Approximate subgroups of linear groups

Emmanuel Breuillard 
Terence Tao introduced a few yeas ago the notion of an approximate subgroup of an ambient group. Roughly speaking these are finite symmetric subsets that are almost closed under multiplication. In this talk I will discuss a conjectural statement regarding the rough classification of such objects and describe a recent joint work with Ben Green and Terence Tao proving it for approximate subgroups of linear groups. I will then explain some consequences regarding the growth of groups and expander graphs. 