Let Fn be a free group of rank n with free basis x1, ⋯, xn. Let {y1, ⋯, yk} be a set of k ≤ n elements of Fn, where each yi is represented by a word Yi(x1, ⋯, xn) in the generators xj. Let ∂ yi/∂ xj ...

Let D ⊂ ℂ be a bounded domain, whose boundary B consists of k simple closed continuous curves, and H∞ (D) be the algebra of bounded analytic functions on D. We prove the matrix-valued corona theorem ...