The structure and classification of the so-called free group II1 factors, arising as vN-algebras of the free groups Fn with n generators, 2 ≤ n ≤ ∞, have been the subject of much interest for 80 years by now. But despite many remarkable results and the development of several insightful techniques, some of the most basic questions concerning this fundamental class of II1 factors remained open:
(1) LFn ≃ LFm iff n = m;
(2) F(LFn)=1 when n < ∞;
(3) infinite generation of LF∞;
(4) existence of non freely complemented maximal amenable MASAs in LFn;
(5) do LFn embed in any non-amenable II1 factor.
I will comment on the progress made on these problems and possible approaches to solve them. | 
