Alex Lubotzky, Stability and testability of permutations' equations

Let A and B be two permutations in Sym(n) which "almost commute"- are they a small deformation of permutations that truly commute?
More generally, if R is a system of word-equations in variables X=(x_1,....,x_d) and A=(A_1,...., A_d) permutations which are nearly solution; are they near true solutions?
It turns out that the answer to this question depends only on the group presented by the generators X and relations R.
This leads to the notions of "stable groups" and "testable groups". We will present a few results and methods which were developed in recent years to check whether a group is stable\testable.
We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic and with IRS's ( =invariant random subgroups).
A number of open questions will be presented.