About CURVE
CURVE is an international research project focusing on representation varieties of 3manifold groups, geometric structures, and aspects of efficient computation. Our research is centered around the shape and Ptolemy coordinates, which are special coordinates on representation varieties coming from triangulations. These coordinates are 3dimensional analogues of FockGoncharov coordinates on higher Teichmüller spaces and have been studied independently and from different perspectives by GaroufalidisGoernerThurstonZickert, BergeronFalbelGuilloux, and DimofteGabellaGoncharov.
Our goal is to understand structural properties of the coordinates, to decide which representations give rise to geometric structures, and to expand the possibilities for exact computation using advanced tools from computer algebra.
Open problems
 Identify the representations that correspond to geometric structures. A PGL(2,C)representation corresponds to a hyperbolic structure if all shapes have positive imaginary part. There should be a similar condition for spherical CRstructures and real flag structures.
 Understand the mysterious duality between the shape and Ptolemy coordinates.
 Investigate the finite set of volumes of boundaryunipotent representations. Walter Neumann has conjectured that the Bloch group is generated by hyperbolic manifolds. If this conjecture is true, each volume would be an integral linear combination of volumes of hyperbolic manifolds.
 Generalize the shape and Ptolemy coordinates to other Liegroups and higher dimensional manifolds.

