Rough diffusion equations with singular forcing

I will discuss recent efforts to place the work of Otto/Weber on quasi-linear SPDE's in an abstract framework close to the theory of regularity structures. This leads to general tools on integration, reconstruction, and multiplication which constitute partial progress towards a general theory of singular SPDE's with variable diffusion coefficients. As a first application, we use these tools to establish a priori bounds for rough diffusion equations driven by a more singular forcing than in Otto/Weber. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.