Abstract : I shall begin with the description of the lattice points counting problem in Euclidean spaces. Namely, establishing an (asymptotic) error estimate for the number of points that the lattice of integral points has in a Euclidean ball of large radius, as the radius goes to infinity.
This problem has a very long history and vast literature is available in obtaining error estimates for Euclidean dilates of the unit ball as well as various other convex bodies.
In the first half of the talk, I shall sketch the Fourier spectral method which originated with Minkowski for Euclidean balls. This classical method does not give the best possible error estimates for the balls, but it gives optimal error in the class of Euclidean dilates of convex bodies with surfaces having non-vanishing Gaussian curvature at all points.
In the last half of the talk, I shall discuss the lattice points counting problem in the context of the Heisenberg groups, for families of balls corresponding to certain radial and homogeneous norms, including the canonical Cygan-Koranyi norm.
In the end, I shall briefly mention the scope of this method discussing more general nilpotent Lie groups.
This talk is based on my joint work with Amos Nevo and Krystal Taylor which is available on arXiv. Most of this talk should be accessible to those who are familiar with basic functional analysis.
School of Mathematical Sciences
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