Hence proved, 92 years on

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 letterrsquos contents On December 22 2012 which marked Ramanujanrsquos 125th birth anniversary mathematician Ken Ono of Emory University Atlanta Georgia backed Ramanujanrsquos conjecture with proof after working on it for 10 years USbased mathematician Ken Ono who solved Srinivasa Ramanujanrsquos 1920 conjecture Pic CourtesyKen 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 Ramanujanrsquos mathematical problem which you have solved The mock theta functions of Srinivasa Ramanujanrsquos 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 couldnrsquot back it with proof What made you take up this problem Ramanujanrsquos last letter is one of the deepest mysteries and so it has always been one of the top problems to solve on any mathematicianrsquos list And it sure feels great to have done it Indian mathematician Srinivasa Ramanujan Pic CourtesyOberwolfach Photo Collectionwikipedia 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 Ramanujanrsquos 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 u00a0For many years the functions lay around like orphans and mathematicians across the world could not figure where they fit into mathematics u00a0In 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 Ramanujanrsquos last letter u00a0We didnrsquot 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 docudrama 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 mdash both in terms of science as well as culture Could you tell us something about Ramanujanrsquos letters to GH Hardy Ramanujan wrote several letters to Hardy However his first and last letters are by far the most famous Hardy recognised Ramanujanrsquos genius because of the first letter and Ramanujanrsquos last letter has been an enigma for many decades The last letter was written from his deathbed mdash the one under discussion mdash and mathematicians have been trying to figure it out since 1920 u00a0Ramanujanrsquos letters were mostly about mathematics mdash rather his mathematics mdash 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 Fermatrsquos last theorem was open for 350 years before it was solved in the 1990s by Andrew Wiles In what fields will the solution to Ramanujanrsquos problem help The mathematics of Ramanujanrsquos 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 mdash a region of space having a gravitational field so intense that no matter or radiation can escape mdash better Who worked with you on this problem My former PhD students Amanda Folsom mdash now a faculty at Yale and Rob Rhoades mdash now a faculty at Stanford mdash 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 projectu00a0 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 Onersquos team computed one of Ramanujanrsquos mock modular forms for values very close to 1 They discovered that the outputs rapidly balloon to vast 100digit negative numbers while the corresponding modular form balloons in the positive direction Onorsquos 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 1The result confirms Ramanujanrsquos 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