Online courses
TCC 2023/24 course: Rigidity and flexibility for discrete structures
This lecture course covers the topic of Geometric Rigidity Theory, the study of whether a framework – a realisation of a graph in Euclidean space with edges as stiff straight bars and vertices as universal joints – is rigid or flexible. The course covers a mixture of combinatorics, geometry and algebra, but does not require any more than a basic undergraduate-level knowledge of the three topics. While the topic is currently utilised in a wide variety of real-world applications, this course focuses on the pure mathematical theory of the topic.
An introduction to the course can be found here.
The course will cover the following 6 main topics in the following order (the final two topics will be covered if there is time available at the end of the course):
Topic 1: Rigidity and infinitesimal rigidity for frameworks
This topic will cover what it means for a framework to be rigid by exploring various equivalent definition for framework rigidity, including local rigidity and continuous rigidity. After this, we then introduce the stronger notion of infinitesimal rigidity. Infinitesimal rigidity is a so-called "generic property", in that either almost all realisations of a graph are infinitesimally rigid or almost all realisations are not. This property is the first step in defining combinatorial rigidity properties for graphs.
Topic 2: Combinatorial rigidity part I – Generic rigidity
This topic covers the combinatorial properties of graphs for which almost all realisations are infinitesimally rigid. The simplest of these properties is Maxwell’s counting condition for framework rigidity. A more complicated one is the matroidal structure that can be associated to rigidity in a fixed dimension. By describing graph extension moves that preserve rigidity, we prove the Geiringer-Laman theorem; an exact combinatorial characterisation of the graphs that are rigid in the Euclidean plane. From this result we can prove further sufficient conditions for rigidity, including Lovasz-Yemini's theorem: every 6-connected graph is rigid in the Euclidean plane. We also cover combinatorial properties of rigidity in 3-dimensional space and higher, including the Maximality Conjecture and a recent sufficient condition for rigidity of Lindemann.
Topic 3: Equilibrium stresses and global rigidity
This topic will explore the framework property of global rigidity, where every realisation with the same edge lengths as a framework is congruent to it. The main tool for analysing global rigidity are weighted Laplacian matrices formed by equilibrium stresses of the framework (edge weightings that satisfy a balancing condition at every vertex of the framework). Using equilibrium stresses we are able to prove that global rigidity is also a generic property.
Topic 4: Combinatorial rigidity part II – Generic global rigidity
This topic will mainly cover the exact combinatorial characterisation for global rigidity in the Euclidean plane. We will also construct various examples to highlight that both rigidity and global rigidity characterisations are difficult in higher dimensions.
Topic 5: Graph and framework rigidity in non-Euclidean spaces
This topic covers the properties of rigidity in spherical and hyperbolic geometries. The key tool for doing so is the notion of coning, wherein a rigid framework is converted into a rigid framework one dimension higher by introducing a new vertex adjacent to all others. We also cover some of the pitfalls that arise when we try to consider global rigidity in hyperbolic spaces.
Topic 6: Rigidity for triangulated surfaces
This topic covers the key rigidity theory results for the class of frameworks formed from triangulated surfaces. The main result we prove is Cauchy’s rigidity theorem: every convex triangulated sphere is rigid. By introducing notions of simplicial complexes, we also prove Fogelsanger's theorem: the 1-skeleton of any triangulated surface is generic in 3-dimensional Euclidean space. This result was recently extended by Cruickshank, Jackson and Tanigawa to prove that, so long as the resulting graph is 4-connected and the original surface was not a sphere, then the graph will also globally rigid in 3-dimensional Euclidean space.