##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Ken Ono

#### University of Wisconsin, Madison

## Freeman Dyson's challenge for the future: The mock $\tensy \theta$ functions

##### Abstract:

Ramanujan's legacy to mathematics is well documented with connections to some of the deepest subjects in modern number theory: Deligne's proof of the Weil Conjectures, the proof of Fermat's Last Theorem, the birth of probabilistic number theory, the introduction of the "circle method" among others. Although most of Ramanujan's mathematics is now well understood, one baffling enigma remained. In his last letter to Hardy (written on his death bed), Ramanujan gave 17 examples of functions he referred to as "mock $\theta$ functions". Over the next fifty years, ad hoc works by many number theorists (such as Andrews, Atkin, Cohen, Dyson, Selberg, Swinnerton-Dyer, Watson...) clearly pointed to the importance of these strange functions. Their work motivated 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." -- Freeman Dyson, 1987 Over the last year, Kathrin Bringmann and I have written a series of three papers on this enigma. Extending recent work of Zwegers, we solve Dyson's challenge in terms of harmonic Maass forms. We fully developed the arithmetic (i.e. p-adic and Galois theoretic) and analytic properties of all such forms, and we have applied these results to solve open problems in additive number theory.

Host: Harold Stark

### December 5, 2006

### 3:00 PM

### AP&M 6402

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