Turbulence and the problem of classifying homeomorphisms

Self-homeomorphisms of the interval [0, 1] can be classified up to conjugacy using certain countable structures as invariants.
The problem becomes much more complex if one is to replace the interval [0, 1] with the square [0, 1]^2. 
Indeed it is a theorem of Hjorth that there exists no definable way to classify self-homeomorphisms of the square [0, 1]^2 by using  “countable structures” as invariants.  
In this series of talks, we will discuss Hjorth’s theory of turbulence and we will use it to prove the anti-classification result due to Hjorth.  
Time permitting we will also discuss possible extensions.