# Julie Rowlett: Trigonometric eigenfunctions, strictly tessellating polytopes, and crystallographic groups

Time: Wed 2021-12-01 13.15 - 14.15

Lecturer: Julie Rowlett (Chalmers)

### Abstract

This talk is based on joint work with my students, Max Blom, Henrik Nordell, Oliver Thim, and Jack Vahnberg. Trigonometric eigenfunctions are, as the name suggests, functions which can be expressed as a finite linear combination of sines and cosines. In 1980, Pierre Bérard proved that a certain type of polytopes in n-dimensional Euclidean space, known as an alcoves, have trigonometric eigenfunctions for the Laplace eigenvalue equation with the Dirichlet boundary condition. In 2008, McCartin proved that in two dimensions, all such alcoves are of four types: rectangles, equilateral triangles, 30-60-90 triangles, and 45-45-90 triangles. In our work, we connect these results with the notion of ‘strictly tessellating polytope.’ We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic on \(\mathbb{R}^n\); the polytope strictly tessellates \(\mathbb{R}^n\); the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the Laplace eigenfunctions are trigonometric functions.

This talk is intended for a general mathematical audience including students!

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Last changed: Nov 28, 2021