Complex analysis is one of the central areas of modern mathematics, and deals with holomorphic functions, which are solutions of the Cauchy-Riemann partial differential equations. The classical theory of functions of one complex variable was developed in the nineteenth century by such giants as Cauchy, Riemann and Weierstrass, and has had a profound impact not only within mathematics, but in applications to physics and engineering. The twentieth century saw the development of the theory of functions of several complex variables in various directions. One of these approaches is to use precise estimates on the Cauchy-Riemann equations in spaces of square integrable functions and differential forms to obtain quantitative understanding of holomorphic function theory on complex manifolds of higher dimensions. Referred to as the “$L^2$-theory of the $\overline{\partial}$-problem”, this method originated in the 1960s in the work of Hörmander, Kohn, Andreotti-Vesentini and since that time has evolved into a very powerful and flexible tool to construct analytic objects.

This program aims to introduce graduate students, postdocs and early-career researchers to this method and its applications. Student participants will be expected to know the material covered at the M.Sc. level courses in this area. The program will cover the basics of the theory through lectures, collaborative problem sessions and presentations. Applications of complex analysis and $L^2$ methods to other areas of mathematics will be especially emphasized. This program will be of interest to researchers working in the areas of Complex Analysis, Complex Geometry (both Algebraic and Differential), Partial Differential Equations, Operator Theory, etc.

**Eligibility Criteria**:

Good knowledge of complex analysis and calculus of several variables (including differential forms) at the MSc level.