I will talk about ideas and applications of Poincare-Dulac normal form reductions in the context of dispersive PDEs. In particular, I discuss an infinite iteration of normal form reductions in the following contexts:

[(i)] quadratic derivative NLS (dNLS): By performing an infinite iteration of normal form reductions, we show that dNLS is reducible to the linear Schr\"odinger equation, indicating integrability of dNLS in some mild sense. As a byproduct, we construct an infinite sequence of quantities invariant under the flow of dNLS.

[(ii)] Construction of a modified energy of infinite order: In establishing an energy estimate, it is often useful to study a modified energy (of finite order) by add a correction term to an energy functional. By applying an infinite iteration of normal form reductions to the (non-Hamiltonian) equation satisfied by the $H^s$-energy functional, we construct a modified energy of infinite order. While the method is robust and is applicable to general dispersive PDEs, we take the cubic fourth order NLS as an example and discuss some applications.