Momentous contributions

THIS HAS reference to the interesting article by Prof. Krishnaswami Alladi `Brief Life of a Mathematician' (TheHindu, December 31). There are quite a few knowledgable mathematicians who hold the view that perhaps Ramanujan would have made many more momentous contributions to Mathematics (like Newton or Gauss or Von Neumann) if he had chosen to work in collaboration with the algebraist, Burnside, whose pioneering book is referred to by Alladi.

It is possible that Ramanujan might not have known about Burnside, as the latter was not a university professor, but was an instructor in the Naval Academy of Great Britain.

But at least two of his problems engaged the attention of algebraists for nearly half-a-century. One of them is to get at a purely group-theoretic proof of the theorem that a group of order paqb, where p and q are primes and a and b are natural numbers, is solvable.

Actually Burnside established this result in his book using the representation theory of groups and the theory of group characters that goes with it. But it was Thompson, who, decades later, gave a `character-free' proof, which however is very involved.

In fact Alladi extols Thompson as the greatest group-theorist since Galois. The other of Burnside's problems, which, is popularly known as The Burnside problem is to find an answer to the question `Is every finitely generated group, each of whose elements is of finite order, necessarily finite?' This was settled in the negative by two Russian mathematicians, Golod and Shafarevitch in 1964.

They proved this as a corollary to a more general theorem, another of whose corollaries gives a negative answer, again, to the so-called Kurosh problem, which is, `Is every finitely generated nil algebra nilpotent?'

The last chapter of Herstein's book `Non-Commutative Rings' (which is a Carus Mathematical Monograph) contains a proof of the Golod-Shafarevitch theorem in clear detail. Herstein believes that this theorem deserves closer scrutiny as it can yield further dividends.

It is not idle to speculate that Ramanujan would have made substantial contributions to Algebra if he had known the then current problems. For, even with a cursory knowledge of the theory of functions of a complex variable, culled out perhaps from Carr's `Synopsis', he seemed to have discovered most of the results in Loney's `Trigonometry' Part II and was very deeply dejected when he came to know that all his theorems were known already.

One wonders how profound his contributions would have been, had he been exposed to the theory of the space of linear operators of Banach, who, incidentally, was in the habit of scribbling his thoughts in his inebriation, on the paper towels in a pub of Warsaw and throwing them out in a waste paper basket, which his wife preserved carefully, probably by tipping the bartender.

Again one is left to speculate about the possible startling results Ramanujan would have come out with in the theory of Hilbert spaces. These ideas of Banach and Hilbert were taking shape when Ramanujan was in his Twenties, which, according to Hardy, is the right age of creativity for a mathematician, as pointed out by Alladi at the very beginning of his article.

Alladi has drawn attention to how both Ramanujan and Galois died young and also how their final outbursts of outstanding ideas were communicated in letters, which was, understandably, the only means of communication open to them.

There is however an important difference in their demeanour. Galois' life is an example of the co-existence of genius and stupidity, as indicated by E. T. Bell in his book `Men of Mathematics'.

But we know that Ramanujan was humble and attributed his extraordinary mathematical ability to the grace of the Goddess of Namakkal in Tamil Nadu. In other words, genius and piety went hand in hand in his case.

K. SITARAM

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