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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.'"
the audience utters a collective "wha?"
"Come on, let's go drink till we can't feel feelings anymore."
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)
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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.
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..
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.
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.