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Distribution of values of linear maps on quadratic surfaces.

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Interactions between Dynamics of Group Actions and Number Theory

We discuss the distribution of values of linear maps on quadratic surfaces. The problem can be set up in the framework of unipotent dynamics and then Ratner’s Orbit Closure Theorem can be used to establish conditions sufficient to ensure that the set of values is dense. We also indicate how a quantitative version of this result follows from equidistribution properties of unipotent flows on homogeneous spaces. A crucial ingredient in the latter is a non-divergence result for certain spherical averages. In order to establish this result we show how one can use ideas of Y. Benoist and J.F. Quint that were developed in the context of random walks on homogeneous spaces.

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

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