The topological group $\mathcal{D}^k(\mathbb{S})$ of diffeomorphisms of the unit circle $\mathbb{S}$ of Sobolev class$H$^{$k$}, for$k$large enough, is a Banach manifold modeled on the Hilbert space $H^k(\mathbb{S})$ . In this paper we show that the$H$^{1}right-invariant metric obtained by right-translation of the$H$^{1}inner product on $T_{\rm id}\mathcal{D}^k(\mathbb{S})\simeq H^k(\mathbb{S})$ defines a smooth Riemannian metric on $\mathcal{D}^k(\mathbb{S})$ , and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the$H$^{1}right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.

In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of the boundary is given either as a list of pixels, or by a Turing machine.

If$n$is a non-negative integer, then denote by ∂^{-$n$}$H$^{∞}the space of all complex-valued functions$f$defined on $\mathbb{D}$ such that$f$,$f$^{(1)},$f$^{(2)},...,$f$^{($n$)}belong to$H$^{∞}, with the norm $$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$$ We prove bounds on the solution in the corona problem for ∂^{-$n$}$H$^{∞}. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space$H$^{∞}.

We prove the vanishing of the Koszul homology group$H$_{μ}(Kos($M$)_{μ}), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.

The purpose of this erratum is to point out a mistake in a previous article by the authors and to explain how this affects the results of that article.