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.