The 90years old Mystery of Ramanujan's last work - Unlocked

The Discovery of Mathematical Beauty " Mock Theta Function" on the Death bed.

On 22nd December, the world celebrated the 125th birth anniversary of Srinivasa Ramanujan the man known for natural genius how hailed as an all-time great mathematician, like Euler, Gauss or Jacobi. In his honour, the Indian government declared 2012 as the Year of Mathematics. I pay my tribute to the great mathematical genius who unraveled the many dimensions of Mathematics by leaving behind 4000 original theorems, despite his lack of formal education and a short life-span.– truth, beauty, elegance, abstractness, certainty, fun.

While on his death bed, the brilliant Indian mathematician SrinivasaRamanujan cryptically wrote down functions he said came to him in dreams, with a hunch about how they behaved. Now 100 years later, researchers say they've proved he was right.

"We've solved the problems from his last mysterious letters. For people who work in this area of math, the problem has been open for 90 years," Emory University mathematician Ken Ono said.

Ramanujan, a self-taught mathematician who was born in a rural village in South India and had no formal training in pure mathematics, but from the age of 10, he started mastering the works of other mathematicians, such as S.L. Loney’s trigonometry, and the works of Bernoulli and Euler, to whom the Cambridge mathematician and Ramanujan’s mentor-collaborator G.H. Hardy compared him with. Spent so much time thinking about math that he flunked out of college in India twice.

In his formative years, after having failed in his F.A. (First examination in Arts) class at College, during this period (1903-1914), he kept a record of the final results of his original research work in the form of entries in two large-sized Note Books.

Ramanujan then met deputy collector V. Ramaswamy Iyer, who had recently founded the Indian Mathematical Society, and asked him for a job as a clerk in the revenue department. Iyer, instead, gave him letters of recommendation to meet other mathematicians in Madras. Iyer later said, “I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.

Eventually, through his mathematician friends, both Indian and British, Ramanujan sent mathematicians letters describing his work, and one of the most preeminent ones, English mathematician G. H. Hardy, recognized the Indian boy's genius and invited him to Cambridge University in England to study. While there, Ramanujan published more than 30 papers and was inducted into the Royal Society.

For a brief window of time, five years, he lit the world of math on fire, which unfortunately overlapped with the first World War years, he published 21 papers, five of which were in collaboration with Prof. G.H. Hardy and these as well as his earlier publications before he set sail to England are all contained in the ``Collected Papers of Srinivasa Ramanujan''.

But the cold weather eventually weakened Ramanujan's health, and when he was dying, he went home to India.

Ramanujan developed much of his mathematics in isolation while still in India, and his works mostly only the results, because he never wrote down the proofs to his conjectures on paper, but is said to have worked on them with chalk and slate because paper was expensive are contained in three  notebooks, totalling about 640 pages in all. A fourth notebook of 87 unorganised pages, called the ‘Lost Notebook’ was rediscovered in 1976. It is important to note that though Ramanujan took his ``Note Books'' with him he had no time to delve deep into them. The 600 formulae he jotted down on loose sheets of paper during the one year he was in India, after his meritorious stay at Cambridge, are the contents of the `Lost' Note Book found by Andrews in 1976.

His works have inspired many, including Prof. Hardy himself, to explore and create new branches of mathematics.

It was in Jan 1920, four months before his demise, that he described mysterious functions that mimicked theta functions, or modular forms, in a letter to Hardy. The letter contained no news about his declining health but only information about his latest work : ``I discovered very interesting functions recently which I call `Mock' theta-functions. Unlike the `False' theta-functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as ordinary theta-functions. I am sending you with this letter some examples ... ''. Try to imagine the quality of Ramanujan's mind, one which drove him to work unceasingly while deathly ill, and one great enough to grow deeper while his body became weaker. I stand in awe of his accomplishments; understanding is beyond me.

As per the explanation provided by Emory University mathematician Ken Ono, Unlike other trigonometric functions such as sine and cosine which are having a regular repeating pattern, the repeating pattern of theta function is much more complex and subtle than a simple sine curve. Theta functions are also "super-symmetric," meaning that if a specific type of mathematical function called a Moebius transformation is applied to the functions, they turn into themselves. Because they are so symmetric these theta functions are useful in many types of mathematics and physics, including string theory.

Ramanujan defined 17 Jacobi theta function-like functions F(q) with lql<1 which he called "mock theta functions" that looked like theta functions when written out as an infinite sum (their coefficients get large in the same way), but weren't super-symmetric. Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.

These functions are q-series with exponential singularities such that the arguments terminate for some power  tn. In particular, if f(q)  is not a Jacobi theta function, then it is a mock theta function if, for each root of unity , there is an approximation of the form.

                                                                                                               

as t-0+ with q=pe-t  (Gordon and McIntosh 2000).

If, in addition, for every root of unity p there are modular forms hj(p)(q)and real numbers such that

is bounded as  q radially approaches p, then f(q) is said to be a strong mock theta function (Gordon and McIntosh 2000).

Ramanujan found an additional three mock theta functions in his "lost notebook", but he died before he could prove his hunch. But more than 90 years later, Ono and his team proved that these functions indeed mimicked modular forms, but don't share their defining characteristics, such as super-symmetry. The expansion of mock modular forms helps physicists compute the entropy, or level of disorder, of black holes.

In developing mock modular forms, Ramanujan was decades ahead of his time, Ono said; mathematicians only figured out which branch of math these equations belonged to in 2002.

The complete list of mock theta functions of order 3 are

 "Ramanujan's legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died," Ono said.

The findings were presented last month at the Ramanujan 125 conference at the University of Florida, ahead of the 125th anniversary of the mathematician's birth on Dec. 22.

Article: The 90years old Mystery of Ramanujan's last work - Unlocked by Rahul Kumar.

Major Source of article: http://www.livescience.com/25597-ramanujans-math-theories-proved.html