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Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sphere packing in a given dimension is a fascinating and difficult problem: the answer is known only up to dimension 3.
This book thus discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory, centering on the study of extreme lattices, i.e. those on which the density attains a local maximum, and on the so-called perfection property.
Written by a leader in the field, it is closely related to, though disjoint in content from, the classic book by J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, published in the same series as vol. 290.
Every chapter except the first and the last contains numerous exercises. For simplicity those chapters involving heavy computational methods contain only few exercises. It includes appendices on Semi-Simple Algebras and Quaternions and Strongly Perfect Lattices.

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1 General Properties of Lattices.- 2 Geometric Inequalities.- 3 Perfection and Eutaxy.- 4 Root Lattices.- 5 Lattices Related to Root Lattices.- 6 Low-Dimensional Perfect Lattices.- 7 The Voronoi Algorithm.- 8 Hermitian Lattices.- 9 The Configurations of Minimal Vectors.- 10 Extremal Properties of Families of Lattices.- 11 Group Actions.- 12 Cross-Sections.- 13 Extensions of the Voronoi Algorithm.- 14 Numerical Data.- 15 Appendix 1: Semi-Simple Algebras and Quaternions.- 16 Appendix 2: Strongly Perfect Lattices.- References.- List of Symbols.

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From the reviews:

"It is worth saying at the outset that Perfect lattices in Euclidean spaces is a state-of-the-art research monograph (with exercises) by one of the leading experts in this rapidly developing field ... . Martinet´s book appears in the same Springer series as Conway and Sloane´s epochal Sphere packings, lattices and groups and it will be similarly appreciated by researchers in this area as a carefully written, historically aware and authoritative companion volume focusing on local methods in lattice theory." (Nick Lord, The Mathematical Gazette, Vol. 88 (512), 2004)

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