Abstract: Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? This can be seen as a question about a property the equation XY=YX.
Studying analogous problems for more general equations (or sets of equations) leads to the notion of "locally testable groups" (aka "stable groups").
We will take the opportunity to say something about "local testability" in general, which is an important subject in computer science. We will then describe some results and methods developed (in a work in progress), together with Alex Lubotzky, to decide whether various groups are locally testable or not.
This will bring in some important notions in group theory, such as amenability, Kazhdan's Property (T) and sofic groups.