Igor Khavkine
I will discuss the Killing operator (K_{ab}[v] = \nabla_a v_b + \nabla_b v_a) on a (pseudo-)Riemannian manifold as an overdetermined PDE and its (formal) compatibility complex. It has been observed that this compatibility complex and its cohomology play an important role in General Relativity. In 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 potential importance, the full compatibility complex is known in only very few cases. I will discuss some strategies and some work in progress to attack this problem.
