The aim of this talk is to introduce the notion of bi-filtered manifolds, which are smooth or complex analytic manifolds, whose tangent space is equipped with a descending filtration indexed by lexicographically ordered pairs of integers (i,j)<(0,0) and compatible with the Lie bracket. While being the generalization of a classical notion of filtered manifolds due to Tanaka, this new notion of bi-filtered manifolds provides a uniform approach to studying some non-regular parabolic geometries, which become regular in this new bi-filtered sence.
We give interpretation of bi-filtered manifolds in terms of cone sturcutres on filtered manifolds, define an alalog of their symbol and its Tanaka prolongation, prove the existence of a natural Cartan connection under the assumption of existence of invariant normalization conditions. As an application, we present several examples of geometric structures that can be effectively treated by this appraoch.