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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.'"
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 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)
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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..
Differential Operator
Vector Bundle
Fredholm Operator
Cokernel
Now, armed with all those definitions of all the unfamiliar terms in that paragraph, complete with links to the terms used in the definitions (which are themselves complete with links to all the terms used in the definitions of the definitions, ad nauseum), you've got all you need to understand those two paragraphs! Isn't the Internet great?
main(c,r){for(r=32;r;) printf(++c>31?c=!r--,"\n":c<r?" ":~c&r?" `":" #");}
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.
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.
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....
"My opinions are my own, and I've got *lots* of them!"
Heh ok, what is the kernel of d/dx?
It is the space of functions f(x) with df(x)/dx = 0. This means that f(x) = c. The dimension of the space of functions f(x) = c for some c is 1-dimensional.
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 ---
You can read the actual theorem from Chapter 3 of Peter Gilkey's book available here. 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. Unfortunately, this doesn't seem to capture how impressive the theorem really is.
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/globan /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 curvature on a few manifolds to illustrate the Gauss Bonnet theorem showing the types of relationships included in the theorems. This has not yet been done after 40 years, but the award would be a fine reason to do it now.
Perhaps someone (maybe a summer student)at the geometry center could make some of the pictures for such a cooperative online introduction. It might do wonders for math advertising.