The mini-course provides an introduction into the area of the corona problem for the algebra $H^\infty$ of bounded holomorphic functions on the open unit disk $\Di\subset\Co$. The structure of the course is as follows.

It contains the formulation of the corona problem and an overview of some classical and recent results in this area.

Banach-valued $\bar\partial$ equations on the disk. We recall some classical results about interpolating sequences for $H^\infty$ functions and apply them to obtain bounded solutions of some Banach-valued $\bar\partial$ equations on $\Di$. We give a sketch of the proof of the formulated result.

Carleson corona theorem

Using results of the previous part, we prove the Carleson corona theorem by reducing it by an algebraic technique known as the Koszul complex to specific $\bar\partial$ equations on $\Di$.

Structure of the maximal ideal space $\mathfrak M(H^\infty)$ of $H^\infty$.} We start with the classical results of Hoffman describing Gleason parts (analytic disks and one-pointed parts) of $\mathfrak M(H^\infty)$. We describe the topological structure of the set $\mathfrak M_a$ of all analytic disks. Then using the classical Carleson construction and the methods developed in part (2), we prove the result of Su\'{a}rez stating that the set of one-pointed Gleason parts $\mathfrak M_s$ is totally disconnected.

Banach-valued corona problem.} We formulate the general Banach-valued corona problem for the algebra $H^\infty(A)$ of bounded holomorphic functions on $\Di$ with values in a commutative unital complex Banach algebra $A$. Using the methods and results of parts (2) and (4), we prove the corona problem for algebras $H_{\rm comp}^\infty(A)$ of holomorphic functions on $\Di$ with relatively compact images in $A$.

Operator completion problem for $H^\infty$.

We formulate some basic facts of the Oka-Grauert theory for $H_{\rm comp}^\infty$ spaces related to the classical Sz.\,Nagy operator corona problem. Then we present some results about extension of $H_{\rm comp}^\infty$ operator-valued functions on $\Di$ to invertible ones.