# Khavkine I. Applications of compatibility complexes and their cohomology in relativity and gauge theories, talk at The Workshop on Integrable Nonlinear Equations, 18-24 October 2015, Mikulov, Czech Republic (abstract)

I will discuss the Killing operator (${\displaystyle K_{ab}[v]=\nabla _{a}v_{b}+\nabla _{b}v_{a}}$) as an overdetermined differential operator and its (formal) compatibility complex. It has been recently observed that this compatibility complex and its cohomology play an important role in General Relativity. In more general gauge theories, an analogous role is played by the "gauge generator" operator and its compatibility complex. An important open problem is to explicitly compute the tensorial form of the compatibility complex on (pseudo-)Riemannian spaces of special interest. Surprisingly, despite its importance, the full compatibility complex is known in only very few cases. I have recently reviewed one of these cases, constant curvature spaces, where this complex is known as the Calabi complex, in arXiv:1409.7212.