Rigidity theory: when does a geometric embedding (a framework) of a graph admit a continuous deformation into a non-congruent arrangement without changing the edge lengths or breaking the connectivity of the graph? When such a deformation is possible the framework is flexible and if no such motion exists the framework is (continuously) rigid. For arbitrary embeddings this is known to be NP-hard but for generic embeddings there is a full combinatorial theory in dimension <3 complete with efficient algorithms. In higher dimensions no such combinatorial description is known.

The rigidity problem is often attacked first by linearisation, that is, differentiating the quadratic edge length equations and considering the resulting infinitesimal version of rigidity. This is completely described by the rank of the rigidity matrix associated to a framework and is known to be equivalent to continuous rigidity for generic frameworks in any dimension.

Rigidity corresponds to the real algebraic variety of solutions to the edge length equations being finite modulo trivial motions (translations and rotations). When this variety is a singleton, i.e. the embedding is unique, the framework is globally rigid. As before there is a complete combinatorial theory in 1 and 2-dimensions but no such characterisations for dimensions 3 and higher.

Frameworks on surfaces: with John Owen and Stephen Power. Typically we choose a surface embedded in Euclidean space which is regular in an appropriate sense and constrain the vertices of the framework to the surface while allowing the edges to cut through the surface. For such surfaces (particularly spheres and cylinders) there are combinatorial descriptions of rigidity depending on the dimension of the space of tangential isometries.

Symmetric frameworks on surfaces: with Bernd Schulze. Suppose that the action of an element of the orthogonal group in 3D moves a framework supported on a surface in such a way that the result is still supported on the surface. Further suppose the embedding is generic aside from the equations defining the surface and those defining the symmetry. An interesting example is half turn symmetry of frameworks on a cylinder. There are essentially two different half turn symmetries worth considering: half turn about a plane coaxial to the cylinder and half turn about a plane orthogonal to the cylinder. We are investigating Laman type theorems in these settings using inductive constructions of orbit graphs.

Globally rigid frameworks on surfaces: with Bill Jackson and Tom McCourt. A framework is globally rigid if every other realisation of the graph with the same edge lengths is congruent to the original. In the plane there is a combinatorial characterisation due, in its various parts, to Hendrickson, Connelly and Jackson and Jordan. We are investigating analogues of that theorem for frameworks on surfaces.

Mixed dimensional Constraints: 2-colour the vertices of a graph then embed the graph on a union of 1 and 2-dimensional manifolds subject to the colouring. For such situations we ask the typical rigidity theory question - is there a purely combinatorial description of minimal rigidity? For surprisingly simple examples, e.g. a line at 45 degree angle to a plane, there are counterexamples to a typical Laman type theorem.