  In this thesis we deal with spectral invariants for polygons and closed
orbisurfaces of constant Gaussian curvature. In each case our method is to
study the heat kernel and the asymptotic expansion of the heat trace. First, we
investigate hyperbolic polygons, i.e. relatively compact domains in the
hyperbolic plane with piecewise geodesic boundary. We compute the asymptotic
expansion of the heat trace associated to the Dirichlet Laplacian of any
hyperbolic polygon, and we obtain explicit formulas for all heat invariants.
Analogous results for Euclidean and spherical polygons were known before. We
unify these results and deduce the heat invariants for arbitrary polygons, i.e.
for relatively compact domains with piecewise geodesic boundary contained in a
complete Riemannian manifold of constant Gaussian curvature. It turns out that
the heat invariants provide much information about a polygon, if the curvature
does not vanish. For example, then the multiset of all real angles (i.e. those
which are not equal to pi) and the Euler characteristic of a polygon are
spectral invariants. Furthermore, we compute the asymptotic expansion of the
heat trace for any closed Riemannian orbisurface of constant curvature, and
obtain explicit formulas for all heat invariants. If the curvature does not
vanish, then it is possible to detect interesting information about the
topology and the singular set of an orbisurface from the heat invariants.
