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

Thursday 26th June 2014 - 13:30 to 14:30
INI Seminar Room 2
Let $G=\text{SL}(2,\mathbb R)\ltimes(\mathbb R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of SL$(2,\mathbb Z)\ltimes(\mathbb Z^2)^{\oplus k}$. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the homogeneous space $\Gamma\backslash G$. The proof involves spectral analysis and use of Weil's bound on Kloosterman sums. I will also discuss applications to effective results for variants of the Oppenheim conjecture on the density of $Q(\mathbb Z^n)$ on the real line, where $Q$ is an irrational indefinite quadratic form. (Based on joint work with Pankaj Vishe.)
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons