We show that solutions of Thurston's equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston's equation and a commutative ring associated to triangulated 3-manifolds. It is shown that solutions to homogeneous equations over the real numbers are critical points of an entropy function.

A family of coordinates h for the Teichmller space of a compact surface with boundary was introduced by the second author. Mondello showed that the coordinate 0 can be used to produce a natural cell decomposition of the Teichmller space, invariant under the action of the mapping class group. In this paper, we show that, for any h 0, the coordinate h produces a natural cell decomposition of the Teichmller space.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices. This verifies affirmatively a conjecture of John Minor.

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

A spherical polyhedral surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedral surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obtained. A characterization of spherical polyhedral surfaces in terms of the triangulation and the angle assignment is established.