Boris Venkov’s Theory of Lattices and Spherical Designs

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 universities. 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