University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Effective Ratner equidistribution for SL$(2,mathbb R)ltimes(mathbb R^2)^{oplus k}$ and applications to quadratic forms

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

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Interactions between Dynamics of Group Actions and Number Theory

Let $G= ext{SL}(2,mathbb R)ltimes(mathbb R2){oplus k}$ and let $Gamma$ be a congruence subgroup of SL$(2,mathbb Z)ltimes(mathbb Z2){oplus k}$. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the homogeneous space $Gammaackslash G$. 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.)

This talk is part of the Isaac Newton Institute Seminar Series series.

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