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Ramanujan?s ?Lost Notebook? Astounds Americans
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18 February 2005 -- Death snatched India’s Srinivasa Ramanujan when his genius had just blossomed - at age 32 in 1920.
Eighty-five years later, it’s still taking several expert mathematicians in America a lifetime to decipher just a portion of his incandescent genius. Remarkably too, some of that work is getting financial support from the National Security Agency.

Just consider what’s happening now

Prof. George Andrews of Pennsylvania State University, one of the world's most eminent mathematicians, is conducting a series of seminars from January 25 until March 22, 2005 at the University of Florida. Andrews, who has spent the past 30 years studying Ramanujan’s considerable output, is providing insights into ‘Number Theory and Combinatorics’ through six lectures on topics related to Ramanujan's ‘Lost Notebook.’ The lost notebook arises from the last year of Ramanujan's life and contains approximately 650 assertions without proofs.

According to Prof. Krishnaswami Alladi, Chairman of the Department of Mathematics at the University of Florida, who recently visited Ramanujan’s birthplace, along with Prof Andrews, the lectures will seek to unveil ‘what did Ramanujan have up his sleeve?’ Alladi’s research is in Number Theory, an area where Ramanujan has made spectacular contributions. He is also the Editor-in-Chief of The Ramanujan Journal, an international publication devoted to all areas of mathematics influenced by Ramanujan.

In his introduction to the first seminar on January 15th, Andrews described the life of Ramanujan, the discovery of his Lost Notebook, and attempted to describe some of his surprising achievements. The third lecture was delivered on February 15. Three more are scheduled.

Essentially, Ramanujan's legacy consists of 4,000 formulas on 400 pages filling 3 volumes of notes, all densely packed with theorems of incredible power but without any commentary or, which is more frustrating, any proof. In 1976, however, a new discovery was made. One hundred and thirty pages of scrap paper, containing the output of the last year of his life, was discovered by accident in a box at Trinity College. This is now called Ramanujan's "Lost Notebook."

Commenting on the Lost Notebook, mathematician Richard Askey says, "The work of that one year, while he was dying, was the equivalent of a lifetime of work for a very great mathematician. What he accomplished was unbelievable. If it were a novel, nobody would believe it." To underscore the difficulty of the arduous task of deciphering the "notebooks," mathematician Jonathan Borwein and Peter Borwein have commented, "To our knowledge no mathematical redaction of this scope or difficulty has ever been attempted."

Prof. Andrews shot to fame in the 1970s when he discovered Ramanujan's Lost Notebook at the Wren Library in Cambridge University and wrote a series of important papers in Advances in Mathematics in which he explained Ramanujan's spectacular results in the context of current research, and in that process made fundamental improvements as well.

"There is still much to understand about the implications of many results in the Lost Notebook and their connections with current research which is one of the reasons to edit the Lost Notebook," said Professor Andrews. The first of these volumes will appear in 2005 and at least two more volumes will be forthcoming. "The mathematical content of the Lost Notebook is so immense, that it is difficult to predict at this time how many volumes it will take to completely edit it," he added.

According to Prof. Alladi, during the 1987 Ramanujan Centennial, the printed form of Ramanujan's Lost Notebook by Springer-Narosa was released by Prime Minister Rajiv Gandhi, who presented the first copy to Janaki Ammal Ramanujan, the late widow of Srinivasa Ramanujan, and the second copy to Professor Andrews in recognition of his contributions.

Ramanujan’s mathematical wizardry is also the reason why University of Illinois Prof. Bruce Berndt, an analytic number theorist with strong interests in several related areas of classical analysis, has devoted 31 years of his research to proving the claims left in three notebooks and a "lost notebook" by the Indian genius upon his death in 1920. Twenty-one students have completed doctoral theses under Berndt's direction, and currently, five Ph.D. students are writing their dissertations under his direction. Most are focusing on material in the lost notebook or on research inspired by Ramanujan.

The three original notebooks contain approximately 3300 results. The project of finding proofs for these claims took Berndt over twenty years to accomplish, and an account of this work can be found in his books, Ramanujan's Notebooks, Parts I-V, published by Springer--Verlag in the years 1985, 1989, 1991, 1994, and 1998. Also during this time, Berndt and Robert A. Rankin wrote Ramanujan Letters and Commentary and Ramanujan Essays and Surveys, both published jointly by the American and London Mathematical Societies in 2001.

Berndt’s research in this direction continues, as he and Andrews plan to publish volumes on Ramanujan's "lost" notebook, analogous to those published on the ordinary notebooks. They are currently "editing" Ramanujan's Lost Notebook, which will be published by Springer later this year.

When an interviewer for Frontline asked Brendt if he found any of Ramanujan’s results difficult to decipher, he admitted: “Oh yes. I get stuck all the time. At times I have no idea where these formulae are coming from. … There are times I would think of a formula over for about six months or even a year, not getting anywhere. Even now there are times when we wonder how Ramanujan was ever led to the formulae. There has to be some chain of reasoning to lead him to think that there might be a theorem there. But often n this is missing. To begin with, the formulae look strange but over time we understand where they fit in and how important they are than they were previously thought to be.”

RAMANUJAN’S PASSION

Born in India in 1887, Ramanujan was a mathematical genius whose work continues to surprise mathematicians into the 21st century. His work is filled with surprises. At the Ramanujan centenary conference at the University of Illinois, it was physicist Freeman Dyson who proclaimed, "That was the wonderful thing about Ramanujan. He discovered so much, and yet he left so much more in his garden for other people to discover."

Born into poverty, Ramanujan grew up in southern India, and although he had little formal training in mathematics, he became hooked on mathematics. He spent the years between 1903 and 1913 cramming notebooks with page after page of mathematical formulas and relationships that he had uncovered.

Ramanujan's life as a professional mathematician began in 1914 when he accepted an invitation from the prominent British mathematician G.H. Hardy to come to Cambridge University. He spent 5 years in England, publishing many papers and achieving international recognition for his mathematical research.

Though his work was cut short by a mysterious illness that brought him back to India for the final year of his life, Ramanujan's work has remained a subject of considerable interest.

The 600 formulae that Ramanujan jotted down on loose sheets of paper during the one year he was in India, after he returned from Cambridge, are the contents of the `Lost' Note Book found by Andrews in 1976. He was ailing throughout that one year after his return from England (March 1919 - April 26, 1920). The last and only letter he wrote to Prof. Hardy, from India, after his return, in Jan. 1920, four months before his demise, contained no news about his declining health but only information about his latest work: ``I discovered very interesting functions recently which I call `Mock' theta-functions. Unlike the `False' theta-functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as ordinary theta-functions. I am sending you with this letter some examples ... ''.

The following observation of Richard Askey is noteworthy: ``Try to imagine the quality of Ramanujan's mind, one which drove him to work unceasingly while deathly ill, and one great enough to grow deeper while his body became weaker. I stand in awe of his accomplishments; understanding is beyond me. We would admire any mathematician whose life's work was half of what Ramanujan found in the last year of his life while he was dying''.

As for his place in the world of Mathematics, this is what Berndt says: `` Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100''.

In 1957, with monetary assistance from Sir Dadabai Naoroji Trust, at the instance of Professors Homi J Bhabha and K. Chandrasekaran, the Tata institute of Fundamental Research published a facsimile edition of the Notebooks of Ramanujan in two volumes, with just an introductory para about them.

The formidable task of truly editing the Notebooks was taken up in right earnest by Berndt in May 1977 and his dedicated efforts for nearly two decades has resulted in the Ramanujan's Notebooks published by Springer-Verlag in five Parts, the first of which appeared in 1985.

Between 1903 and 1914, before Ramanujan went to Cambridge, he compiled 3,542 theorems in the notebooks. Most of the time Ramanujan provided only the results and not the proof. Berndt says: "This is perhaps because for him paper was unaffordable and so he worked on a slate and recorded the results in his notebooks without the proofs, and not because he got the results in a flash."

The three original Ramanujan Notebooks are with the Library of the University of Madras, some of the correspondence, papers/letters on or about Ramanujan are with the National Archives at New Delhi and the Tamil Nadu Archives, and a large number of his letters and connected papers/correspondence and notes are with the Wren Library of Trinity College, Cambridge. The Ramanujan Institute for Advanced Study in Mathematics of the University of Madras is situated at a short distance from the famed Marina Beach and is close to the Administrative Buildings of the University and its Library. Mrs. Janakiammal Ramanujan, the widow of Ramanujan, lived close to the University's Marina Campus and died on April 13, 1994. A bust of Ramanujan, sculpted by Paul Granlund was presented to her and is now with her adopted son Mr. W. Narayanan, living in Triplicane near Chennai.

Ramanujan's story has been told recently by mathematician Ian Stewart, writing in the magazine "New Scientist". Stewart brings out strongly the special nature of Ramanujan's genius, which is his amazing intuition of the correctness of a complicated mathematical result. He points out what a lot this has to do with Ramanujan's lack of a formal education. Because of this, many of Ramanujan's proofs have serious gaps in them. They do not follow step by step in the way that a conventional proof would. In fact, later mathematicians have often had to wrestle long and hard to produce proofs that are completely watertight: to dot all the i's, and cross all the t's, as it were. The incredible thing is that for all their lack of technical rigor, almost all of Ramanujan's results turned out to be correct.

DREAM COME TRUE

Alladi says that his visit to Ramanaujam’s birthplace was a dream come true. “What an inspiration to see this small humble home from where so many significant mathematical discoveries poured forth ” says Alladi. He informs that Shanmugha Arts, Science, Technology and Research Academy (SASTRA), a private university whose main campus is located in the town of Tanjore, purchased the home of Ramanujan in 2003, and has since maintained it as a museum. Research chairs established at the SASTRA centre at Kumbakonam — two by the Department of Science and Technology, Government of India and one by City Union Bank Ltd. — encourage research in the field of mathematics in honor of Ramanujan.

Meanwhile the founding has been announced of the "Ramanujan Prize for Young Mathematicians from Developing Countries" by the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, in cooperation with IMU, and with support from the Niels Henrik Abel Memorial Fund, Norway. The Prize will be awarded annually for the highest mathematical achievement by young researchers from developing countries, which conduct their research in a developing country. The recipient must be less than 45 years old. Work in any branch of the mathematical sciences is eligible for the prize. The Prize amount will be $10,000. The goal is to make the selection of the first Prizewinner in 2005.

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