The famous Carleson corona theorem asserts that the open unit disk is dense in the maximal ideal space of the algebra of bounded holomorphic functions on it (denoted $H^\infty$). Similar statements for the algebra of bounded holomorphic functions on a polydisk and for slice algebras for $H^\infty$ remain the major open problems of multivariate complex analysis. For instance, the answer to the last problem would be obtained if one were able to show that $H^\infty$ has the Grothendieck approximation property. This problem posed by Lindenstrauss in the early 1970th is also unsolved. The strongest result in this direction was proved by Bourgain and Reinov in 1983 and asserts that $H^\infty$ has the approximation property up to logarithm. In the talk I will present a proof of the corona theorem for slice algebras for $H^\infty$, describe topological structure of the maximal ideal space of $H^\infty$ and as a corollary present some results on $Sz$. Nagy operator-valued corona problem for $H^\infty$.