Curved $A_{\mathrm{\infty }}$-algebras and Chern classes

We describe two constructions giving rise to curved $A_{\mathrm{\infty }}$-algebras. The first consists of deforming $A_{\mathrm{\infty }}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a vector bundle on a polyhedron $X$, we exhibit a curved $A_{\mathrm{\infty }}$-structure on the complex of matrix-valued cochains of sufficiently fine triangulations of $X$. We use this structure as a motivation to develop a homotopy associative version of Chern-Weil theory.