Abstract:
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.