April 30, 2006

Chandrakant Khare and Fermat's Last Theorem

I had seen Simon Singh on Book-TV some years back talk about this book, Fermat's Enigma regarding the quest to solve Fermat’s Last Theorem (FMT) but though its been at the back of my mind, I never read up more about it till yesterday when I found a great website, Larry Freeman‘s blog on the subject.

The blog is a great site for amateur readers like me to read about ‘the story behind FMT and Wiles' proof in a way accessible to the mathematical amateur.’ You can also read about the intrigue and excitement that caught even the fancy of the media when
Andrew Wiles, with the help of Richard Taylor, proposed that he had a solution to FMT in 1995. (If you insist on reading it, here are the papers, all 129 pages of it – most mathematicians also do not follow it – so, do not say you were not forewarned!)

In 2006, another mathematician, Chandrashekhar Khare (previously at TIFR and now Associate Prof at University of Utah), has provided a very significant result (here is the paper) that builds on the work done by Wiles.

The Slashtdot entry on this says:"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. "

Actually, Khare does not provide another proof of the FMT but proved what is known to experts as the ‘level-one Serre conjecture’. This conjecture was posed in 1972 by the Fields medallist Jean-Pierre Serre, and belongs to the field of Arithmetic Algebraic Geometry.

From: http://plus.maths.org/latestnews/jan-apr05/serre/

Chandrashekhar Khare, a mathematician from the University of Utah, has announced that he has Serre's conjecture is in a sense a parent of Fermat's last theorem: mathematicians have known for some time that if the first is true then so is the second. In fact, it is a certain part of the conjecture which implies Fermat's last theorem, and this part was proved by Khare and his collaborator J.P. Wintenberger, and independently by the mathematician Dieulefait.

Fermat's last theorem as well as the conjecture by Serre, are ingredients of a wider program to unify various areas of mathematics, known as Langlands philosophy, (conceived by the mathematician Robert Langland in the 1960's and consists of a set of conjectures concerning the intimate relationship between number theory, geometry and algebra). The idea behind such unifying theories is that it should be possible to directly translate every concept in a given area of maths into all the other areas of maths……and the relationship between the objects should be the same in both areas.

See more on other recent work on Serre’s conjectures

Recommended Books from Larry Freeman‘s blog: Fermat Last Theorem

Recommended Reading from Larry Freeman‘s blog: Fermat Last Theorem

Other references gleaned from other resources on the web – in addition to those already hyperlinked in the text above:

Read about Serre's work in the Plus article En-Abeled.

Sidenote: I debated titling this only as 'Fermat's Last Theorem' but I think FMT has got lots of media coverage over the years (including a PBS dedicated episode on Nova called The Proof, which dealt with the investigation of the theorem and its solution by Wiles) while Khare, being an Indian and having worked mainly in India until recently has not (outside of mathematicians involved in number theory...so, as an Indian, I put him right up there on the title along with Fermat ;)

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