While on his death bed, the brilliant Indian mathematician Srinivasa Ramanujan cryptically wrote down functions he said came to him in dreams, with a hunch about how they behaved. Ramanujan is one of the outstanding math minds of history who contributed to the theory of number, of partitions and the theory of continued fractions. It was on his deathbed in 1920 that Ramanujan described mysterious functions that mimicked theta functions, or modular forms, in a letter to the great English pure mathematician G. H. Hardy. Ramanujan died before he could prove his hunch.
But more than 90 years later, Emory University mathematician Ken Ono and his team proved that these functions indeed mimicked modular forms, but don’t share their defining characteristics, such as super-symmetry. Ono commented “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,”
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. Ono has commented “Ramanujan’s legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died.”
There are many anecdotal reports on the importance of visual thinking by scientists. Perhaps the best known visual discoveries are the benzene ring and the helical structure of DNA. Even more vivid was Albert Einstein’s explanation how human reasoning includes visual thinking.
- “The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined. …. This combinatory play seems to be the essential feature in productive thought before there is any connection with logical construction in words or other kinds of signs which can be communicated to others”.
- Albert Einstein in a letter to Jacques Hadamard.
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. Like trigonometric functions such as sine and cosine, theta functions have a repeating pattern, but the pattern 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’s story is one of the great romantic tales of mathematics. It is an account of triumph and tragedy, of a man of genius who prevailed against incredible adversity and whose life was cut short at the height of his powers. Ramanujan, a self-taught mathematician born in a rural village in South India, spent so much time thinking about math that he flunked out of college in India twice.
Educated at the Government College at Kumbakonam, he obtained a scholarship for the University of Madras, but after his marriage in 1909, became a clerk in the Madras Port trust. But he 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. He left India by boat 3/17/1913 and arrived in London 4/14/1913 in spite of great difficulties caused by his caste as a Braman. His diet and religious observations were compromised by the great difference in life style to which he had to adapt.While there, Ramanujan published more than 30 papers and was inducted into the Royal Society.
Hardy helped him get into his original work, but the isolation of his life and difficulties in diet caused him great stress. He became ill in 1917, and in January 1918, tried to kill himself by throwing himself in front of a train. Rescued, he struggled on until he was able to return to India in 1919. Ramanujan’s notebooks were published in 1957, and his “Collected Papers” published in 1927 and 1962. He died 4/26/1920, Kumbakonam.
So fasten your shoulder straps for an Indian Adventure with this 1987 documentary for the Channel 4 ‘Equinox’ science series. It is the extraordinary story of how in 1914 the self-taught maths genius Srinivasa Ramanujan was brought from Madras to Trinity College, Cambridge, by GH Hardy, who called their relationship ‘the one truly romantic episode of my life’.