Abstract

A lattice in R^n is called well-rounded, abbreviated WR, if it contains n linearly
independent shortest vectors. Such lattices have many symmetries and play a central role
in discrete optimization, extremal lattice theory, and related number theoretic problems.
WR lattices corresponding to integral quadratic forms are of specific interest in the
arithmetic context. In this talk, I will discuss some results on distribution properties
of integral WR lattices, mostly concentrating on the planar case. In particular, I will
present some counting estimates on the number of WR sublattices of planar lattices. I will
also talk about the special class of integral WR lattices, coming from ideals in number
fields, giving a partial characterization of number fields for which such lattices exist.

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