Contemporary Mathematics

Volume 587, 2013

http://dx.doi.org/10.1090/conm/587/11672

Boris Venkov’s Theory of Lattices and Spherical Designs

Gabriele Nebe

1. Introduction

Boris Venkov passed away on November 10, 2011, just 5 days before his 77th

birthday. His death overshadowed the conference “Diophantine methods, lattices,

and arithmetic theory of quadratic forms” November 13-18, 2011, at the BIRS

in Banff (Canada), where his important contributions to the theory of lattices,

modular forms and spherical designs played a central role. This article gives a

short survey of the mathematical work of Boris Venkov in this direction.

Boris Venkov’s first work on lattices was a new proof [20] of the classification of

even unimodular lattices in dimension 24, in 1978, which was reprinted as Chapter

18 of the book “Sphere packings, lattices and groups” [24]. This was the first

application of the theory of spherical designs to lattices shortly after their definition

in the fundamental work [25] by Delsarte, Goethals, and Seidel. In the same spirit a

combination of the theory of spherical designs with modular forms allowed Venkov

to prove that all layers of extremal even unimodular lattices form spherical designs

of strength 11, respectively 7, if n ≡ 0, 8 (mod 24). Since then, lattices became an

important tool for the construction and investigation of spherical designs (see for

instance [6] or Section 5). Boris Venkov’s work on the connection between lattices

and spherical designs finally led him to the definition of strongly perfect lattices.

His lecture series in Bordeaux and Aachen on this topic (see [8]) initiated many

fruitful applications of this theory, some of which are collected in [30]. Strongly

perfect lattices provide interesting examples of locally densest lattices, so called

extreme lattices. The definition of strongly perfect lattices was very successful, for

instance it allows us to apply the theory of modular forms to show that all extremal

even unimodular lattices of dimension 32 are extreme lattices. It also permits us

to apply representation theory of the automorphism group to show that a lattice

is extreme. The notion of strong perfection has been generalized to other metric

spaces such as Grassmanians or Hermitian spaces and also to coding theory.

Boris Venkov spent a great part of his mathematical life visiting other univer-

sities. When I asked him whether he has a complete list of his visits for proposing

him for his Humboldt Research Award in 2007, he answered “A complete list of

my visits would be too long. It contains also exotic visits like Tata Institute in

Bombay, Universidad Autonoma in Mexico or Universidad de Habana, Kuba.” He

1991 Mathematics Subject Classification. Primary 01A70; Secondary 00-02, 11H06, 11H50,

11H56, 11H71, 11F11, 11F46.

c 2013 American Mathematical Society

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