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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. "
www.math.utah.edu/~shekhar/papers.html
is in any way relevant why?
I have a feeling a lot of excellent math departments will be looking to hire this guy from Utah.
503 - Service Unavailable. There is insufficient bandwidth in the server room to supply you with a copy of this paper.
I know I'm poised to make a huge breakthrough, unfortunately I can never seem to make it over that last hurdle, which is, you know.. to make the actual breakthrough.
Starsucks
At least this Indian mathematician is still alive. :)
Even better, at least this Indian mathematician has a name.
If htis pans out as well as it looks like it will, this guy will be a full professor in no time flat.
Reject Fear - Embrace Hope
Could somebody explain what this is about, and what this would mean? There isn't any concrete information on that in TFA ...
Besides, this is kinda vaporware. Why is this even news? Why not talk about it once it's done?
EagerEyes.org: Visualization and Visual Communication
So he's involved with outlining a two-part solution... and he's completed one part of it. That's sort of an actual accomplishment, isn't it?
I mean, I'm poised to win the lottery. He's actually doing things.
Don't disappoint your bird dog. Go to the range.
He has proved what is known to specialists in the field as the `level-1 case of the Serre conjecture.' In earlier work done with the French mathematician, J.P. Wintenberger, in December 2004, Dr. Khare outlined a two-part general strategy to prove the Serre conjecture fully. The present result is a first key step.
Wikipedia page for Serre conjecture
the underline appears all the way through " to make a significant breakthrough in the field of number theory with his solution "
even though the word "solution" leads to a different link than all of the preceding words.
Pretty exciting stuff! (Relatively speaking, of course :-)
It was proved in 1995 by English mathematician Andrew Wiles.
Wikipedia page of the theorem
I don't follow the field close enough to know its relation to Serre's multiplicity conjectures.
that Serre's Conjecture was already proven?
The best education consists in immunizing people against systematic attempts at education. - Paul Feyerabend
...hundreds of new mathemtical theorems are discovered by people around the world. Many of these become peer reviewed and published. So why is this particular one on the front page? It's basically unknown outside of mathematical circles and is posted on a web site where any crackpot can post. Shall we start having stories about JSH on sci.math?
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Um ... the only one related to crypto would be the theorem that a^p mod p == a if p is prime and a is co-prime to p.
That's not only not the famous Fermat Last Theorem but it's also trivially provable with basic number theory.
Tom
Someday, I'll have a real sig.
Just to speculate on a possible "what use" question that might arise, I can't help but notice the line This is one of the central themes of modern research in number theory and is devoted to the study of the relation between the symmetries of number theory and geometry. . If I may be so bold, anything that ties the study of pure math to geometry probably has implications for quantum mechanics. These objects may lie embedded in higher dimensions, and probably settle into stable configurations from near infinite possibilities. But they still have to satisfy some allowable mathematical model. This is just the type of thing that may allow us to better predict what those allowable states could be.
Letter To Iran
This is the real problem beyond Fermat
an ill wind that blows no good
*ducks*
Because I love to watch hot math action.
No! no! Introduce a Lemma!
Ya that's it, Proof by Counter-Example, that's the way I like it.
With this kind of progress, we should have FTL engines by the end of next year.
Being Indian is totally irrelevant to the story
*sigh*
But the story isn't using "Indian" in a racist way. It's merely an addition, perhaps to shed some "interesting" light on his background outside of his area of research. Not everything that mentions somebody's ethnicity is racist.
You sound like one of those overly-PC people who make things difficult for everyone, just for the sake of trying to live up to some misplaced "holier than thou" moral code.
Person1: "See those kids playing? One of them is my niece."
Person2: "Which one?"
Person1: "The black-haired one."
Person2: "There are six of them."
Person1: "The one in the blue shirt."
Person2: "That leaves four..."
Person1: "Ummm, the one with the sandals..."
Person2: "Three..."
Person1: "...and the red ball."
Person1: "Oh, you mean the black girl? Cute kid."
All your football coach are belong to us.
Go Gators.
Waiting on a math major to give a long-winded set of analogies to make this somehow releevant to the masses....
If religous zealots don't believe in Evolution, then why are they so worried about bird flu?
Can any expert confirm this or explain why this is relevant?
Yes, Fermat's Last Theorem was proven by Andrew Wiles in the early nineties.
This result would (apparently) supply another proof. Like the first, it would rely on quite complex and modern mathematics, but a slightly different sort than before.
The thing is that Fermat's Last Theorem is not especially important to mathematics; it's mostly a historical curiosity. However, it is a simple enough equation that anyone with a smattering of mathematics can understand: all you need to understand is exponentiation and addition operations, what an equation is, and what integers are. Plus, the story about Fermat's boast makes good press. These things make the equation famous.
So, the fact that this may prove Fermat's Last Theorem is icing on the cake, but for mathematicians the importance of the result is in its major implications for a vast field of research (algebraic geometry).
If it is actually proven, that is. I have seen enough popular accounts of some mathematician "on the verge of proving X" to not put much trust in such things. Wiles was wise to work in secret.
I wonder, is there a second Serre's Conjecture, or do people not do research any more to see if their work has already been done? Every link I can find for Serre's Conjecture or Quillen-Suslin Theorem indicates that it has already been proved (Quillen got the Fields medal in 1978).
"There are a dozen opinions on a matter until you know the truth. Then there is only one." - CS Lewis (paraprhase)
I'm a pure mathematician and I think this story is both uninteresting and irrelevant. It's not nerdy at all. It's a parochial feel-good story for Indians but unfortunately, because it's available over the world, that's to the Web, it's been mistaken for relevant story about something interesting.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
We need to find some Weapons of Math Instruction...
That's right. All your base.
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...
I heard a better story, but I have no idea if it's true or not.
There was a guy from Jamaica who had to go to the hospital for some reason, and he was driven there by his friend. When filling out the forms, he neglected to fill out the race field, and the receptionist nurse told him that he should check African-American.
He tried to explain to her that he was neither African nor American, even showing her his passport. Eventually he had to point out his (white) friend, who as coincidence has it was of South African descent and an American citizen. An African American, so to speak.
Regrettably, I don't remember how the whole thing ended.
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).