# Hence proved, 92 years on

Jan 06, 2013, 09:37 IST | Phorum Dalal

In 1920, mathematician Srinivasa Ramanujan lay on his deathbed and penned a vague idea about a complex mathematical function and sent it to his mentor and English mathematician, GH Hardy. Ramanujan died three months later and what followed was almost a century of attempts to make sense of the letter’s contents.

On December 22, 2012, which marked Ramanujan’s 125th birth anniversary, mathematician Ken Ono of Emory University, Atlanta, Georgia, backed Ramanujan’s conjecture with proof after working on it for 10 years.

US-based mathematician Ken Ono, who solved Srinivasa Ramanujan’s 1920 conjecture. Pic Courtesy/Ken Ono

Ono is the Asa Griggs Candler professor of Mathematics and Computer Science at Emory University. He is also an editor of 10 mathematics journals. Most notably, Ono is managing editor of the proceedings of the American Mathematical society, and is also on the US National Committee of Mathematics at the US National Academy of Sciences.

Please elaborate on Srinivasa Ramanujan’s mathematical problem, which you have solved. The mock theta functions of Srinivasa Ramanujan’s last letter are about two dozens of bizarre power series which he recorded in his last letter to GH Hardy. Written from his deathbed, Ramanujan conjectured a strange radial property for these functions, but he couldn’t back it with proof.

What made you take up this problem?
Ramanujan’s last letter is one of the deepest mysteries and so it has always been one of the top problems to solve on any mathematician’s list. And it sure feels great to have done it.

Indian mathematician Srinivasa Ramanujan. Pic Courtesy/Oberwolfach Photo Collection/wikipedia

It took you 10 years to prove the property of the functions. What were the challenges you faced during the process?
The biggest problem was that Ramanujan did not give a definition of what a mock theta function should be. He offered 17 examples, which he could not prove. Ramanujan’s letter was vague and short. Mathematicians could not ask him questions about his ideas because he died three months after writing the letter. His secret died with him.  For many years, the functions lay around like orphans, and mathematicians across the world could not figure where they fit into mathematics.  In 2002, Dutch mathematician Sander Zwegers finally understood that these functions are pieces (technically known as holomorphic projections of mass forms, a very exotic type of function) of mass forms. However, this understanding was not enough to answer the questions in Ramanujan’s last letter.  We didn’t know how to formulate the problem before this understanding came through. In a way, this made it possible for us to attack it.

Tell us about the film which is being made on the same topic.
Yes, Nandan Kudhyadi is directing a docu-drama, which will release in March 2013. I have a speaking role in it and am also one of the scientific consultants. The film is about the legacy of Ramanujan, his life, his mathematics and most importantly his legacy to the world — both in terms of science as well as culture.

Could you tell us something about Ramanujan’s letters to GH Hardy?
Ramanujan wrote several letters to Hardy. However, his first and last letters are by far the most famous. Hardy recognised Ramanujan’s genius because of the first letter, and Ramanujan’s last letter has been an enigma for many decades. The last letter was written from his deathbed — the one under discussion — and mathematicians have been trying to figure it out since 1920.  Ramanujan’s letters were mostly about mathematics — rather, his mathematics — and are scattered. Some are in Chennai and others are in Cambridge.

Tell us something about other similar conjectures in maths that have taken years to be solved?
Many old problems in maths still remain unsolved. There are problems which are centuries old, and, perhaps, the most famous is the Riemann Hypothesis, a conjecture proposed by proposed by mathematician Bernhard Riemann in 1859. Fermat’s last theorem was open for 350 years before it was solved in the 1990s by Andrew Wiles.

In what fields will the solution to Ramanujan’s problem help?
The mathematics of Ramanujan’s last letter has already been applied to physics, polymer chemistry, probability, number theory, algebraic geometry and representation theory since 2002.

What aspects of maths or physics will the present discovery help understand better?
The discovery is likely to help scientists understand the black hole — a region of space having a gravitational field so intense that no matter or radiation can escape — better.

Who worked with you on this problem?
My former PhD students Amanda Folsom — now a faculty at Yale - and Rob Rhoades — now a faculty at Stanford — worked with me on this problem.

Was this a funded project?
Yes, the US National Science Foundation and the Asa Griggs Candler Fund funded this project.

The discovery
Ramanujan functions stated that their outputs would be very similar to those of modular forms when computed for the roots of 1, such as the square root -1. Ten years ago, mathematicians formally defined this other set of functions, now called mock modular forms. But still no one deciphered what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1. Ken One’s team computed one of Ramanujan’s mock modular forms for values very close to -1. They discovered that the outputs rapidly balloon to vast, 100-digit negative numbers, while the corresponding modular form balloons in the positive direction. Ono’s team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. In other words, the difference in the value of the two functions, ignoring their signs, is tiny when computed for -1.The result confirms Ramanujan’s incredible intuition

What are functions?
Functions are equations that can be drawn as graphs on an axis, like a sine wave, and produce an output when computed for any chosen input or value

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