Math 762 — Spring 2001 Discrete Geometry: Rigid and Flexible Structures

Prerequisites: A good command of linear algebra such as Math 221, or Math 431, including a willingness to consider quadratic forms and projective geometry. This will be accessible to any first year graduate student or a senior-level undergraduate.

Topics: The following is a selection of subjects. A proper subset will be chosen as a function of those present. The basic objects of study will be configurations of points in Euclidean space with various constraints, especially distance constraints. The inspiration is from structural engineering, but there will be no direct dependence.

1. Global consequences of distance constraints on point configurations.
    
2. Quadratic energy functions associated to the space of all configurations. These correspond to certain "stress" matrices.
    
3. Equilibrium stresses and their relations to rigidity.
    
4. Affine transformations and their effect on equilibrium stresses.
    
5. Examples of globally rigid structures - "tensegrities".
    
6. Rigidity theorems related to polyhedra in three-space.
    
7. Infinitesimal and static rigidity of bar-and-joint frameworks.
    
8. Prestress rigidity of tensegrity frameworks.
    
9. Applications to graph theory including the Colin de Verdiere number associated to a finite graph.
    
10. Generic rigidity and generic global rigidity. -- This can be used to determine when a generic configuration is determined uniquely, globally up to congruence by a small set of distance equality constraints.
     
11. Symmetric tensegrities using representation theory for finite groups.
     
12. Maxwell-Cremona theory -- the correspondence of planar stresses in frameworks and lifts in three-space.
     
13. Finite mechanisms that are surfaces in space and the volume bounded by them. -- Mathematical bellows do not exist.