For more information about this meeting, contact Ae Ja Yee, Robert Vaughan, Mihran Papikian.

Title: | Period functions for quantum modular forms |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Trung Hieu Ngo, University of Michigan |

Abstract: |

A quantum modular form is a function on the rational numbers which is not necessarily modular but whose failure to be modular gives rise to an interesting analytic function. There are two important places where one finds a quantum modular form. First, it arises from a certain transform of the period function of a Maass form (thanks to Ramanujan, Andrews, Dyson, Hickerson, Cohen, Bruggeman, Lewis, Zagier,...). Second, it arises from the radial limit of both a mock theta function and a partial theta function (thanks to Rademacher, Ramanujan, Rhoades, Zagier, Zwegers,...). It is a conjecture of Dyson that the two instances belong to a unified framework.
With Yingkun Li and Robert C. Rhoades, we discover and study a q-hypergeometric series construction, called renormalization, which tie together a duality in each of the two instances. We will describe a new application of renormalization results and quantum modular forms to counting integer partitions with prescribed properties. This is achieved via certain circle method technique, executed by Hardy-Ramanujan and Wright amongst others.
The talk includes a motivated introduction to quantum modular form (with more emphasis on the perspective of Bruggeman-Lewis-Zagier). No background knowledge is assumed and graduate students are welcome. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 02 / 13 / 2014 |

Time: | 11:15am - 12:05pm |