Полиномиальные автоморфизмы, гипотеза о якобиане, гипотезы Диксимье и Концевича

The talk is devoted to the famous Jacobian conjecture: $JC_n$: Has any polynomial mapping of $C^n\to C^n$ with constant Jacobian a polynomial inverse? Diximer conjecture $(DC_n)$: $\mathrm{End}(W_n)=\mathrm{Aut}(W_n)$, where $W_n=C[x_1,\dots,x_n,\partial x_1,\dots,\partial x_n]$. It was well known that $DC_n$ implies $JC_n$. Recently, together with Kontzevich, the author proved that $JC_{2n}$ implies $DC_n$. This is related to Kontzevich conjecture, saying that $\mathrm{Aut}(W_n)$ is isomorphic to the group of polynomial symplectomorphisms of $C^{2n}$. These questions are related to describing $\mathrm{aut}(\mathrm{aut}(W_n))$. Recently author proved that the group of algebraic $\mathrm{Aut}(\mathrm{Aut}(C^n))$ contains only inner automorphisms.