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.'"

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Atiyah-Singer Index Theorem

[amigoro]

From MathWorld:

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 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 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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Atiyah Singer index theorem

[Ats]

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..

Here you go, a layman's explanation.

[Anonymous Coward]

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.

Re:Here you go, a layman's explanation.

[sqlgeek]

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