The Morera Problem in Clifford Algebras and the Heisenberg Group

In an open subset Ω of the complex plane ℂ the Morera theorem gives a simple looking necessary and sufficient condition for a continuous function f to be holomorphic in Ω. Namely the vanishing of all the integrals ∫_γf(z) dz, where γ is an arbitrary Jordan curve in Ω whose interior also lies in Ω. The Morera problem consists in finding relatively small families Γ of Jordan curves such that the vanishing of the corresponding integrals still ensures that the conclusion of Morera’s theorem still holds. In this lecture we discuss a number of recent results obtained in the case one considers functions defined in two fairly different settings, the Clifford algebras and the Heisenberg group.