On Distribution of Integral Well-Rounded Lattices in the Plane

Authors

Lenny Fukshansky, Claremont McKenna CollegeFollow

Document Type

Lecture

Department

Mathematics (CMC)

Publication Date

12-2010

Abstract

Let Λ be a lattice of full rank in the N-dimensional Euclidean space R^N for N ≥ 2. The minimum of Λ is defined as |Λ| := min{║x║ : x ∈ Λ \ {0}}, where ║║ stands for the usual Euclidean norm on R^N, and the set of minimal vectors of Λ is defined to be S(Λ) := {x 2 Λ : ║x║ = |Λ|}. The lattice Λ is called well-rounded (abbreviated WR) if the set S(Λ) spans R^N. WR lattices are important in discrete optimization, in particular in the investigation of sphere packing, sphere covering, and kissing number problems. Moreover, certain questions about distribution of WR lattices, along with the covering conjecture of Woods for WR lattices, came up recently in McMullen’s celebrated work on Minkowski’s conjecture. In this talk, we will discuss some results on distribution of integral WR lattices in the plane.

Rights Information

© 2010 Lenny Fukshansky

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Recommended Citation

Fukshansky, Lenny. "On Distribution of Integral Well-Rounded Lattices in the Plane." Special Session: Arithmetic of quadratic forms and integral lattices at the First Joint Meeting of AMS and Sociedad Mathematica de Chile, Pucon, Chile. 15-18 December 2010.