Volume optimisation on triangulated 3-manifolds

In 1978, Thurston introduced an affine algebraic set to study hyperbolic structures on triangulated 3-manifolds. Recently, Feng Luo discovered a finite dimensional variational principle on triangulated 3-manifolds with the property that its critical points are related to both Thurston's algebraic set and to Haken's normal surface theory. The action functional is the volume. This is a generalisation of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary. Combining the work of Luo, Futer-Gueritaud, Segerman-Tillmann and Luo-Tillmann, this gives a new, finite dimensional variational formulation of the Poincare conjecture, and is expected to give insights that lead to a discrete interpretation of the 3-dimensional Ricci flow.