Appears in collection : Mathematics on the Crossroad of Centuries - A Conference in Honor of Maxim Kontsevich's 60th Birthday

Fukaya categories are constructed from moduli spaces of discs of virtual dimension 0 and 1. The higher dimensional moduli spaces can in principle be used to define variants the Fukaya category which have coefficients in spectral rings, lifting the current construction over discrete rings. However, outside the setting of exact symplectic manifolds, the problem of curvature arises, and one needs to resolve the anomaly problem without appealing to the tools of ordinary algebra. I will discuss joint work with Blumberg on formulating this generalisation, with McLean and Smith on showing that Lagrangian Floer theory fits within this new framework, and with Bottman on Floer-theoretic constructions that yield mirror varieties in this setting.