Sérgio Luís Zani

On the global solvability for overdetermined systems

We consider a class of systems of two smooth vector fields on the 3-torus associated to a closed 1-form. The global solvability of such systems is completely determined by the connectedness of the sublevel and superlevel sets of a primitive of this 1-form in the minimal covering. The main point of this talk is addressing the case when the 1-form is non-exact and its periods are rationally dependent. This is a work in collaboration with A. Bergamasco, A. Kirilov, and W. Nunes.

Global Analytic Regularity for Structures of Co-Rank One

We consider real analytic involutive structures 𝒱, of co-rank one, defined on a real analytic paracompact orientable manifold M. To each such structure we associate certain connected subsets of M which we call the level sets of 𝒱. We prove that analytic regularity propagates along them. With a further assumption on the level sets of 𝒱 we characterize the global analytic hypoellipticity of a differential operator naturally associated to 𝒱. As an application we study a case of tube structures.

GLOBALLY ANALYTIC HYPOELLIPTIC VECTOR FIELDS ON COMPACT SURFACES

We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold. In particular, we show that such vector fields exist only on the torus.