Effective Ratner equidistribution result for $\mathrm{SL}(2, \mathbb R)\ltimes \mathbb R^{2k} $ and applications to quadratic forms.

by Prof. Pankaj Vishe (University of York, UK)

Thursday 14 Aug 2014, 16:00 → 17:00 Asia/Kolkata

AG-69 (Colaba Campus)

Let $G=\mathrm{SL}(2,\mathbb R)\ltimes \mathbb R^{2k}$ and let $\Gamma$
be a congruence subgroup of $\mathrm{SL}(2,\mathbb Z)\ltimes\mathbb Z^{2k}$.
We give an effective equidistribution result for a family of 1-dimensional
unipotent orbits in $\Gamma\backslash G$. The proof involves Specral methods
and bounds for exponential sums. We apply this result to obtain an effective
Oppenheim type result for a class of indefinate irrational quadratic forms.
This is based on a joint work with Andreas Strombergsson.