Freeman Dyson's challenge for the future: The mock theta-functions

Document Type

Presentation Abstract

Presentation Date

9-17-2007

Abstract

The legend of Ramanujan is one of the most romantic stories in the modern history of mathematics. It is the story of an untrained mathematician, from south India, who brilliantly discovered tantalizing examples of phenomena well before their time. Indeed, the legacy of Ramanujan's work (as a whole) is well documented and includes direct connections to some of the deepest results in modern number theory such as the proof of the Weil Conjectures, and the proof of Fermat's Last Theorem. However, one final problem remained. In his last letter to Hardy (written on his death bed), Ramanujan gave examples of 17 functions he referred to as "mock theta functions". Without a definition and without good clues, number theorists were unable to make any real sense out of these peculiar functions. Nevertheless, these examples make important appearances in many disparate areas of mathematics, a fact which inspired Freeman Dyson to proclaim:

Mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group- theoretical structure... This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions.

--Freeman Dyson, 1987

In this lecture I will outline the history of Ramanujan's final enigma, and I will discuss the solution obtained in joint with Kathrin Bringmann. The result appears in a series of four papers in Inventiones Mathematicae, Annals of Mathematics, Proceedings of the National Academy of Sciences, and the Journal of the American Mathematical Society.

Additional Details

Monday, 17 September 2007
4:10 p.m. in Math 103
3:30 p.m. Refreshments in Math Lounge 109

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