Seminar

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 Bérard proved that a certain type of polytopes in n dimensional Euclidean space, known as alcoves, always 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) the first eigenfunction for the Dirichlet Laplacian on a polytope in R^n is trigonometric (2) a polytope strictly tessellates R^n (3) a polytope is an alcove.  Moreover, we prove that if the first eigenfunction of a polytope is trigonometric, then all the eigenfunctions are trigonometric.  There is not really any microlocal analysis in this talk, so people from both programs will be able to follow it.  Why choose a topic with no MLA for the MLA seminar you may ask?  We conjecture:  the only bounded domains in R^n which have a complete set of trigonometric eigenfunctions are strictly tessellating polytopes (=alcoves).  I suspect that one may be able to prove this conjecture by adding microlocal analysis to the mix.  This is the reason I chose this topic because perhaps this talk could lead to a collaboration to prove the conjecture!