A Bratteli-Vershik model for $\mathbb{Z^2}$ actions, or how cohomology can help us make dynamical systems

The Bratteli-Vershik model is a method of producing minimal actions of the integers on a Cantor set. It was given by myself, Rich Herman and Chris Skau, building on seminal ideas of Anatoly Vershik, over 30 years ago. Rather disappointingly and surprisingly, there isn't a good version for $\mathbb{Z}^2$ actions. I'll report on a new outlook on the problem and recent progress with Thierry Giordano (Ottawa) and Christian Skau (Trondheim). The new outlook focuses on the model as an answer to the question: which cohomological invariants can arise from such actions? I will not assume any familiarity with either the original model or the cohomology. The first half of the talk will be a gentle introduction to the $\mathbb{Z}$-case and the second half will deal with how to adapt the question to get an answer for $\mathbb{Z}^2$