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Science

Atiyah and Singer to Share the 2004 Abel Prize 127

Posted by michael from the you-cut-i-choose dept.
sbar writes "The 2004 Abel prize-winners have been announced.From the website: 'The Atiyah-Singer index theorem is one of the great landmarks of twentieth century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory. Its authors, both jointly and individually, have been instrumental in repairing a rift between the worlds of pure mathematics and theoretical particle physics, initiating a cross-fertilization which has been one of the most exciting developments of the last decades.'"
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Atiyah and Singer to Share the 2004 Abel Prize More

Atiyah and Singer to Share the 2004 Abel Prize

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  • Suddenly, (Score:5, Funny)

    by acxr is wasted ( 653126 ) * on Saturday March 27, 2004 @05:07AM (#8688168)
    the audience utters a collective "wha?"

    • Re:Suddenly, (Score:5, Funny)

      by Anonymous Coward on Saturday March 27, 2004 @05:28AM (#8688210)
      Dr. Hibbert: Homer, I'm afraid you'll have to undergo a coronary bypass operation.
      Homer: Say it in English, Doc.
      Dr. Hibbert: You're going to need open-heart surgery.
      Homer: Spare me your medical mumbo-jumbo.
      Dr. Hibbert: We're going to cut you open and tinker with your ticker.
      Homer: Could you dumb it down a shade?
    • The new tower of Babel.

      Different professions find that the language is not up to the task of quickly and concisely describing what they do, so they re-use words giving them new meanings, invent new ones and in the process make it difficult for the layman to understand WTF they are talking about. Sometimes deliberately but more often simply due to convenience.

      In order to even have a chance of understanding, you'd have to know the meanings of the underlying language, otherwise it's just babble.

      It's worth n
    • Yeah, followed closely by "Is there a Linux port?"
    • the audience utters a collective "wha?"

      No doubt. I read that blurb three times, and every time I heard this *WHOOOSH!* sound right above my head. This story gets a "+1 - Inscrutable" rating from me.

      • *WHOOOSH!* sound right above my head

        Congratulations, you're much more mathematically talented than average. Most folks reading this theorem see a small speck flying in the stratosphere, then hear a faint *whooosh* several seconds later.

        Heck, I took a 400-level topology course way back when, and I still couldn't fully parse the abstract [wolfram.com].

        • Re:Suddenly, (Score:2, Informative)

          by H*(BZ_2)-Module ( 536932 )
          You can read the actual theorem from Chapter 3 of Peter Gilkey's book available here [www.emis.de]. The Aitiyah-Singer Index Theorem is Theorem 3.9.5 on page 233, right at the top of the page. There is a nice explanation of it in easy to understand terms on MIT's press release [mit.edu]. Unfortunately, this doesn't seem to capture how impressive the theorem really is.

    • Re:Suddenly, (Score:3, Funny)

      by decimal0 ( 721072 )
      the singer Aaliyah? But I thought she was dead!

  • Fantastic news (Score:3, Funny)

    by Anonymous Coward on Saturday March 27, 2004 @05:10AM (#8688172)
    Now all they have to do is derive a theorem that can solve the conundrum that is, how to share the trophy between them equally each week which as you all know contains a number which, wait for it.. is not divisable by two without remainder!

    The real work has yet to be done.
    • If it's not too troublesome, why not cut off Monday? I don't have much use for it! It's just one less day I have to wait to get to the weekend.

      Or, they could each share it the first six days equally (1 for 3 days, the other for the other three days). Then, on Sunday, they could give it to ME! :)
    • Ah, we may not be able to locate the exact position of the trophy, but we can now determine the probability of its position and velocity!

  • my contribution (Score:5, Informative)

    by mandalayx ( 674042 ) * on Saturday March 27, 2004 @05:10AM (#8688173) Journal
    Honestly I wish I knew what this was about, but I don't. So I'll defer to greater authorities. Perhaps someone can explain in a Feynman-esque manner?

    Atiyah [wikipedia.org] is of The University of Edinburgh and is one of the founders of K-theory, a branch of topology. He won the Fields in 1966 (sic). Singer [wikipedia.org] is of MIT, and is an institute professor [wikipedia.org], which is supposed to be a big deal.

    I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery. -Descartes

    Interesting quote they left. Perhaps a more classy way of saying that their margin was too small to write another wonderful proof in?

    • Re:my contribution (Score:4, Informative)

      by S3D ( 745318 ) on Saturday March 27, 2004 @05:38AM (#8688223)
      I'm not quite up to the task, my math is not quite strong, but basically it is about differential equation with partial derivatives (differential operator strictly speaking) on some curved space (manifold). The equation is elliptic (that basically mean it's formula in some coordinate system could be written similar to the formula of elliptic curve, with dervivatives instead of powers, more in depth-it's about specter of the operator) The theorem is about some important property of this equetion/operator could be derived only form topology of the curved space and some highed degree coefficients of the equation. Basically how underlying topology of the space influence this equation. It's told this theorem very important for physics, especially particle physics, which deal a lot with differential operators on the curved spaces...
      You are welcome to correct me if I'm wrong here...
      • The theorem is about some important property of this equetion/operator could be derived only form topology of the curved space and some highed degree coefficients of the equation. Basically how underlying topology of the space influence this equation. It's told this theorem very important for physics, especially particle physics, which deal a lot with differential operators on the curved spaces...

        So I guess they're saying that, despite decades-- nay, CENTURIES-- of mathematicians striving to keep their f

    • Here you go, a layman's explanation. (Score:5, Informative)

      by Anonymous Coward on Saturday March 27, 2004 @06:26AM (#8688300)
      None of this is accurate, but it'll give you some sense.

      The theorem basically states that there is a deep connection between analytic properties of a manifold and the topological invariants of the manifold.

      A "manifold" is a mathematical "space". Think of it as a big playdough that you can put things on. You put things call "vector bundles" on them (imagine sticking little arrows on your playdough. For those with some math background, these vector bundles,roughly, are just functions.

      Imagine you have two different set of vector bundles on it (i.e. 2 different set of functions)

      A "partial differential operator" will eat the function from one set, and spit out a function from the other. An "elliptic PDO" does this uniquely, and can be inverted (i.e. you can eat either set.)

      Usually, the geometry of your playdough manifold will determine the number of such PDOs.

      Now, there is an "index" associated with the elliptic PDOs. The index is the difference between (roughly) the number of PDOs inside the "kernel" (ok this is too hard to explain what is a kernel) and the number that is NOT in the kernel.

      Usually, given a manifold, it is easy to compute the index without knowing the exact details of your vector bundles and manifold etc (it is hard to find the exact number in/outside the kernel).

      There is also a thing call "topological invariants" associated with your playdough. A topological invariant is any mathematical quantity that does not change if you mash around the playdough manifold *without making new holes that go through*. For example, the Euler Characteristic is one such number. A rough guide is the number of holes of a pretzel. Pretzels with same number of holes will have the same Euler Char (though they might look very different).

      What atiyah and singer found is that there is a deep connection between the Index of the analytic operators on a smooth compact complex manifold without boundary and its topological invariants.

      "smooth" means there is no "kink" or rough edges of your playdough (a cube is not smooth, but a sphere is). Compact means it is finite in size. Without boundary means it is not bounded by a border (The surface of a sphere is compact and has no boundary, a piece of paper is compact but is bounded by its edges).

      Complex means that the functions that live on the manifold can have complex numbers.

      That's all I can figure out. Anybody who knows better should feel free to correct me.

      • What atiyah and singer found is that there is a deep connection between the Index of the analytic operators on a smooth compact complex manifold without boundary and its topological invariants.

        *WHOOOSH!*
        There it goes again. What IS that thing?

      • Re:Here you go, a layman's explanation. (Score:5, Informative)

        by sqlgeek ( 168433 ) on Saturday March 27, 2004 @10:15AM (#8688841)
        The concept of a kernel isn't all that hard. In math you're commonly looking for mappings (functions) between things that are too complicated to understand and things that aren't. You want to find the relationship between the the complicated one and the intelligible one. Seem reasonable? Ok.

        Now in group theory you're looking at very simple algebraic structures, such as: 1. how the integers act under addition, 2. how the positive real numbers act under multiplication, 3. how a book could be put back onto the shelf (i.e backward, upside down, etc). In spite of the fact that in group theory you're only looking at a single operator (addition, multiplication, moving a book around) on a set of elements (integers, positive reals, a book) groups can actually get very complicated. So, in group theory we often want to map a more complicated group to a simpler group.

        Now, in each of the above groups there is an "identity" element in the group: zero in addition of integers, 1 in multiplication of positive reals, and with the book the identity corresponds to picking the book up and then putting it back just the way you found it. If we map a complicated group to one of these simpler groups, then the _kernel_ is the set of all elements of the complicated group that map to the identity of the simpler group.

        Here's an example.

        Complicated group: integers under addition

        Simple group: the numbers 0 and 1 with respect to addition modulo 2 (i.e. 0+0=0, 0+1=1, 1+1=0)

        Mapping: even numbers map to 0, odd numbers map to 1.

        Identity of simple group: 0 (N+0=N, right?)

        Kernel of mapping: all even integers (in the complicated group), because all even integers (in the complicated group) map to zero (in the simple group)

        That wasn't so bad, now was it?

        Scott
      • It's trivially simple, if you've taken Math 18000 at Star Fleet Academy. Basically, Atiyah and Singer demonstrated, back in the 21st century, what we now know was the basis for a rigorous proof that Britney Spears' brain surface area is mappable to that of the common ground squirrel. This, in turn, led to the development of the infinite probability drive which we use daily.
      • There's an even clearer and simpler way to explain it (from the press release [abelprisen.no], as reporters have to explain it to a lay audience):

        We describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas involving their rates of change, so-called differential equations. Such formulas may have an "index", the number of solutions of the formulas minus the number of restrictions which they impose on the values of the quantities being comput

    • Or maybe a fancy way to say that he couldn't just come out and say that he didn't believe in God given the times he lived in.

      IIRC, his tomb said "to hide well is to live well" or somethign to that effect.

    • and is an institute professor, which is supposed to be a big deal.
      I checked the link and I understand the significance, but come on! If they want to elevate a member of their faculty, you'd think that the bright people at MIT could come up with a title that sounds like it says more than, "Yeah, he teaches here".

  • Did a little google on this thing (Score:5, Funny)

    by Belzu ( 735378 ) on Saturday March 27, 2004 @05:16AM (#8688184) Journal
    And here is a somewhat clear and concise explanation:
    "In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is a basic general result that came at the end of a long development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. There have been a number of subsequent developments, in particular in the work of Alain Connes.

    We start with a compact smooth manifold (without boundary) and an elliptic operator E on it. Here E is a differential operator acting on smooth sections of a given vector bundle. The property of being elliptic is expressed by a symbol s that can be seen as coming from the coefficients of the highest order part of E; s is a bundle section and required to be non-zero. E.g. for a Laplacian s is a positive-definite quadratic form.

    By some basic analytic theory the differential operator E gives rise to a Fredholm operator. Such a Fredholm operator has an index, defined as the difference between the dimension of the kernel of E (solutions of Ef = 0, the harmonic functions in a general sense) and the dimension of the cokernel of E (the constraints on the right-hand-side of an inhomogeneous equation like Ef = g)."
    Which leads me to wonder:
    HUH!?
    • Here E is a differential operator acting on smooth sections of a given vector bundle.
      That almost sounds like ..... pr0n for mathematicians!

    • Re:Did a little google on this thing (Score:5, Informative)

      by coats ( 1068 ) on Saturday March 27, 2004 @07:34AM (#8688410) Homepage
      In very informal English:
      A
      manifold is a generalization of a surface (on a surface like a torus, you can move in two independent directions at any point; on an n dimensional manifold, you can move in n independent directions. Space-time is a 4-manifold.) Manifolds are the most general sorts of objects you can write differential equations and integrals on.

      Elliptic differential equations (very informally speaking) are differential equations that act like the equations for equilibrium problems.

      THEOREM Elliptic differential equation systems have finite dimensional solution sets (Hodge, Fredholm). (That dimensionality is an integer)

      THEOREM That dimensionality is a topological invariant of the manifold. (de Rham).

      The Heat Equation solution technique for elliptic differential equations leads to the computation of an integral over the manifold (not sure the best reference here, probably Friz John from NYU-Courant). (The result of that integral is a real number.)

      Theorem (Atiyah-Singer) The (real-number) integral coming from the Heat Equation solution technique is the same as the (integer) topological invariant coming from the dimensionality of the solution space.

      This says that the topology ("how many holes in the torus?") is intimately tied up with the solvability of differential equations (an entirely different branch of math); moreover, the differential equations (as occur in mathematical physics) have solution properties that generate integers (tying in to quantum mechanics).

      • Ok, I'm impressed. After reading through about a dozen of the highest rated explanations of this theorem yours was the most satisfying. (I can't judge correctness of course, I'm clueless here).

        Since you did so well, here's another challenge :-).

        The theorem is said to have played a role in futhering understanding of particle physics. What was that role? What theoretic physical systems are mappable to this theorem?

        And then for the extra-credit problem ...

        What is the connection between the vast complexity

  • Atiyah-Singer Index Theorem (Score:5, Informative)

    by amigoro ( 761348 ) on Saturday March 27, 2004 @05:23AM (#8688201) Homepage Journal
    From MathWorld [wolfram.com]:
    A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an n-dimensional compact differentiable C^infinitiy boundaryless manifold.

    And this [encyclopedia4u.com] is the least technical definition I have come across so far.

    Trawling thru the USENET I found:
    The Atiyah-Singer expression is:

    { ch(V|X^g)(g) * U(N^g) * Td(X^g) / det (1-g | (N^g)*) } [X^g]


    where X is a G-manifold for G cyclic, generated by g, ch()(g) is an equivariant Chern character for trivial G-spaces, U is a combination of characteristic classes which "accounts for" the normal bundle N^g of X^g (the fixed set of X) in X, Td is the Todd class, and the determinant is evident.

    Apparently the INVARIANCE THEORY, THE HEAT EQUATION, AND THE ATIYAH-SINGER INDEX THEOREM [www.emis.de] is a good source too.

    And This book:
    "The Atiyah-Singer index theorem and Elementary number theory" F. Hirzebruch and D. Zagier (Publish or Perish)

    Moderate this comment
    Negative: Offtopic [mithuro.com] Flamebait [mithuro.com] Troll [mithuro.com] Redundant [mithuro.com]
    Positive: Insightful [mithuro.com] Interesting [mithuro.com] Informative [mithuro.com] Funny [mithuro.com]

    • Are you sure you don't miss a term after the second (N^g)* ?

      This formula is way over my head (high in the stratosphere) but I quite sure that a multiplication takes 2 arguments.

      By the way, "G-manifold for G cyclic, generated by g, ch()(g) is an equivariant Chern character for trivial G-spaces" convinced me of the superiority of the Gnome desktop over KDE.

    • Re:Atiyah-Singer Index Theorem (Score:1, Informative)

      by Anonymous Coward
      The whole mess of a formula you give there is a fixed point formula coming from equivariant index theory... that's why it seems so complicated. The simplest form of the A-S theorem is that for any Spin^c manifold M and vector bundle E on M one has

      Index (D_E) = ( Td(M) ch(E), [M] )

      where D_E is a specific differential operator constructed using the vector bundle E and the Dirac operator on M. (It turns out that as far as index theory goes every elliptic differential operator can more-or-less be rewritten as

  • Rich rewards for everybody (Score:5, Interesting)

    by jandersen ( 462034 ) on Saturday March 27, 2004 @05:26AM (#8688209)
    First of all, I'm surprised to see this mentioned in this list. Not because it isn't an essential and relevant result, but because most people here simply don't have a clue about abstract mathematics.

    As many people have experienced, studying the higher mathematics is incrediby rewarding, intellectually, especially the parts that have nothing to do with numbers (ie. most). Even if you don't get into the intricacies of stringent proofs of theorems, it is still a world of such incredible wonder. Are you fascinated by science fiction and fantasy? Then mathematics should be able to captivate you; personally I can't think of anything more mindblowing than such things as topology, geometry and algebra.

  • I Hear Ya (Score:3, Funny)

    by Talisman ( 39902 ) on Saturday March 27, 2004 @05:30AM (#8688215) Homepage
    "Its authors, both jointly and individually, have been instrumental in repairing a rift between the worlds of pure mathematics and theoretical particle physics, initiating a cross-fertilization which has been one of the most exciting developments of the last decades."

    Tell me about it. I was just talking to my voluptuous Swedish masseuse girlfriend about the Atiyah-Singer index theorem and she was all like, "Oooohhhhhh take me NOW!" but in a Swedish accent and stuff.

    One of the most exciting developments in the last decade, indeed.

    Talisman
  • instrumental in repairing a rift between the worlds of pure mathematics and theoretical particle physics, initiating a cross-fertilization which has been one of the most exciting developments

    This is by far the dumbest pickup line since "do you want to see my japanese paintings at home". Math theories definitely won't help you drop hints to girls in a night club you know...

  • Atiyah Singer index theorem (Score:5, Informative)

    by Ats ( 88113 ) on Saturday March 27, 2004 @05:53AM (#8688252) Homepage
    This is an attempt to write a simplified introduction, which hopefully doesn't contain too many outright errors. The errors may be due to both oversimplification and the fact that I am only studying this subject myself, so corrections are welcome.

    The Atiyah-Singer index theorem provides a link between algebraic topology, the study of 'large-scale', structural properties of manifolds, and advanced calculus on manifolds. So in order to precisely understand what the theorem states, some background in those two areas is essential. But I'll try to give some examples of the concepts that it deals with.

    The index of a differential operator A is the difference dim (ker A) - dim (coker A), where dim means dimension, ker A means the kernel of A, and coker means the cokernel of A. The kernel and cokernel are somewhat analogous to their meanings in linear algebra, for an n x n square matrix A, just as differential operators and matrices have many analogous properties. In linear algebra the kernel is also sometimes known as nullspace, the space of vectors x with A x = 0. The cokernel is slightly more involved. For a matrix A, it is the orthogonal complement of its range, the space of y such that A x = y for some x. With some linear algebra you can prove that for an n x n matrix A, dim (ker A) - dim (coker A) = 0.

    But with differential operators it is more complex. To take an example of the real line, R, and the differential operator d/dx, the kernel clearly has dimension one, whereas the cokernel has dimension zero, which is rather easy to see intuitively, but could require some work to prove carefully, I'm not sure. Anyway, the Atiyah-Singer index theorem deals instead with multidimensional differential operators, and pseudodifferential operators instead of differential operators. The pseudodifferential operators are a superset of differential operators, defined via Fourier analysis.

    I don't know how the topological side could be illustrated that well...The topological invariants that appear on the other side of the theorem are in some ways similar to describing the deformation invariant structure of a manifold by counting holes on it, but the topology that the index theorem deals with is vastly more general and powerful and doesn't necessarily have much to do with holes anymore.

    The index theorem has been used for example in particle physics where the topology of the spacetime manifold can be used to obtain information about the Dirac operator for fermions, which is an elliptic (pseudo)differential operator, the operator class that the index theorem deals with.

    There is a good book by Booss and Blecker on the subject: "Topology and Analysis: the Atiyah-Singer Index Formula and Gauge-Theoretic Physics
    ", geared toward physical applications. Too Amazon doesn't seem to have it. "Spin geometry" by Lawson and Michelsohn is also pretty good. I am reading those two books at the moment..
    • The Atiyah-Singer index theorem provides a link between algebraic topology, the study of 'large-scale', structural properties of manifolds, and advanced calculus on manifolds

      And them boys got the Navel Price for studying manifolds? There ain't nothing special 'bout them, just bolt them to the block, stick the exhaust pipe right on the other end, and tighten the collar real good, and that's it. Awh been doin' that fer twenty years now ya know, and I ain't never got no price...

    • I have one small correction... (Score:5, Funny)

      by bloggins02 ( 468782 ) on Saturday March 27, 2004 @08:55AM (#8688604)
      the kernel clearly has dimension one

      For some reason, your word processor has randomly substituded the word "clearly" in your discussion of topology and differential equations.

      Microsoft has confirmed this to be a problem with certain math professors and graduate students.

      Solution:Installation of Girlfriend 1.0 or Real Life 2.37 or higher appears to correct the problem

      Temporary Workaround: If the above programs are not available, automatically replacing the word "clearly" with "confusingly" seems to retain the sentence's grammatical structure and enforce its true meaning.
  • I didn't know Aliyah the Singer was a Nobel Prize winner!

    (Comes from not reading the articles, ever!)

  • From what I heared... (Score:4, Informative)

    by k-hell ( 458178 ) on Saturday March 27, 2004 @06:25AM (#8688296)
    A fundamental problem with solving complex system of differential equations is that it is often nearly impossible to solve them. So what the Atiyah-Singer index theorem answers is how many solutions the system of differential equations has. I.e., it can tell us if the system has any solutions at all, and that the answer only depends on the shape of the geometric area where the model resides (thus, it is purely a topologic answer). As you can imagine, applying this theorem can save a lot time.

  • topology (Score:5, Insightful)

    by cancerward ( 103910 ) on Saturday March 27, 2004 @06:25AM (#8688297) Journal
    I only know Atiyah as the author of a textbook on commutative algebra, which was a graduate course I hated.

    There's a lot of incomprehension in the comments about higher mathematics. The fact that all four of the Clay Mathematics Institute Research Fellows [claymath.org] this year are not native Americans indicates the truth of the AeA's comment on math teaching in American schools. [slashdot.org] I note that all of the fellows are in topology or closely related areas. My doctorate is in combinatorics, "the slums of topology", so I'm probably not qualified to explain the Atiyah-Singer theorem to y'all!

    • The fact that all four of the Clay Mathematics Institute Research Fellows this year are not native Americans indicates the truth of the AeA's comment on math teaching in American schools.

      ... and the fact that at least one of them shares his first name with me, and is only a year older than me indicates the extent to which I under-performed in college.

      :-|

      Actually, it probably doesn't, but heck, I'm having mixed feelings about this particular piece of news:- should I feel proud that another Akshay has sc

  • maybe this will help, probably not (Score:4, Informative)

    by 1iar_parad0x ( 676662 ) on Saturday March 27, 2004 @07:12AM (#8688377)
    I think I can summarize the collective "wha" by saying, I do really appreciate postings on abstract mathematics, but I don't have a clue what your talking about. In fact, I could have a PhD in mathematics and be a respectable researcher and only have a foggy notion.

    With that said, I included a couple of links below:

    Wikipedia's explanation on the problem [wikipedia.org]

    an insanely terse definition with a bibliography of the originally sited papers [wolfram.com]

  • Good news! (Score:4, Interesting)

    by azaris ( 699901 ) on Saturday March 27, 2004 @07:35AM (#8688414) Journal

    For all those not initiated to deeper mathematics, there's a simpler online proof [www.emis.de] that uses the heat equation instead to prove the Atiyah-Singer Index Theorem.

    Of course, the first chapter alone is over 80 pages of functional analysis, but still...

  • Practical Application of the Atiyah-Singer Theorum (Score:5, Funny)

    by steveoc ( 2661 ) on Saturday March 27, 2004 @08:52AM (#8688593)
    In response to some of the negative 'So What ?' comments, I shall use AC's brilliant explanation to deduce a practical application of this most excellent theorum.

    You need to look past the obvious sometimes, young Grasshoppers. Lets apply the Atiyah-Singer Theorum to a night club scenario.

    A nightclub, is a bounded 3-D dimensional space, which may be inhabited by (amongst other things), a collection of personages, which are nothing more than manifolds in a 4-D continuum.

    The Atiyah-Singer theorum proposes that there is a deep connection between the index of the manifold, and the topological nature for each personage.

    Having a rich understanding of the index of the vector bundles for these manifolds can then allow you to derive the underlying topology of these unbounded mainfolds.

    The underlying aim of being in the Night Club, for our purposes, is to ultimately deduce the underlying topology of the subject, without having to physically remove their clothes, or subject them to X-rays or invasive procedures.

    By applying the Atiyah-Singer theorum in this case, we can compare the vector normals for surface vectors around the chest area of the subject. You will quickly note that some subjects have a more or less constant vector normal for this section, whilst others have an interesting flowing perturbation of the surface, yieling a set of vectors which significantly alter the index of the entire manifold.

    Other more subtle clues abound .. but generally if you are able to observe and compute the vector normals, then by appling Atiyah-Singer, we now have the ability to deduce topological invariants, as well as the probable vectors of these invariant-holding bounded manifolds in the 4-D continuum.

    As AC explained in the pretzel example, topological invariants include things like the number of holes in the preztel. And here is the crux of the matter, my learned friends.

    We can now select from a set of 4-D manifolds, those manifolds which are most likely to offer up a set of invariants for a finite space of time in the near future space-time continuum, because amongst all of the nightclub inhabitants, our superior mathematical abilities allow us to quickly compute indeces and probabilities, as well as quantum outcomes.

    Your choice of invariants is entirely up to you, each to his / her own, I say.

    This, ladies and gentleman, is why great mathematians of both sexes and persuasions, manage to get laid as often and as varied as they so choose, whilst the dumb-ass jocks of the world have to make do with watching football, getting drunk with their mates, or mindlessly burning rubber on public roads.

    Its pure Darwinism in action.

    • All humans are topologically equivalent to the torus.

      You can grab a person by the mouth and ass hole and then diffeomorph them into a torus by evening out their digestive system.

      Therefore all humans are S(1)xS(1)

      unless they have something stuck up their ass, in which case they are S(2)

      __________________
      Is it any coincidence that the doughnut and coffee mug are also topologically equivalent?

      __________________
      I shall now refer to the surjective mapping from S(1)xS(1) -> S(2) as the "butt-plug" project

  • Don't feel bad about not understanding the details of this. I have a masters degree in math (and know a good deal about topology and analysis) and this stuff is still mostly jibberish to me. This is very deep stuff. But the way it interconnects math and physics is very interesting.

  • Intro to Topology (Score:3, Informative)

    by sqlgeek ( 168433 ) on Saturday March 27, 2004 @09:36AM (#8688701)
    And Atiyah has an absolutely wonderful little (very little) book that covers some of the foundations of topology in an accessible, non-rigorous manner. It is the single book that I would hand to anyone who wanted to know what topology was, but didn't want to learn how to read/write proofs.

    Ok, I'm back from the bookshelf, and I was entirely mistaken. The book I was refering to above is by Paul Alexandroff and is called _Elementary Concepts of Toplogy_. The book right beside it (also very small) is in fact by Michael Atiyah -- _The Geometry and Physics of Knots_. It is not at all a book for non-mathematicians, but for the record, covers interrelations between knot theory, topological invariants and differential geometry in an astounding breadth for such a slim volume. Wonderful stuff.

    Scott

  • The Abel Prize (Score:5, Informative)

    by kisak ( 524062 ) on Saturday March 27, 2004 @10:46AM (#8688967) Homepage Journal
    The Abel Prize is named after the brilliant Norwegian mathematician Niels Henrik Abel [abelprisen.no] that died at the age of 26, after living a life with little money and little support. It is quite amazing that at that young age Abel was able to produce results that put a lasting mark on modern math. Another of the "young dead" in the history of mathematics is Galois [st-and.ac.uk], who died at the age of 21 and is remembered for results that expanded on earlier work of Abel. Because of these two and also many other mathematicians who did their best work at very young age, math has got the reputation of being the young man's science (or young woman for that matter, even if there seems to be a male dominance in math still in these days).

    The Abel prize [abelprisen.no] is introduced as a sort of "Nobel Prize of math" where people are rewarded for results and achievements that have shown themselves to be of lasting value in the field. Alfred Nobel did not want there to be a Nobel Prize in math, since he himself saw little scientific value of math! The most prestigious prize in math before the Abel came into being is the Fields medal [st-and.ac.uk], but this prize is only given to younger mathematicians (belove the age of 40) that has made break-through results and also show promise for the future. The Fields medal is handed out every 4 years while the Abel will be handed out every year (first prize was handed out last year).

    Must have been ironic for Abel if he were to know that such a huge money prize is to be given out in his name, when his whole life he had to live in poverty and fight to get time and money to do his scientific work. The irony of Abel's life is also that Abel himself finally got a professorship in Berlin but too late; the letter was sent to him two days after his death.

    • From the website:

      The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.

      That's about $909k USD [yahoo.com].
  • At first I thought this post was about the ayatollah khomeini starting a singing career.
  • I and I expect a few others here are quite interested in theorems such as this, however we run into a bit of a problem. We cannot understand them. So my question is, to those of you who hold advanced maths degrees, where can we go to find out about the world of abstract mathematics. Where are good introductory websites? What are good introductory texts? Inquiring minds want to know!
    • Look at some of the books by Keith Devlin or Martin Gardner.

    • Re:Where to learn about abstract mathematics? (Score:1, Informative)

      by Anonymous Coward
      For this particular topic (the Atiyah Singer theorem) it is going to require some work because upper undergraduate explanations and examples have not yet been published. http://www.math.uni-bonn.de/people/strohmai/globa n /about.html gives an idea. So does http://math.bu.edu/people/sr/webbook/node2.html However, what you really want, it seems to me, is an illustrated lecture on the four theorems which are prior examples of Atiyah Singer, each itself illustrated by specific examples, such as the integrated
  • It turns out that Singer is my "PhD grandfather"; he was my PhD advisor's PhD advisor. (He is the "PhD grandfather" of many matheticians.) The Atiyah-Singer index theorem is a tremendous accomplishment and this prize is a good way to recognize the importance of their theorem.
    Some of the comments here bring to mind a complaint I have, even if these comments are funny (e.g. "Now all they have to do is derive a theorem that can solve the conundrum that is, how to share the trophy between them equally each
    • It is just envy and ignorance on the part of us mere mortals. In twenty years of formal education, I had one decent math teacher--in the seventh grade. He quit half-way through the year to go to work for private industry. Not to say some of the teachers were not good mathematicians, they just were not good enough teachers to get through my thick head, although I studied diligently. On the other hand, many of the math teachers were also athletic coaches. What does that tell you about restraining school-
      • In high school, my cross country/track coach was a math teacher. As a coach, he was great (not like a football coach but just kind of "crazy" (in a fun way)) but I never had a math class from him and cannot say if he was good or bad. In K-12, my best math teacher was my ninth grade algebra teacher. He had been a wrestler (Olympic ?) in eastern europe and he was "tough"; he started my interest in math. (I found out later from my folks that he followed my academic career and, for example, attended some event
        • I remain envious and in awe of people with a deep understanding of math! Certainly, I did not mean to disparage an entire class of people, that is to say "coaches." I meant to say that most of my math teachers could not get through to me, and that U.S. school systems tend to under-fund their math departments. I always felt shortchanged in my math education, and that made, other, dependent subjects more difficult. I tried to rectify my deficiencies in college, but that was a disaster. My first college tr
  • you got to be pretty smart to even begin to understand what these two guys did that was so smart
  • Apparently Atiyah and Singer have become entangled.

  • Here [wolfram.com] is some more information.
  • Comment removed based on user account deletion
  • Why do all the explanations about what this story is about sound like a conversation in engineering between data and geordi, right before the imminent threat will destroy the ship?
  • What? The noBEL prize... wait... no

    Wait.. is that singer? Is this a grammy... wait... no

    What... it's an Abel award. What is that? An award you get when your brother kills you (/biblical reference).

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