Slashdot Mirror
Going Beyond Fermat's Last Theorem
amjith writes "An Indian mathematician, Chandrashekhar Khare, is poised to make a significant breakthrough in the field of number theory with his solution of part of a major outstanding problem in algebraic number theory. He is currently an associate professor in Mathematics Department of University of Utah. "
This site does not have a very good record with mathematical breakthroughs that it runs on the front page. Just to give some examples:
1) A year and a half ago Slashdot ran a story (along with most of the MSM) about a Swedish girl having solved the 16th Hilbert problem. That turned out to be a completely bogus claim - she had, in fact, proved nothing.
2) Slashdot ran with there being infinitely many twin primes. The proof was flawed.
3) No, the Riemann hypothesis (the most coveted result in all of Mathematics) has not been proved.
Those are just the examples I can remember off hand. There have been several more, and I cannot think of a single one that has turned out to actually be true. So please take vague stories about being "poised to make a great story" from local press with a pretty hefty grain of salt...
Glancing over the responses so far, I've come across several links to "the" Serre conjecture. Of course, since this is Slashdot (Land of the Karma Whore) it also looks like not a one of those referred to the conjecture relevant to this discussion.
The particular conjecture of Serre that matters here focuses on the two-dimensional representations over a finite field of the Galois group Gal(Qbar/Q). Now since that's not particularly illuminating, let me say a bit more...
First, Qbar denotes the algebraic completion of the rational numbers -- that is, all the stuff you need to add to the rationals so that you can do stuff like factor polynomials with rational coefficients. So things like sqrt(2) are in Qbar, but transcental numbers like pi aren't.
Gal(Qbar/Q) is the group of symmetries of Qbar over Q -- the ways you can map it to itself while still preserving multiplication and addition, and leaving the rational numbers inside Qbar alone. For instance, complex conjugation gives an element of the Galois group.
Now one way to understand any group of symmetries is by looking at its "linear representations" -- basically, ways of assigning matrices to each of the symmetries so that matrix multiplication matches up with the composition of symmetries.
The conjecture talked about here claims to describe (in some sense) all such (irreducible) representations of Gal(Qbar/Q), at least if you limit yourself to 2x2 matrices and coefficients in a finite field.
This is similar to the Langlands Correspondence, which (among other things) deals with representations of Gal(Qbar/Q) by complex matrices (though not just 2x2).