Special Issue "Geometry of Numbers"
A special issue of Mathematics (ISSN 2227-7390).
Deadline for manuscript submissions: closed (30 June 2017).
Interests: integral and orthogonal representations of finite groups; lattices, spherical designs and modular forms, and their analogues in coding theory
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
Manuscript Submission Information
Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.
Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.
Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.
- 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