Jet determination in higher codimension

Let $M$ be a real-analytic submanifold of ${\mathbb C}^N$ of codimension $d$. Consider the set of germs of biholomorphisms $F$ at a point $p\in M$ such that $F(M)\subset M$. By the work of Cartan, Tanaka, Chern and Moser, if $d=1$, every such $F$ is uniquely determined by its first and second derivatives at $p$ provided that the Levi form of $M$ is non-degenerate at $p$. In this talk, we will discuss some recent developments on the generalization of this phenomenon to higher codimension.