Srinivasa Ramanujan · Mathematics
Ramanujan’s final gift - the mock theta functions of his last letter to Hardy in 1920, understood only in the twenty-first century through Zwegers’s work of 2002 - together with his death at thirty-two and the mathematics, and physics, that his intuition is still generating.
In early 1919 Ramanujan, gravely ill after five years in England, returned home to India. He continued to work from his sickbed, and on 12 January 1920, about three months before his death, he wrote what would be his last letter to Hardy. In it he announced a discovery: a new class of functions he called mock theta functions. Theta functions were among the most important and beautiful objects in classical mathematics, and Ramanujan had found functions that behave almost like them, sharing many of their striking properties, yet subtly refusing to be genuine theta functions at all. He listed seventeen examples, grouped them by an ‘order,’ and gave tantalising properties, but, characteristically, no definition precise enough to say exactly what a mock theta function was. Then he died, at thirty-two, leaving the mathematical world a final riddle it would take eighty years to read.
For most of the twentieth century, mock theta functions remained a beautiful mystery. Mathematicians such as G. N. Watson kept the subject alive and proved many of Ramanujan’s specific assertions, but the fundamental question - what is a mock theta function, really? - stayed open. The breakthrough came in 2002, when a young Dutch mathematician, Sander Zwegers, in his doctoral thesis, finally revealed the hidden structure. Zwegers showed that Ramanujan’s mock theta functions are pieces of a larger, previously unrecognised kind of object: they can be ‘completed,’ by adding a specific non-holomorphic correction term, into genuine modular forms of a new type now called harmonic Maass forms. In other words, the mock theta functions were the visible fragments of a structure that classical theory had no name for, and Ramanujan had glimpsed those fragments eight decades before the structure itself was understood. His last letter had been a message from the mathematics of the future.
Ramanujan died on 26 April 1920, at the age of thirty-two, in Kumbakonam. In his few working years he had been elected a Fellow of the Royal Society and a Fellow of Trinity College, the first Indian to receive either honour, and had produced a body of work whose depth is still being measured. He was a devout man who attributed his insights to the goddess Namagiri and who, his associate and biographer S. R. Ranganathan reports, once told him that an equation meant nothing to him unless it expressed a thought of God. Whatever one makes of the theology, it names something true about his mathematics: he pursued it as a search for beauty and hidden order rather than for use or fame. The mock theta functions are the perfect emblem of his life - offered in his last months, more beautiful than anyone could yet understand, and pointing, as it turned out, far beyond the horizon of his time.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that in his last letter (1920) Ramanujan described mysterious ‘mock theta functions’ that no one understood until Zwegers in 2002 placed them within modular-forms theory, and that he died at thirty-two with his mathematics still unfolding. Explain why this final idea was so far ahead of its time.
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