Singularities and Bi-complexes for PDEs

Many moduli spaces in geometry and physics, like those appearing in symplectic topology, quantum gauge field theory (e.g. in homological mirror symmetry and Donaldson-Thomas theory) are constructed as parametrizing spaces of solutions to non-linear partial differential equations modulo symmetries of the underlying theory. These spaces are often non-smooth and possess multi non-equidimensional components. Moreover, when they may be written as intersections of higher dimensional components they typically exhibit singularities due to non-transverse intersections. To account for symmetries and provide a suitable geometric model for non-transverse intersection loci, one should enhance our mathematical tools to include higher and derived stacks. Secondary Calculus, due to A. Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation and is centered around the study of the Variational Bi-complex of a system of equations. In my talk I will discuss a generalization in the setting of (relative) homotopical algebraic geometry for possibly singular PDEs.
This is based on a series of joint works with Artan Sheshmani and Shing-Tung Yau.