RAMANUJAN @ 125

The story of mathematician genius Srinivasa Ramanujan is a story of human triumph and an example of what genius can accomplish against the odds, Robert Kanigel, Ramanjuan's biographer said on Monday.

Delivering a lecture hosted by the Organising Committee of Ramanujan 125, TNQ Books and Journals and the Institute of Mathematical Sciences, Mr. Kanigel said that while illuminating the genius of Ramanujan, “we should also momentarily hold the spotlight briefly” on the scores of people who did not make it due to adverse circumstances.

After giving a bare-bone account of the Ramanujan story, Mr. Kanigel who for his research visited India, which remains a jumble of mostly happy experiences and memories, and the beautiful but slightly forbidding Cambridge, took several questions from the audience.

Responding to a question on the everyday applicability of Ramanujan's theorems, Mr. Kanigel pointed out that Ramanujan's work was in pure mathematics and he took delight in exploring numbers without attaching any larger purpose.

Saying that he felt good when his book was translated into German, Italian or Greek, Mr. Kanigel also admitted to a certain sadness that it was not available yet in Tamil. His delight would know no end if “The Man Who Knew Infinity” were to be available in Ramanujan's own language, he said.

Offering his reasons for the title of the book, Mr. Kanigel said it defined though in a narrow sense the intimacy with which Ramanujan worked with numbers and theorems; it was as if “he knew infinity as his homeland.”

Mr. Kanigel was amused by a suggestion to get Italian filmmaker Bernardo Bertolucci to make a film based on his biography and to a questioner, who wanted to know whether Ramanujan would have felt more comfortable and enjoyed greater creative freedom in the U.S. than in Cambridge, he said he would like to think that Ramanujan in the U.S. might have felt more at home and a little “less stiff.”

When someone wanted to know whether working on the biography on Ramanujan had in any way changed him as a person, he remarked in a lighter vein that the impact would have been more on his wife when he got immersed in work.

N. Ram, Editor-in-Chief of The Hindu, said Mr. Kanigel's biography on Ramanujan, which was first published in 1991 and has undergone several printings, was perhaps his best and most influential works.

Among the many features of this book was its perfectly legitimate practice of the art of narrative journalism, or the use of a novelistic imagination in bringing a story alive without ever crossing the line between fact and fiction, he said.

The biography counters the flatness of the picture of Ramanujan's origins and life in 19 century south India, locates him in everyday life, brings alive the emotional geography of family relationships and the imprint on a young man of the cultural ethos, religious values and rituals of the times.

Mariam Ram (TNQ Books and Journals) and Balasubramanian of the Institute of Mathematical Sciences also participated.

collector's edition of Ramanujan notebook

The familiar face of Srinivasa Ramanujan looks up in relief from two classy hardback volumes in black. Inside is proof of the mathematician's genius. For both form and content, there can be no doubt that the second edition of Notebooks of Srinivasa Ramanujan is a collector's edition.

The notebooks of the ‘man who knew infinity,' originally believed to have been written down on loose sheets of paper, have kept mathematicians busy for over a century, scrambling to provide derivations for the results. An effort by the Tata Institute of Fundamental Research, Mumbai, the two volumes contain over 700 pages in Ramanujan's neat hand, some of his finest work. The books also have a foreword from Bruce Berndt, American mathematician known for his work in explaining the results Ramanujan postulated in his notebooks.

It is believed that Ramanujan actually worked out the problems on a slate in an attempt to save paper, using the sheets only to note down results. Since the discovery of the notebooks, mathematicians have wrestled with the results trying to arrive at plausible derivations. Most of the work has now been solved, thanks to Professor Brendt, according to M.S. Raghunathan, Vice-Chairman of the National Committee supervising the Ramanujan 125 year celebrations.

The books were launched at a function in which Prime Minister Manmohan Singh participated here on Monday. The release was possible with help from the archiving and digitising team at the Roja Muthiah Research Library (RMRL) in Chennai.

The production quality is of excellent standard, a huge improvement over the first edition published in 1957, again by the TIFR.

A brief overview of combinatorics in ancient and medieval India

The beginnings of combinatorics in India date back to Bharata's Natyasastra and the last chapter of the work Chandahsastra (Sanskrit Prosody) by Pingala (c. 300 BCE). Pingala deals in a few cryptic sutras with the combinatorics underlying the metres of Vedic Hymns and classical Sanskrit poetry. Metre is the basic rhythmic structure of a verse, here characterised by a finite sequence of syllables, some short and some long. We recall that syllables are irreducible units of speech, some being short like ka and some being long like kaa. We define the length of a metre as the number of its constituent syllables. We shall abbreviate a short syllable by l and a long syllable by g. For example, the length of a metre gll is 3.

Pingala gives a method of enumeration, which is called prastara, of metres of a given length n (which indeed are 2n in number), using a recursive procedure.

For example, the prastara of metres of length 1 is g, l (which are displayed as an array with two rows). According to Pingala, the prastara of metres of length 2 is obtained from the above by adding a g to the right of the prastara above, and then an l to this prastara, so that we get gg; lg; gl; ll (which are displayed as an array with four rows). The prastara of metres of length 3 is the array comprising the eight rows ggg; lgg; glg; llg; ggl; lgl; gll; lll which is obtained from the prastara of metres of length 2 above by adding first a g and then an l to the right. The above recursive procedure is then continued.

In this remarkable manner, Pingala's method leads one to a construction of the binary expansion of integers, by setting g = 0 and l = 1 in the successive rows of the prastara, the fifth row of the prastara of length n being a mnemonic for the binary expansion of i — 1.

For example, the 5th row ggl in the prastara of the metres of length 3 stands for the binary expansion 0:1 + 0:2 + 1:22 of 4. The last row, that is, the (2n — 1)-th row, of the prastara of metres of length n is ll; … l which yields the formula 1 + 2 + + 2n - 1 = 2n — 1:

With the advent of Prakrit and Apabhramsa poetry, came the idea of extending the above theory to matra metres where the value of a long syllable g is assumed to be 2 and that of a short syllable l is assumed to be 1, the value of the metre being defined to be the sum of the values of its constituent syllables. The construction of the prastara of metres of value n is achieved as above by a recursive procedure which is more subtle. The prastara of metres of value 1 is l; that with value 2 is g; ll; the prastara of metres of value 3 is obtained from these two by adding a g to the right of the prastara of metres of value 1 to get lg, and an l to the right of the prastara of metres of value 2 to get gl; lll; thus the prastara of metres of value 3 is lg; gl; lll. Similarly, we can write down the prastaras of metres of value n, using the prastaras of metres of value n _ 1 and n _ 2. If sn is the number of elements of in the prastara of metres of length n, we have s1 = 1; s2 = 2, and for n _ 3, sn = sn_1 +sn_2. This relation was noticed by Virahanka (c.600 CE).

The study of matra metres thus led the ancient Indian mathematicians to the sequence sn = 1, 2, 3, 5, 8, …, (what is generally known as the Fibonacci sequence, though discovered centuries before Fibonacci). As in the case of binary expansions, we obtain now unique expansions for natural numbers in terms of the Fibonacci numbers.

Combinatorics evolved in time not merely to apply to Sanskrit prosody but to many other problems of enumeration: in medicine by Sushruta, perfumery by Varahamihira, music by Sarngadeva (which shall be briefly discussed below), and so on.

Sarngadeva (c.1225 CE), who lived in Devagiri in Maharashtra, under the patronage of King Singhana, wrote his magnum opus Sangitaratnakara, a comprehensive treatise on music which gives in its first chapter a prastara enumerating all the 7! = 7_6_5_4_3_2_1 = 5040 permutations of the swaras S;R; G;M; P;D;N. The prastara of a single swara is S, that of two swaras S;R, is SR;RS; the prastara of three swaras, S;R;G is SRG;RSG; SGR;GSR;RGS;GRS:

More generally, the prastara of all the seven swaras, starts with the swaras in the natural order SRGMPDN and ends in the last or the 5040th row with the swaras in the reverse order, NDPMGRS, and the intermediate rows are constructed by a rule formulated by Sarngadeva.

It is indeed a remarkable fact that if we start more generally with n elements a1; a2; _ _ _ an and arrange their permutations in a prastara following Sarangadeva's rule, the ith row of the prastara is a mnemonic for a unique expansion of i; 1 _ i _ n!, as a sum of factorials, i = 1 _ 0! + c1 _ 1! + c2 _ 2! + _ _ _ + cn_1 _ (n _ 1)!; (with the convention 0! = 1), where the coefficient cj of j! lies between 0 and j _ 1. In particular, we have the beautiful formula n! = 1 _ 0! + 1 _ 1! + 2 _ 2! + _ _ _ + (n _ 1) _ (n _ 1)!; implicit in the work of Sarngadeva.

Indian combinatorics continued to flourish till the 14th century, when the celebrated mathematician Narayana Pandita wrote his comprehensive Ganitakaumudi (“Moonlight of Mathematics") in 1356 CE, placing the earlier work on combinatorics in a general mathematical context. He has in this great work a chapter on magic squares entitled Bhadraganita, where among other things, he constructs a class of 384 pan-diagonal 4 _ 4 magic squares with entries 1; 2; 3; _ _ _ ; 16.

We recall that in a magic square, the numbers in the rows, columns and diagonals sum to the same magic total. In a pan-diagonal magic square, the broken diagonals also yield the same magic total.

Succinctly put, pan-diagonal magic squares have the remarkable property that they can be considered as a magic squares “on the torus". It is of interest to note that Rosser and Walker proved in 1936 (the proof was simplified by Vijayaraghavan in 1941) that there are only 384 pan-diagonal 4 _ 4 magic squares with entries 1; 2; _ _ _ ; 16.

Curiously enough, Ramanujan, in his Notebooks of probably his earliest school days, has the magic square

1 14 11 8

12 7 2 13

6 9 16 3

15 4 5 10

This turns out to be one of the 384 magic squares considered by Narayana Pandita. (Notice that rows, columns, diagonals and broken diagonals add to 34. For example, 5+3+12+14 = 12 + 9 + 5 + 8 = 34).

Xenophanes, the founder of the Eleatic School of Philosophy, had the well known dictum - Ex nihilo nihil fit, “Out of nothing, nothing comes." One wonders whether after all Ramanujan was indeed influenced somewhat by the mathematical tradition of his ancestors.

Ramanujan mathematics centre to be set up in Chennai

The Ramanujan Mathematical Society (RMS) will hold a series of activities in 2012 (National Mathematical Year) to mark the 125th birth anniversary of mathematician Srinivasa Ramanujan.

RMS president and chair of the organising committee M.S. Raghunathan said on Monday a mathematics centre named after Ramanujan would be set up in Chennai. It would have a host of facilities, including a museum.

Efforts were on to bring out the biography of Ramanujan by Robert Kanigel in regional languages. A documentary, tracing the history of mathematics in India, would also be made.

The activities were planned to involve different sections, including school-goers, college students, students pursuing their research in mathematics and others interested in developments in the discipline.

In the programmes involving school students, for instance, talks and activities would focus on helping them get rid of their fear of the subject, Professor Raghunathan said.

The year-long celebrations of Ramanujan birth anniversary would culminate in an international conference of mathematicians in New Delhi in December next, he said.