topic of open mathematical conjectures, my favorite still has to be Goldbach's Conjecture. It's tantalizingly simple; it states that any even integer greater than 4 can be expressed as the sum of two prime numbers. It seems

    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

  • I (a Ph.D mathematician) highly recommend viewing the Nova special on Andrew Wiles and Fermat's Last Theorem. The documentary is engaging, has a wonderful human element (can you imagine realizing your life's dream and then turning to face the rest of your life....what do you do?), and presents the mathematical ideas in a fashion that actually keeps you interested and is faithful to what's actually going on without being confusing (or condescending).

    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!

  • There's but one mistake in your comment -- the most common mistake in intelligencia today. "GL", "Sp" et al are not acronyms. "GL" is an initialism and "Sp" is an abbreviation. Acronyms refer to word/phrase shortenings that are pronounced as words (ie. "Nortel" for Northern Telecom).

    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 was the guy who did that (prove the epsilon conjecture). Frey was the first to come up with a possible link between STW and FLT. Then Serre outlined a possible proof and called it the epsilon conjecture. And like I said Ken Ribet was the one to finish the proof of the connection. With that in hand, we knew that STW => FLT, and in fact already a part of STW would be sufficient.

    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.
  • Last semester one of my Math/Comp. Sci. profs showed a really good Nova special on the solving of Fermat's last theorem. For those of you don't know, this simply came from a note that was written in the margin of one of Fermat's books in his library that he could solve this. No proof included. Anyone who says they think they know how he solved it would only be guessing. You get a lot of really good background information on alot of the names involved in this article such as Wiles, Shimura, and Taniyama.

    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.
  • That psycho math/physics professor sounds like any other crackpot. Warning: if someone is confusing as hell, and pretends to be a math genius, just ignore him.

    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.
  • In fact, if you read physics papers by people like Witten, you'll get smacked around your ears with elliptic curves.
  • I read the article and I still don't know anything more than I did before. Dr. David Whitehouse obvously doesn't understand what this thereom is. He gave no laymans terms for what this means, no analysis, nothing. The best he could do was quote a literal definition that only a mathmatician would understand, say that it relates numbers to shapes, and claim that it was one of the most important mathmatical discoveries of the 20th century.

    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.

  • NO!

    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.
  • Frey came up with the initial idea, Serre worked it out further, and indeed Ribet finished the proof of the implication. It seems extremely unlikely that Taniyama has ever expected just the slightest connection between his conjecture and FLT. It would be interesting to know if he thought the conjecture would be proved before the end of the millennium.
  • And I hope that if Littlewood returns in 500 years and asks about the Riemann hypothesis, we will be able to confirm its truth.
  • Oh and 31 is not even, and 13+17 = 30 :)
  • I very highly doubt that this problem has been proved to be undecidable.

    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.
  • 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 ?

    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 :)

  • Well said. Of course, I cannot follow the mathematics all that well... having struggled through under-graduate calculus with less than outstanding results.

    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.

  • what is this proven theorem going to allow us to do?

    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!
  • That's completely idiotic. Understanding STW on a level deeper than "all chipmunks are really woodchucks in disguise" would require several years of graduate mathematics, and those several years would have to be doing the right type of mathematics. People, even smart ones, need to accept that there are simply some things that they couldn't understand, even if they worked very hard for a very long time.

    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
  • Although proofs and such can be very comforting to know about, engineers (and some scientists)
    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...

  • by Anonymous Coward on Monday November 22, 1999 @05:55PM (#1512568)

    It's a bit bold to regard Langland's program (not proposition) as a GUT.

    1. Shimura-Taniyama originated the idea about a deep connection between modular forms are related to elliptic curves.
    2. Weil made it plausible and precise but no one likes Weil (PBS) so sometimes his name is not added to the STW conjecture.
    3. Frey thought that STW-->FLT by using a solution to FLT to create an elliptic curve that probably wasn't modular.
    4. Serre made the framework of Frey's idea precise in his epsilon conjecture
    5. Ken Ribet proved Serre's epsilon conjecture establishing that STW-->FLT
    6. Wiles almost proved STW
    7. Wiles former student Taylor was brought in to help fill in an essentially small gap (something about deformations of Galois representations, wasn't it?)
    8. 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.
  • We don't call something "a science" because we like it. At least I don't.

    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.
  • The mathematical association of america has some nice information on this:
    http://www.maa.org/mathland/math trek_11_22_99.html [maa.org]
  • 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 ?
    A great method of killing a dinner party dead: remark casually that "$\int_0^\infty\exp(-x^2)\,dx" = \sqrt{\pi}/2$". Somebody stated that this simple fact has more impact on our life than all works by Shakespeare, Mozart and Leonardo da Vinci combined.

    Please moderate this post down for your protection.
    --

  • my favorite open conjecture is another simple one; I forgot the name, but it states that you start with any integer, and keep doing this operation: if the number is even, divide by two, if not, multiply by three and add one. if the conjecture is right, whatever the number you started with, you end up in the cycle 1-4-2-1. as far as I know, no-one knows where to even start proving it.

  • 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.
  • First, let me say that I heartily agree that math is one of the best degrees you could have to do computer programming or any other kind of work that is heavily oriented toward problem solving.

    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
  • check out this link to treasure troves if you want to know more about it... http://www.treasure-troves.com/math/Taniyama-Shimu raConjecture.html
  • Well, a consequence of Popper's position is that there is no such thing as knowledge of universals.

    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.

I've never been canoeing before, but I imagine there must be just a few simple heuristics you have to remember... Yes, don't fall out, and don't hit rocks.

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