Algebraic Approach to Electric Impedance Tomography of Surfaces

We consider the problem on determination of the (conformal class) of a surface $M$ with the metric $g$ and the boundary $\Gamma$ from its DN map $\Lambda: \ f\mapsto \partial_\nu u^f|_{\Gamma}$, where $u^f$ is a harmonic extension of $f\in C^{\infty}(\Gamma)$ into $M$ and $\nu$ is the exterior normal vector. We discuss

• the algebraic approach (proposed by M.I.Belishev) for such a determination. Its key idea is that $M$ is conformally equivalent to the spectrum of the algebra $A(M)$ of holomorphic functions on $M$; the latter being determined (up to isomorphism) by the boundary data.
• generalisations of the above approach to the case of non-orientable $M$ or to the case in which the DN map is given only on an arbitrarily small segment of the boundary.
• the characterization of DN maps, i.e., the necessary and sufficient conditions for $\Lambda$ to be a DN map of some surface. In the algebraic approach, such conditions are obtained from basic properties of holomorphic functions (such as algebraic closeness, the argument principle, etc).
• the stability of solutions, i.e., the continuous dependence (in the Teichmüller metric) of the conformal class of $M$ on its DN map $\Lambda$.

The talk is based on the joint works with M.I.Belishev.