Dynkin diagrams are a ubiquitous combinatorial structure with connections to, among many other things, polyhedra and Kleinian singularities. In this talk,we will survey many of the places where Dynkin diagrams appear. The talk should be accessible to all graduate students.

Motivated by properties of widely used minerals called Zeolites, there has been a rapid development of work towards predicting flexibility or rigidity of periodic structures. Given data bases for possible artificial zeolites, and the observations that functioning zeolites are flexible - there is a strong interest in ways to test a computer model prior to building the crystals in the lab. Several of these recent papers have given necessary (and sometimes sufficient) conditions for periodic generic frameworks to be infinitesimally rigid. Other recent papers have explored when symmetry in a finite framework shifts the framework from rigid to flexible.

Building on these two foundations, recent work with Bernd Schulze (TU Berlin) and Elissa Ross (York University) has examined necessary conditions for rigidity of periodic frameworks with added symmetry. Again, there are circumstances, such as inversive symmetry in a crystal, which convert the count for generic rigidity into an orbit count which guarantees flexibility.

We will present an overview of these results, with a few animations and tables, as well as the core technique of orbit rigidity matrices. We conclude with an array of unsolved problems. Related papers are on the arXiv.