# Conformal Parallel Vector Fields and Inverse Problems

# Conformal Parallel Vector Fields and Inverse Problems

**In-Person Talk **

***Please note the change in time and location***

A celebrated theorem of Dos Santos Ferreira-Kenig-Salo-Uhlman states that Calderón inverse conductivity problem can be often solved in manifolds admitting a so called conformal parallel vector field. However, it is not trivial to decide whether a given manifold admits such a vector field, being a property of the conformal class. I will present necessary condition can be obtained in terms of the classical conformal invariant tensors, Weyl and Cotton-York, which allow to classify all Thurston geometries. Examples based on product surfaces and Lie Groups show that such condition it is no sufficient. However a detailed analysis of the algebraic structure of the tensors in combination with Frobenius theorem yields an almost complete understanding of how many conformal parallel vector fields a manifold might admit in dimensions 3 and 4. It is still open whether a sufficient condition in terms of conformally invariant quantities might exist.