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Going beyond Fermat's last theorem

T. Jayaraman

An Indian mathematician makes an astounding advance in number theory

Prof. Chandrashekhar Khare in Mumbai. — Photo: Paul Noronha

MUMBAI: 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.

In a paper posted on the Mathematics Arxiv on the web in April 2005 and subsequently sent for publication to a leading mathematics journal, the 37-year-old mathematician based at the University of Utah 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.

According to Professor Dipendra Prasad of the Department of Mathematics at the Tata Institute of Fundamental Research (TIFR), Mumbai, the result is "one of the outstanding results in recent times in this field." He added that before the joint work of last year and Dr. Khare's current result, "it was unclear to the experts in the field that the conjecture would become a theorem in the near future." Before he moved to the United States in 2004 to take up a position on the University of Utah faculty as Associate Professor, Dr. Khare was on the TIFR faculty for nearly a decade.

Experts in the field emphasise that the attempt to prove the Serre conjecture — named after the eminent French mathematician, Jean-Pierre Serre, who originally formulated it in the early 1970s — has been the driving force behind many recent developments in number theory. As Professor Serre himself noted many years ago, his conjecture, if proved in generality, would imply the proof of Fermat's last theorem.

Dr. Khare's work reaps the harvest of seeds sown by Andrew Wiles and his co-worker, Richard Taylor, en route to proving Fermat's last theorem. Speaking to this correspondent after outlining his results at a TIFR seminar, Dr. Khare recounted that little progress had been made towards the proof of the conjecture till Professor Wiles' great work, and the realisation by Dr. Taylor that their methods could be used to tackle the solution of this outstanding problem. Asked about the reaction of experts to his proof, Dr. Khare said that Professor Serre himself was "excited and happy" to see his result, and that others also had reacted positively. As to the time frame of the final result, he noted that the present result had emerged earlier than he had originally anticipated.

He added that it was difficult to predict when it would eventually be completed.

According to Dr. Khare, the full proof of the Serre conjecture will provide a new impetus to the advance of the `Langlands Programme.' 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. Together with Professor Wiles' earlier work, in his view, the full proof will open new lines of research not explored so far.

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