CURVE is an international research project focusing on representation varieties of 3-manifold 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 3-dimensional analogues of Fock-Goncharov coordinates on higher Teichmüller spaces and have been studied independently and from different perspectives by Garoufalidis-Goerner-Thurston-Zickert, Bergeron-Falbel-Guilloux, and Dimofte-Gabella-Goncharov.
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.
- 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 CR-structures and real flag structures.
- Understand the mysterious duality between the shape and Ptolemy coordinates.
- Investigate the finite set of volumes of boundary-unipotent 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 Lie-groups and higher dimensional manifolds.