  We announce a new result which shows that under either Dirichlet, Neumann, or
Robin boundary conditions, the corners in a planar domain are a spectral
invariant of the Laplacian. For the case of polygonal domains, we show how a
locality principle, in the spirit of Kac's "principle of not feeling the
boundary" can be used together with calculations of explicit model heat kernels
to prove the result. In the process, we prove this locality principle for all
three boundary conditions. Albeit previously known for Dirichlet boundary
conditions, this appears to be new for Robin and Neumann boundary conditions,
in the generality presented here. For the case of curvilinear polygons, we
describe how the same arguments using the locality principle fail, but can
nonetheless be replaced by powerful microlocal analysis methods.
