A proof of the Kontsevich conjecture on noncommutative birational transformations

I will talk about our recent proof (arXiv1305.1965) of the Kontsevich conjecture dated back
at 1996, and mentioned at the 2011 Arbeitstagung talk on 'Noncommutative identities'
(arXiv1109.2469). This conjecture says that certain transformations given by matrices over
free noncommutative algebra with inverses ('free field' due to P.Cohn) are periodic, on
the level of orbits of the left/right diagonal action. Namely, let (M_{ij}), 1≤i,j≤3 be a
matrix, whose entries are independent noncommutative variables. Let us consider three
'birational involutions' I1:M→M^{−1}; I2:M_{ij}→M_{ij)^{−1},∀i,j; I3:M→M^t.
Then the composition Φ=I1∘I2∘I3 has order three.