Crystallographic Groups, Strictly Tessellating Polytopes, and Analytic Eigenfunctions
Journal article, 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

47A75

Secondary 20F55

51M20

Author

Julie Rowlett

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

Max Blom

Student at Chalmers

University of Gothenburg

Henrik Nordell

Student at Chalmers

University of Gothenburg

Oliver Thim

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

Jack Vahnberg

Student at Chalmers

University of Gothenburg

American Mathematical Monthly

0002-9890 (ISSN)

Vol. 128 5 387-406

Representations of Lie groups. Harmonic and complex analysis on symmetric and locally symmetric spaces

Swedish Research Council (VR) (2018-03402), 2019-01-01 -- 2022-12-31.

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1080/00029890.2021.1890498

More information

Latest update

6/7/2021 1