de Rham Theory in Derived Differential Geometry

Derived differential geometry is a nascent field applying techniques from derived algebraic geometry to the study of spaces with smooth structures. As such, it serves as a natural home for studying objects arising in BV formalism. For instance, concepts such as critical loci of action functionals or their quotients by gauge actions can be naturally interpreted as derived differentiable stacks.
In our work, we build a version of de Rham theory for these spaces and prove a version of the de Rham isomorphism. Due to the highly singular nature of all objects involved, developing such a theory is significantly more challenging than in the usual differential geometry, and thus, we construct our formalism with inspiration from algebraic geometry rather than classical differential topology. As a main application of the developed theory, we obtain a version of the comparison morphism between de Rham and constant sheaf cohomology arising from the corresponding map of stacks. This should enable further developments, with a view towards a fully-fledged theory of shifted symplectic structures for derived differentiable stacks.
The talk is based on a preprint of the same name, arXiv:2505.03978.

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