Abstract: The Inverse Conductivity Problem was first posed by A.P. Calder\'on in 1980. Let $\Omega \subset \mathbb{C}$ be a simply connected domain.  Consider the following Dirichlet problem

\begin{equation*}

\begin{cases}

\nabla \cdot (\sigma \nabla u) = 0 & \text{ in } \Omega \\

u = f & \text{ on } \partial \Omega

\end{cases}

\end{equation*}

and the Dirichlet-to-Neumann operator $\Lambda_\sigma$,

\begin{equation*}

\Lambda_{\sigma} f = \left. \sigma \frac{\partial u}{\partial \nu}\right|_{\partial \Omega}

\end{equation*}

The question Calder\'on posed was: is a conductivity $\sigma \in L^\infty(\mathbb{C})$ uniquely determined by its Dirichlet-to-Neumann operator $\Lambda_\sigma$?  The focus of this talk is the solution found by Kari Astala and Lassi P\"aiv\"arinta in 2006, where they study a Beltrami equation related to the conductivity problem through a change of variable.  Using the properties of quasiconformal maps and a D-bar method, they define a nonlinear Fourier transform and use its properties to obtain uniqueness.  This is also a shameless advertisement for Francis Chung's upcoming topics course on inverse problems.