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# Discontinuity property of a certain Habiro series at roots of unity

## May 30 at 9:00 – 10:00 CEST

**International Seminar on Automorphic Forms**

Toshiki Matsusaka (Kyushu University)

The object of this talk is a family of q-series originating from Habiro’s work on the Witten-Reshetikhin-Turaev invariants. The q-series usually make sense only when q is a root of unity, but for some instances, it also determines a holomorphic function on the open unit disc. Such an example is Habiro’s unified WRT invariant H(q) for the Poincaré homology sphere. In 2007, Hikami observed its discontinuity at roots of unity. More precisely, the value of H(ζ) at a root of unity is 1/2 times the limit value of H(q) as q tends towards ζ radially within the unit disc. In this talk, we give an explanation of the appearance of the 1/2 factor and generalize Hikami’s observations by using Bailey’s lemma and the theory of mock/false theta functions.

You can join the Zoom meeting at https://tu-darmstadt.zoom.us/j/68048280736

The password is the first Fourier coefficient of the modular j-function (as digits).