Abstract: It is a classical problem of complex analytic geometry to determine when a given analytic space has an algebraic structure. For nonsingular surfaces the solution is given by the Enriques-Kodaira classification. For singular surfaces, however we only have some sufficiency results (due to Grauert, Artin, Brenton and others) and the general picture is not clear.

In this talk we present a new (and effective) algebraicity criterion for a class of surfaces containing the complex plane (C^2). We also present a (somewhat unexpected) application to the classical "moment problem", which asks, given a closed subset S of R^n, for characterization of linear functionals on the ring of polynomials(over R) which arise from integration on S with respect to some (Borel) measure on S. We apply our algebraicity criterion to "almost" solve the moment problem for planar semialgebraic sets with a single "tentacle". The latter is based on a joint work with Tim Netzer.