CAPA: A negative result in the spectral determination of triangles

  • Date: –16:15
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64121
  • Lecturer: Gerard Orriols-Gimenez
  • Organiser: Matematiska institutionen
  • Contact person: Rodrigo Schaefer
  • Seminarium

Since Mark Kac asked the question of "hearing the shape of a drum", the problem of spectral determination of domains has become an important topic in geometric analysis. However, very little is known in general, and most results are negative and use the entire spectrum of the Laplacian. It is thus natural to restrict our attention to finite-dimensional moduli spaces, like those of polygons, and a finite number of eigenvalues. In this context, we produce a computer-assisted proof of the fact, supported by the numerical evidence of Antunes and Freitas, that the first, second and fourth Dirichlet eigenvalues of a triangle do not determine it in the class of triangles. We will explain the ingredients of the proof, which use the novel "lightning Laplace solver" technique of Gopal and Trefethen and recent explicit bounds to control the error of the eigenvalues and their position in the spectrum.