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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!?
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?
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.