Equidistribution in Arithmetic: Dynamics, Geometry and Spectra

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A central problem in Number Theory is finding integer solutions to polynomial equations. Some systems of equations have plenty of integer solutions. This research concerns the distribution of the integer solutions inside the continuum of real solutions. A wide-range of conjectures imply that in many cases the distribution of integer solutions mimics a randomly generated set in the ambient space. Systems of equation that have a large enough group of symmetries are called homogeneous and they play a central role in this project. These symmetries can be used to relate different integer solutions to each other and facilitate the introduction of methods from the theory of dynamical systems to this research area. The dynamical methods have been immensely successful in solving long standing problems in number theory. The major objectives of this project are to make progress on long-standing problems about equidistribution in number theory, to expose undergraduate students to number theory and prepare them to graduate studies, and to prepare graduate students to conduct research at the intersection of number theory and dynamics.The focus of this project is the chaotic behavior of periodic subgroup orbits on homogeneous spaces, and the randomness exhibited in the asymptotic behavior of automorphic forms. Early breakthroughs include Linnik's results about the equidistribution of integral points on the sphere and Margulis's solution of the Oppenheim conjecture regarding the values attained by an irrational quadratic form at integer points. To make progress on these topics, this project will fuse and expand results from homogeneous dynamics and analytic number theory. Specifically, the proposed research relies on measure rigidity techniques from homogeneous dynamics, the study of relative trace formulae and the theta correspondence from the theory of automorphic forms, and sieve methods from multiplicative number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.