Crystallographic Groups, Strictly Tessellating Polytopes, and Analytic Eigenfunctions
Artikel i vetenskaplig tidskrift, 2021

The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Berard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of Berard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. To conclude, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.

Primary 20H15


Secondary 20F55



Julie Rowlett

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Göteborgs universitet

Max Blom

Student vid Chalmers

Göteborgs universitet

Henrik Nordell

Student vid Chalmers

Göteborgs universitet

Oliver Thim

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Algebra och geometri

Jack Vahnberg

Student vid Chalmers

Göteborgs universitet

American Mathematical Monthly

0002-9890 (ISSN)

Vol. 128 5 387-406

Representationer av Liegrupper. Harmonisk och komplex analys på symmetriska och lokalt symmetriska rum

Vetenskapsrådet (VR) (2018-03402), 2019-01-01 -- 2022-12-31.


Algebra och logik


Matematisk analys



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