Special Issue "Geometry of Numbers"
A special issue of Mathematics (ISSN 2227-7390).
Deadline for manuscript submissions: closed (30 June 2017).
Geometry of Numbers is a famous and classical area of mathematics founded by Hermann Minkowski to apply the theory of lattices in Euclidean space to important problems in algebraic number theory. Even in this classical orientation it is still a vivant area, as we see from the recent achievements by Curtis T. McMullen on the Minkowski conjecture and the work on Euclidean minima of number fields initiated by Eva Bayer-Fluckiger. But the topic has substantially grown, and the aim of the present Special Issue is to collect original research articles, as well as a few high level survey articles focusing on connections between lattices and number theory. Among the important areas are, from my personal perspective: Arakelov geometry, diophantine approximation, K-theory and the cohomology of arithmetic groups, and of course the classical topics such as lattices and modular forms.
Already in the actual theory of lattices in Euclidean spaces, there are quite a few very remarkable recent results, such as the proof that the maximum density of a sphere packing in dimension 8 resp. 24 is realized by the E8-lattice, respectively the Leech lattice by Viazovska et al., and the discovery of new extremal even unimodular lattices in dimension 48 and 72. Also the counter-example to Woods conjecture given by Regev, Shapira, and Weiss and showing that McMullen’s approach to prove Minkowski’s conjecture fails in higher dimensions, will certainly stimulate the research in this area.
- 11E12 Quadratic forms over global rings and fields
- 11F11 Modular forms, one variable
- 11F75 Cohomology of arithmetic groups
- 11HXX Geometry of Numbers
- 13F07 Euclidean rings and generalizations
- 14G40 Arithmetic varieties and schemes; Arakelov theory
- 20C10 Integral representations of finite groups
Prof. Dr. Gabriele Nebe
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- Lattices in Euclidean spaces
- Sphere packing problem
- Sphere covering problem
- Automorphism groups of lattices
- Connections to modular forms
- Lattices with algebraic structure
- Application of lattices in number theory
- Arithmetic groups and Cohomology
- Arakelov geometry
- Hyperbolic lattices