A basic question in classical gauge theory is to describe moduli spaces of solutions to PDEs arising in that context (such as Bogomolny equations on R^3 or anti-self-duality equations on R^4), at least as complex manifolds, in explicit finite-dimensional terms. In the case of Bogomolny equations, a powerful means to do this is provided by the scattering data assigned to monopoles. While this is a well-known classical fact, it has only been fully appreciated more recently that upon quantization, monopole scattering matrices become R-matrices for shifted Yangians based on finite-dimensional Lie algebras. From the modern perspective, the fact that the monopole S-matrix provides the bridge between monopole moduli and Yangians is its essential feature. In this talk we will explain how this construction can be generalized to moduli spaces of noncommutative instantons on R^4 and R^3 x S^1; the corresponding scattering matrices are semiclassical limits of R-matrices of shifted affine Yangians (for Lie algebras of type A). Variations on the theme lead to deformations of the B-model of topological string theory on backgrounds of the form uv + P(z, w) = 0, that live over the moduli space of noncommutative instantons (the noncommutativity parameter is identified with the topological string coupling). Based on https://arxiv.org/abs/2601.07949