We show that on a smooth compact Riemann surface with boundary (M0, g) the Dirichletto-
Neumann map of the Schrödinger operator â g + V determines uniquely the potential V .
This seemingly analytical problem turns out to have connections with ideas in symplectic
geometry and differential topology. We will discuss how these geometrical features arise and
the techniques we use to treat them.

This is joint work with Colin Guillarmou of CNRS Nice. The speaker is partially supported
by NSF Grant No. DMS-0807502 during this work.