Commutator type and Levi type of a system of CR vector fields

Finite type conditions arise naturally during the study of weakly pseudoconvex hypersurfaces in $\mathbb{C}^n$, which are defined to measure to degeneracy of the Levi form. Let $M$ be a pseudoconvex hypersurface in $\mathbb{C}^n$, $p\in M$, and let $B$ be a subbundle of the CR tangent bundle $T^{(1,0)}M$. The commutator type $t(B,p)$ measures the number of commutators of the sections of $B$ and their conjugates needed to generate the contact tangent vector at $p$. The Levi type $c(B,p)$ is concerned with differentiating the Levi form along the sections of $B$ and their conjugates. It is believed that these two types are the same, which is known as the generalized D'Angelo Conjecture. In this talk, I shall talk about the recent progress on this conjecture, which is based on the joint works with X. Huang and P. Yuan.