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Description

In this talk, we show that the pair correlation of the Laplacian eigenvalues on 3-dimensional rectangular flat tori follows Poissonian statistics. Similar to the earlier work of Eskin, Margulis, and Mozes on 2D flat tori, these eigenvalues are represented by values of positive definite quadratic forms at integer points. In the 3D case, the problem reduces to a special case of the quantitative Oppenheim conjecture for rapidly shrinking intervals. Our approach reformulates the problem in terms of homogeneous dynamics via theta functions on (SL_2(R)/SL_2(Z))^3, and relies on a sharp quantitative estimate for escape of mass in this space. This is joint work with Jens Marklof and Matthew Welsh.