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The mathematical philosopher

Mathematics is the language with which we can understand the working of the universe, and its underlying philosophy and Dr Sujatha Ramdorai works on one such dialect called the algebraic number theory.
The dialect helps study the arithmetic of algebraic number fields through the confluence of the abstraction of algebra and the structure rendered by the number theory to find answers to the mysteries of the numbers as we know them today and is increasingly unfurling new and diverse areas on applications like cryptography used in code encryptions.

Her specific area of work is to explore ways to understand and expand the theory, which has provided tools and techniques to understand and attack the Birch and Swinnerton–Dyer (BSD) conjecture, which is one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute attracting a cash prize of 1 million dollars for the first correct proof.

In mathematics, a conjecture is a proposition that appears to be correct on the basis of incomplete information and for which no proof has been found as yet. It is an assertion that is made with ample numerical evidence to support that assertion, but for which a complete general proof is missing. Once a complete general proof is made available, the conjecture becomes, in mathematical parlance, a theorem.

The Birch and Swinnerton–Dyer conjecture deals with a certain type of equation, defining elliptic curves (the trajectory of the planets movement.) over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions.

It is widely recognized as one of the most challenging mathematical problems and has been proved in a number of cases. The study of elliptic curves in itself has found applications in a range of topics; from Cryptography and encryptions, top string theory, Iwasawa theory is a crucial tool to understand why this conjecture is true. BSD conjecture, in very simplistic terms, predicts that two invariants associated to an elliptic curve, which a priori seem totally unrelated, are in fact closely related, and helps one conclude when the set of solutions to the elliptic curve is finite or infinite.

This theory, named after a leading, influential Japanese mathematician of the last century, is the only known theory that provides tools and techniques to systematically attack this conjecture, as well as other generalisations of it. As a bonus, it also gives a philosophical basis for why one should expect these two invariants to be actually related. The solution of the conjecture that scientists are pursuing is to prove that these two seemingly unrelated invariants are actually related.

For half of the last century Iwasawa theory has been studied in the commutative (directional) context. Dr Ramdorai is studying it in the non-commutative (non-directional) context with hopes to gain better insights to the conjecture.

Though the carrot of the one million dollar prize is attractive enough and added to it is the multifarious applications of the elliptical curve, Dr Ramdorai insists that mathematicians study their subject without any specific aim or benefit in mind. “Pure mathematicians study their subject just because of its intrinsic beauty”.

She is entirely fascinated by elliptical curves, which she says has an enormously rich structure, multilayered, with connections to complex geometry, topology and number theory. What she finds centrally attractive is not their recent application to cryptography, but its richness in terms of the mysteries.

Her approach to mathematics is philosophical. “How can one convey the purity of structure and the accompanying beauty that one encounters as mathematical problems yield themselves to solutions?” she quotes Bertrand Russell in a book ‘Lilavati’s Daughters brought out by the Indian Academy of Sciences and indeed she can talk untiringly about it.

“Mathematics, rightly viewed, possesses not only truth, but also supreme beauty – a beauty cold and austere, like that of sculpture,” she added quoting Russell.

“While the physicist is trying to understand what is behind the closed door, we mathematicians usually know what is behind the closed door and are trying to prove that their guess is right for the physicist to verify when she/he opens the door. We strive to theoretically and empirically prove why our insights or guesses are correct,” she says and she enjoys the churning of her intellect as she waits for that proof to click.

Dr Ramdorai has been deeply inspired by her grandmother who always regretted that she did not have the opportunity to ‘complete her education’ and thereby kept the thirst for knowledge burning in her grandchildren. This went a long way to help her win laurels like the Shanti Swarup Bhatnagar award, ICTP Ramanujan Award, become the Managing Editor of the International Journal of Number Theory (IJNT), Editor of the Journal of Ramanujan Mathematical Society (JRMS), member of the National Knowledge Commission, 2007 2009 and member, Prime Minister’s Scientific Advisory Council (PM_SAC), Government of India, 2009 onwards.

She strongly believes that ‘there is a whole new world in science waiting to be discovered and claimed by women who can play a crucial role in understanding the philosophy and mysteries of the universe.

By Archita Bhatta written in discussion with Dr Sujatha Ramdorai