Julie Rowlett, Chamlers University

Title: When does a domain have a complete set of trigonometric eigenfunctions for
the Laplace eigenvalue equation?

Abstract: This talk is based on joint work with my students, Max Blom, Henrik Nordell,
Oliver Thim, and Jack Vahnberg. In 2008, Brian McCartin proved that the only polygonal
domains in the plane which have a complete set of trigonometric eigenfunctions are:
rectangles, equilateral triangles, hemi-equilateral triangles, and isosceles right triangles.
Trigonometric eigenfunctions are, as the name suggests, functions which can be expressed
as a finite linear combination of sines and cosines. In 1980, Pierre Berard proved that a
certain type of polytopes in n dimensional Euclidean space, known as an alcoves associated
to a root system and its Weyl group, also have a complete set of trigonometric eigenfunctions.
In our work, we connect these results with the notion of `strictly tessellating polytope.
We prove that the following are equivalent: (1) a polytope in Rn has a complete set of
trigonometric eigenfunctions (2) a polytope strictly tessellates Rn (3) a polytope is an
alcove associated to a root system and its Weyl group.

This talk is intended for a general mathematical audience including students! No experience
with any of these mathematical concepts is required, as everything will be explained so that
everyone can follow along.