Abstract

Numerical tools play an essential role nowadays in the study of any mathematical phenomenon for which analytic tools are not available. In spectral geometry, the dependence of the eigenvalues of differential operators on the geometry of the object studied is complex and often not explicit. The usage of numerical simulations provides valuable insights regarding research problems.  In this talk I will discuss how numerical simulations regarding the optimization of spectral quantities with respect to the geometry of the domain can influence and guide the theoretical study. In some situations, numerical simulations using guaranteed error bounds can even provide computer assisted proofs of theoretical results. I will present a recent result in this direction, obtained in collaboration with Dorin Bucur, regarding the local minimality of the regular n-gon when minimizing the fundamental eigenvalue of the Dirichlet-Laplace operator on n-gons of fixed area.  One complex aspect regarding numerical shape optimization is the computation of the sensitivity of the objective function with respect to geometric parameters. Spectral quantities can be encoded using neural networks, providing a natural differentiable structure, thus speeding the optimization process. I will present recent preliminary results on this topic obtained in collaboration with Dorin Bucur and Alexis de Villeroché.