Slashdot Mirror
Atiyah and Singer to Share the 2004 Abel Prize
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.'"
the audience utters a collective "wha?"
"Come on, let's go drink till we can't feel feelings anymore."
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.
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 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 is of MIT, and is an institute professor, 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?
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!?
And this is the least technical definition I have come across so far.
Trawling thru the USENET I found:
The Atiyah-Singer expression is:
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 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 Flamebait Troll Redundant
Positive: Insightful Interesting Informative Funny
Nothing to see here
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.
"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
"Study your math, kids. Key to the universe." -The Archangel Gabriel
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...
"A door is what a dog is perpetually on the wrong side of" - Ogden Nash
1. Write the book "Atiyah-Singer Index Theorem for Dummies"
2. ???
3. Profit
Profit from whom? all three people who'll buy the book? Try writing a book on what Atiyah and Singer might be doing together at night in bed, like "Hot Gay Madrigal in the Middle of my Theorem", you'll make a better seller.
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..
I mean Aaliyah was a great Singer but I don't see why she deserves a prize?
Although her topology was an exciting development...
Anyone with the slightest knowledge of Inverted Topolology and Qantas Therorem Thingamajigs (especially Industrial ones), knows that the singer Aaliyah really had nothing to do with the any of the lastest advancements in the cross fertilization of Superstring theory, period.
I didn't know Aliyah the Singer was a Nobel Prize winner!
(Comes from not reading the articles, ever!)
-- You are in a maze of little, twisty passages, all different... --
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.
There's a lot of incomprehension in the comments about higher mathematics. 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. 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!
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.
Cokernel:
The cokernel of a group homomorphism of Abelian groups (modules, or abstract vector spaces) is the quotient group (quotient module or quotient space....
Yup, these definition were all I needed. Oh, and a doctorate in mathematics. Sadly I bought mine, via the internet, from the university of central capadocia, florence. (UCCF)
My eyes glazed over and I went into Luser mode. good grief, WTF is this all about? Okay, this is news for nerds.... but really, give me the latest SCO stupidity, it's more easily understood.
Wait.....
Has someone worked out a mathematical model for SCO stupidity? Is this is what this is about? Oh my, now that IS impressive! Can someone post the equation, pls?
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
an insanely terse definition with a bibliography of the originally sited papers
What do you mean my sig is repetitive? What do you mean my sig is repetitive? What do you mean....
Not because it isn't an essential and relevant result, but because most people here simply don't have a clue about abstract mathematics.
So... should we move this to an AOL chatroom or what?
For all those not initiated to deeper mathematics, there's a simpler online proof 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...
At least that's what is looked like first thing in the morning...
coffee....
The alternative to limited government is unlimited government.
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 noting that IT professionals are particularly guilty of the practice.
Government of the people, by corporate executives, for corporate profits.
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.
.. 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.
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
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.
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.
The following comment has already dealt with this= 101916 &cid=8688201
http://science.slashdot.org/comments.pl?sid
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.
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 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, 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.
--- guns don't kill people, people with guns kill people ---
Was it Feynnman who said that when you derive a new mathematical result, it feels more like you discovered it than like you invented it?
He said it feels like it was already there.
Already where?
My wife answers that by saying "in the math universe", which is filled with beautiful abstractions and shortcuts instead of the clunky assemblies of matter and stretches of distance that make up our universe.
I read it as Aaliyah the singer won the nobel prize. I was really disappointed with this post.
these days are going to women.
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!
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 week which as you all know contains a number which, wait for it.. is not divisable by two without remainder!").
A person (friend) earned a MS in math from us and, after teaching for a few years, applied for a job in IT in (approximately) 1997. At the job interview, people gave him a very hard time about knowing math and having a masters in math. When they gave him programming tasks, they found out that he was an excellent programmer (and they hired him). (After some time, he left and now heads the IT department for a moderately large legal firm.)
This is just one example which makes me think many programmers (and IT people in general) are afraid of math. When you consider that Don Knuth's PhD is in mathematics and the contributions of mathematicians like Gene Golub, it would seem to be difficult to justify the negative feelings toward mathematics which sometimes occurs in IT. Can you explain this to me?
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" projection. I guess that's more of an insertion than a projection though.
__________________
I've got one more but it's too dirty and I'm tired.
A Usenet Troll Triumphs on Slashdot
you got to be pretty smart to even begin to understand what these two guys did that was so smart
bring it on! --- JFK
Apparently Atiyah and Singer have become entangled.
Here is some more information.
=====
imagetweak.netWeb-based image t
Actually you have a hole between each of your ears and your throat (Eustachian tubes). Also a hole from each nostril to your sinuses and again to your throat, and from each eyesocket to your sinuses as well (try plugging your nose and blowing real hard - you can whistle through the corners of your eyes and really gross out the other kids - not that I'd know).
;).
So the holes piercing your external topology number seven - practically Swiss cheese. And that's not counting people with interesting injuries who might have more somewhere, nor all the internal tunnels (the kidneys alone are mathematically mind-boggling
On the other hand if you lop someone's head off, they're a doughnut. Or if just they have a bad cold.
Now I'm thinking of Homer Simpson in that Halloween episode...
No, I think they know what they're on about. Surely not the first time someone named Singer has been instrumental in repairing a rift...
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).
Jay | http://oldos.org
Does this apply to both Linear and Non-Linear differential equations?
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".
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 trig instructor was a newbie trying to teach a class of about two-hundred surly students. I dropped-out after a few weeks, and re-took the class from a recent Greek immigrant. After all, they invented trigonometry didn't they? He seemed to be a good mathematician but he could hardly speak English (although, he spoke better English, than I spoke Greek!) I barely passed, and totally discouraged, never took a calculus class, or any math beyond trig. Otherwise, I was an outstanding student and might have gone into physics. It always seemed like I "almost got" math. Looking back, if one or two teachers could have opened up the blocked pathways in my brain at the appropriate time, it would have made a world of difference. I copied all the serious Slashdot comments on the Atiyah-Singer index theorem for later study. The funny thing is, I think I understand what the theorem is about and why it is significant--in a general, theoretical sort of way--even though the math is beyond me. Weird. I believe any prejudice against math majors stems from the ignorance/ inadequacies of other people regarding the discipline. Yes, we are in agreement, we need more people with math skills. I'd have a better chance of comparing Poe and Joyce than proving a basic theorem in geometry!
"...while history is usually explicable it is often irrational" --Roger Spiller
The Novikov conjecture (or here) is worth mentioning.