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Swedish Mathematician Lennart Carleson Wins Abel 144

Posted by ScuttleMonkey from the under-recognized-fields-of-study dept.
William Robinson writes "Sci Tech is reporting that Swedish mathematician Lennart Carleson has won the Abel Prize on Thursday for proving a 19th century theorem on harmonic analysis. His theorems have been helpful in creating iPod. Prof Carleson's major contributions have come in two fields - the first has subsequently been used in the components of sound systems and the second helps to predict how markets and weather systems respond to change. One of Carleson's many triumphs was settling a conjecture that had remained unsolved for over 150 years. He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."
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Swedish Mathematician Lennart Carleson Wins Abel More

Swedish Mathematician Lennart Carleson Wins Abel

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  • Except at some negible points? (Score:3, Insightful)

    by Opportunist ( 166417 ) on Monday March 27, 2006 @05:41AM (#15001432)
    My math prof used to shoot chalk pieces onto me for saying something like that!

    That's when I decided that statistics is more my kinda speed.

    • Re:Except at some negible points? (Score:5, Informative)

      by gowen ( 141411 ) <gwowen@gmail.com> on Monday March 27, 2006 @06:01AM (#15001492) Homepage Journal
      Well they mean "almost everywhere", which has a very precise meaning. i.e. except at a set of measure zero (finite or countably infinite set of points.) Of course, that countable set could theoretically be the rationals, so I don't know whether I'd call it negligible.

      • Re:Except at some negible points? (Score:2, Informative)

        by Anonymous Coward

        Well they mean "almost everywhere", which has a very precise meaning.

        The original article on Sci-Tech Today does use the phrase "almost everywhere" instead of the incorrect "negligible". I suppose it is excusable for someone who is not familiar with the area not to realise that it has a precise meaning, so maybe it would have been better to use the more common "presque partout", and let people reach for their French dictionaries.

        • ...so maybe it would have been better to use the more common "presque partout", and let people reach for their French dictionaries.

          Maybe the French is more common in your part of the world, but around here (let's say USA and Ontario) people tend to use almost everywhere (or a.e.) almost everywhere. In the technical sense, I mean.

      • Re:Except at some negible points? (Score:4, Informative)

        by Capt'n Hector ( 650760 ) on Monday March 27, 2006 @07:06AM (#15001631)
        Incorrect. A set of measure zero can be uncountable. (cf the Cantor set)
        • And just to combine stuff we could have a dense, uncountable set of measure zero as well (e.g. the union of the Cantor set and the rationals).
          • Well, since we can have a dense countable set (a fairly unintuitive result), a dense, uncountable, zero-measure set should come as no surprise.
            • I fail to see how the combination "dense" and "countable" (which I don't think is unintuitive, btw) suggests the existence of the combination "uncountable" and "zero-measure".
              • Well, someone else mentioned zero-measure uncountable sets.
                The existence of a dense countable set is well known.

                Given these facts, (and the vacuously true observation that the measure of a union is not greater than the sum of the two measures) do you really think its a tricky leap to find a dense, uncountable set of measure zero?
            • Try this for unintuitive:
              Find a Banach space X and two sets A and B such that A and B are disjoint, convex and dense and A U B = X.

              (Yes, this was an assignment problem, but I've already finished that course)
        • Ooops. You're right. In my defense, it's been 10 years since I did any real analysis.
      • Isn't the set of all computable numbers also countable infinite?
        So you could have a continuous function which diverges from the sum of its Fourier series in all computable points?

      • Well they mean "almost everywhere", which has a very precise meaning. i.e. except at a set of measure zero (finite or countably infinite set of points.)

        It's perfectly possible for a non-countable set to have measure zero, too. It's been a while, but IIRC, the transcendental numbers are the (or at least, a) standard example for this.
        • Some null sets are uncountable (as has been pointed out above). The transcendentals are not null, however, because the transcendentals are the complement of a set which is countable (and null) (the algebraics).

          The Cantor set is the canonical uncountable set of measure zero.
      • As there are an uncountable infinity of points arbitrarily close to each of those negligible points, and we are talking about continuous functions, it appears to me that those points are truly neglibile for all practical purposes.

        Andy
        Hoping that he hasn't misapplied some math that he hasn't used since his degree more than 35 years ago.

    • I find it very ironc that you would say that, because one of the primary motivations behind Lebesgue integration and measure theory is their application to statistics. *shrug*

  • negligible? (Score:1, Insightful)

    by Anonymous Coward
    "He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."

    Negligible? In engineering, maybe. In mathematics, never.

    • Re:negligible? (Score:3, Informative)

      by Anonymous Coward
      It would have been better if they had said "almost everywhere". That is, the set of points at which a Fourier series diverges has Lebesgue measure zero. There is a quasi-converse, due to the Israeli analyst Katznelson, which, given any set of Lebesgue measure zero -- let's call it Bill -- constructs a continuous function whose Fourier series diverges everywhere in Bill. For more info, see Tom Korner's excellent "Fourier Analysis".
    • Your is the attitude that should have caused Euler to with-hold half his discoveries.

      Rigor is very well for the rigorous mathematician. For the rest of us, and particularly for the purposes of talking to the layman (i.e. slashdot), it is a useless pedantry.

      Anyone you might care to name who understands mathematics well enough to be able to understand a distinction of rigor most does not need anyone to tell the difference between what is "propper" and what is easy to say.

  • Young people today (Score:5, Funny)

    by gowen ( 141411 ) <gwowen@gmail.com> on Monday March 27, 2006 @05:44AM (#15001447) Homepage Journal
    <grumpy old man> Young people today. You tell them about a deep result in real analysis, and the only thing they're interested in is how it relates to their iPod. And get off my lawn. </grumpy old man>

    • Re:Young people today (Score:5, Interesting)

      by LarsWestergren ( 9033 ) on Monday March 27, 2006 @06:55AM (#15001617) Homepage Journal
      I'm 31 and have recently started doing a lot of maths in my spare time so that I can get a real computer science and engineering degree one day (I have a degree, but it is CS light... now that I work as a programmer I know how much I'm really lacking), so it is nice to see that at least for some people the old saying by Hardy, "mathematics is a young man's game" isn't true. Carleson is 78 today, and around 40 back when he did the main breakthroughs he is honored for today.

      Hardy's saying is a bit of slight against all female mathematicians too, come to think of it...
      • Well of course in my day we had it tough. I had to get up in the morning at ten o'clock at night half an hour before I went to bed, drink a cup of sulphuric acid, work twenty-nine hours a day down mill, and pay mill owner for permission to come to work, and when we got home, our Dad and our mother would kill us and dance about on our graves singing Hallelujah.

        You people these days just don't get it do they?
      • Did you also have to pause for a bit when the author used the phrase, "a continuous function (one with a connected graph)" Until I pretended I were back in grade school, I was really wondering how we jumped to graphs.

        I guess some folks are normal to reality.
      • Hardy 's saying is a bit of slight against all female mathematicians too, come to think of it...

        Using the word "man" to refer to people in general goes back thousands of years. But, you know, I think you're right. In THIS case, he meant it as a clever, ambiguous attempt at women bashing.

      • Heh, Carleson's age was the first thing I checked, incidentally, and you're right, while this meme of all "good" mathematicians being under 29 was popular, it's fast fading out of style. I should hope so; here I am, 24 years old, with a CS-lite degree, and in a dead-end job, but with a fair bit of mathematical ambition. Easier to plan careers (or consider university programmes) if you don't have a mythical six-year limit hanging over your shoulders. ;-)

        • Easier to plan careers (or consider university programmes) if you don't have a mythical six-year limit hanging over your shoulders.

          It's also good to recognize that you don't need to solve 150 year-old problems, or create powerful new branches of mathematics to have a career. While it would be great to do some earthshaking work, and earn a reputation as a first-class intellect, you can have a decent life doing more humble mathematics. It's a good thing, too, because if only top-notch mathematicians coul

          • you can have a decent life doing more humble mathematics.
            Indeed :-) I was only referring to the oft-quoted stereotype [slate.com] that a mathematician's most productive years are those before he turns 30. Good to see this stereotype being broken steadily.

            • I was only referring to the oft-quoted stereotype that a mathematician's most productive years are those before he turns 30.

              I understood that. And, actually, the stereotype I've always heard isn't about productivity, but about significant contributions. The notion being that any important mathematics will by done by age 30. Older mathematicians have always, I think, produced a sufficient volume of work... but historically the great breakthroughs have come from the young. Until recently, anyway.

              That

  • Wiki Article (Score:5, Informative)

    by zaguar ( 881743 ) on Monday March 27, 2006 @05:45AM (#15001449)
    For those of you who want more than a press release, heres a start :

    Wiki Article on the Breakthrough [wikipedia.org]

  • Well I hope his blonde haired blue eyed and bikini clad assistants got recognised to.
  • Since TFA doesn't mention anything about how his connection to iPods, I don't see how he helped the iPod any more than I did by not joining Apple and botching up the manufacturer's contract for producing crack-resistant LCD screens ... oh wait.
    • I don't know for sure, but this may have been a slight over-generalization. Data compression (especially music) in general is driven by Fourier analysis. The toy example is that to represent a pure sinusoid by straightforward sampling we'd need something on the order of thousands or millions of numbers (depending on the sampling rate and length of the signal). Using Fourier decomposition, strictly speaking, we'd only need 2 numbers: the phase and amplitude. Add as many details as you want to make it pra
    • One use of the Fourier transform is from the time domain to the frequency domain, and vice versa. So for example, in writing a function for EQ on an mp3 player you might use a time function that you wrote in the frequency domain, thus utilizing the Fourier transform to do so. I'm not saying iPods run FFT (Fast Fourier Transforms) real time, but this guy's work really handed down a lot of usefull tools to digital audio: playback, recording, and acoustic testing and research.
  • I am very glad for him.

  • nice work, but no iPod (Score:5, Informative)

    by penguin-collective ( 932038 ) on Monday March 27, 2006 @06:17AM (#15001533)
    The result he proved is nice mathematics, but you don't need it for iPods or audio coding. First of all, for many engineering purposes, it only matters that it works, not that you can prove that it works theoretically. Secondly, audio coding is done over discretely sampled signals, and most of those theorems become simple linear algebra in that case.
    • First of all, for many engineering purposes, it only matters that it works, not that you can prove that it works theoretically.

      - Proclaimed the engineer after one successful run of his simulation/program, before he ran it again with a different set of initial conditions/parameters only to see it fail because he didn't understand the math behind what he was doing.

      There is a Platonic dilemma dealing with the necessity of proof for a mathematical idea to "exist," which is all well and philosophical, but
      • There is a Platonic dilemma dealing with the necessity of proof for a mathematical idea to "exist," which is all well and philosophical, but that's not to say proof shouldn't matter for engineers.

        There is no "dilemma"; knowing a mathematical proof is neither necessary nor sufficient to determine that an engineered system works.

        Proclaimed the engineer after one successful run of his simulation/program, before he ran it again with a different set of initial conditions/parameters only to see it fail because he

  • Obligatory CmdrTaco quote on the subject (Score:4, Funny)

    by grand_it ( 949276 ) on Monday March 27, 2006 @06:19AM (#15001538)
    His theorems have been helpful in creating iPod.

    No wireless. Less decimals than pi. Lame.

  • iPod? (Score:4, Insightful)

    by liangzai ( 837960 ) on Monday March 27, 2006 @06:21AM (#15001542) Homepage
    Yeah, it might be that the MPEG-4/AAC/H.264 algorithms are based in part on Fourier analysis, but I fail to understand how Carleson's theorems have been used in making the iPod. Cupertino is hardly knowledgable in the more esoteric realms of theoretical mathematics, and there is simply no need to incorporate such stuff in an mp4 player.

    This is bad journalism, written by bad reporters who lack the most basic understanding of mathematics and engineering. He just thought it might be cool to clam in an iPod in the mess.
    • It's sad to see that people are abusing their moderation privileges. Labeling the parent flamebait is hardly consonant with the rules of Slashdot, and the perpetrator is most likely a sore loser who cannot participate in a debate with intellectual honesty.

    • Re:iPod? (Score:4, Insightful)

      by glesga_kiss ( 596639 ) on Monday March 27, 2006 @08:06AM (#15001783)
      I fail to understand how Carleson's theorems have been used in making the iPod.

      The iPod reference got this story greenlighted on slashdot. Otherwise it might not have made it. If you want to guarantee acceptance, mention something bad about M$, something good about Linux, or anything about Apple (preferably good, but there is the odd flame article).

      This advice was brought to you by someone with a 100% submission record. (ok, one of one ;-)

    • Re:iPod? (Score:1, Insightful)

      by Anonymous Coward
      You're probably one of those guys that also says that people aren't interested in science anymore. I think that associating his work with the iPod, even if vaguely fraudulent, is a useful helper in placing the work of many mathematicians into context.

      Not only that, I think it's a goodwill gesture borne out of tremendous respect for the work these guys do. Now, I know my share of really cool people that could never have gotten higher educations because of whatever reason, and somehow, they sometimes feel l

      • ...IMHO.

        Why not mention signal processing, that makes it possible to filter out unimportant data from sound so that iPods (and it's likes) can store more music and MP3 and Vorbis files (and their likes) doesn't can be as small as they are. See there, iPod is still mentioned, but not in a way that makes you think there was something special (mabye mechanically) about the construction of an iPod that the world never before had seen.

        When seeing the iPod referenche, I at first thought there mabye was some
    • Narrator: Several years ago, in the basement lab of apple computer, engineers are working on a revolutionary new product. They call it ... the iPod:

      Engineer 1: Ok, the prototype is almost finished, but we have a problem.

      Engineer 2: What's that?

      Engineer 1: Well, we can't prove that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points.

      Engineer 2: ...

      Engineer 1: Come one, man! Focus! If we can't figure this
      • Engineer 2: Wha...? We're getting weird reports in the field from people getting pops and clicks in Britney Spear's music. Only hers.

        Engineer 1: Who wants to listen to ... oh, never mind. What's the problem?

        Time passes. Testing occurs.

        Engineer 1: It turns out that the keyboard her band uses isn't encodable because the particular waveform it produces yields a Fourier series that doesn't converge. Net result -- the keyboard makes the codec explode. Too bad we didn't know that some fourier series don't c

    • I believe this relates to shannon's theorem as used in audio. This states that a continuous waveform may be reconstructed completely from samples taken at greater than twice the highest component frequency of the waveform (Nyquist rate) -- and the waveform can be analyzed for frequency content via fourier analysis. This is EXTREMEMLY important in digital audio -- because that's how it works and how we reconstruct an analog wveform from 1's and 0's.
      Admittedly, throwing the ipod reference in was a troll, bu

  • iPod Reference Misleading (Score:4, Informative)

    by Chrononium ( 925164 ) on Monday March 27, 2006 @06:22AM (#15001548)
    The iPod reference is completely misleading, as simple harmonic analysis is way bigger than just an iPod. It's merely talking about this guy proving that Fourier was basically right, validating harmonic analysis and expanding the horizons for signal processing. That's the biggie: signal processing, not the bloody iPod. The stupid article probably includes iPod just for the sake of hits.

  • WTF does this have to do with iPods?! (Score:5, Informative)

    by pslam ( 97660 ) on Monday March 27, 2006 @06:23AM (#15001550) Homepage Journal
    His theorems have been helpful in creating iPod.

    Oh really? Search Wikipedia entries, the articles, all links - no mention of iPod except in those annoying side adverts. Why? Because it has nothing to do with it

    Credit where credit is due, and none is due here.

    If you want credit, how about: Shannon [wikipedia.org], Fourier [wikipedia.org] and Huffman [wikipedia.org]. Then there's all the folks involved in working out noise masking and all the oddities of human hearing that I don't have the names of.

    I seriously need a "No iPod mentions whatsoever" checkbox for my slashdot profile to pull some more signal out of the slashdot article noise.

    • Search Wikipedia entries, the articles, all links - no mention of iPod except in those annoying side adverts.

      You seem to ascribe to Wikipedia a degree of authority and completeness that even the Encyclopaedia Britannica doesn't claim. Just because a connection isn't documented there doesn't mean it doesn't exist. This is not to say Carlsson's Theorem has anything to do with digital signal processing (it doesn't, of course).

      • You seem to ascribe to Wikipedia a degree of authority and completeness that even the Encyclopaedia Britannica doesn't claim

        I ascribe no such thing. It is merely a readily available and publically readable example which, incidentally, Britannica is not.

  • one with a connected graph. For instance the function equal to sin(1/x) for x != 0 and 0 for x = 0 does have a connected graph but is NOT continuous.
    • If I recall correctly, a connected graph can be related to something to do with topology (haven't done any yet so can't say).

      For those not in the know, continuity at a point p for a function f means

      f(x) -> f(p) as x -> p

      If the function

      f:[a,b] |-> [c,d]

      is continous for all x in the interval [a,b] the function itself is said to be continuous about that interval. This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.

      • Re:Indeed (Score:3, Interesting)

        by The Cow of Pain ( 949595 )
        This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.

        That's not a mis-statement in the case of a real function of a real variable. It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connected (i.e. every two points on the graph can be connected by a continuous path (and by saying 'continuous path' I have of course made the definition self-referential and thus silly, but it is still true)
        • It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connected

          There is a minor difference between what you said and what the article text said, although no one except math PhDs would be likely to care. (For the record: your statement is correct, but the article text is not.)

          A path connected graph is not the same thing as a connected graph. There exist examples of graphs which are connected but not path connected. The article text

    • You obviously don't have an iPOD!
    • Silly article summary, confusing connectedness with path-connectedness.
  • ...was published in the Swedish maths journal Acta Mathematica and is calles On the convergence and growth of partial sums of Fourier series [actamathematica.org].
  • You have to love the new vernacular; where everything is defined in its relation to pop culture and the standard unit of size is the football field.

  • Bah (Score:1)

    by denoir ( 960304 )
    It was just a calculated win on his part. Pro-math has become so phony.
  • His theorems have been helpful in creating iPod.

    Wow ... so math is good for something after all!!!
  • Saying that this has "been helpful in creating iPod" is at least weird.
    It's like saying Einstein's special theory of relativity helped to invent the automobile. After all it deals with motion.

    For practical purposes lot's of convergence theorems for Fourier series had been known before this one and those would be more than enough to show that in practice things would work.

    Take for instance
    http://en.wikipedia.org/wiki/Riesz-Fischer_theorem [wikipedia.org]

    from 1907.

    And before that even others, though this one is quite nice.

    S
    • It's like saying Einstein's special theory of relativity helped to invent the automobile.

            No, his special theory of relativity also helped create the iPod...
  • Is he Swedish or is he a Norwegian living in Sweden? I believe the submission is in error on a non-negligible point!
  • It's worth mentioning that Carleson was on the faculty at UCLA, usually spending at least the winter quarter there (it doesn't take a genius to prefer Los Angeles in February to Uppsala in February.) I think all of the graduate students he advised were in Sweden though, which seems to be the case from the math genealogy site: http://www.genealogy.ams.org/html/id.phtml?id=197 8 1 [ams.org] He did at least intermittently teach the first-year graduate analysis course at UCLA, and made those students suffer (and learn.

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