The method of homotopy formula and the Cauchy-Riemann (d-bar) equation on pseudoconvex domains of finite type in $\mathbb{C}^2$

The homotopy formula is one of the most powerful methods in several complex variables. In the first part of the talk, we will introduce the basics of the method, as well as some recent advances that allow us to solve a wide range of problems from the d-bar equation to the deformation theory of complex structures on manifolds with boundary. In the second part, we will present a new homotopy formula which yields almost sharp estimates in Sobolev and Hölder-Zygmund spaces for the d-bar equation on finite type domains in $\mathbb{C}^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main ingredient of our proof is the construction of holomorphic support functions that admit precise boundary estimates when the parameter variable lies in a thin shell outside the domain.