Dr. Waldo Arriagada is Assistant Professor of Mathematics in the Department of Applied Mathematics and Sciences at Khalifa University.
His research interests include:
One aspect of these topics concerns the study of the geometrical methods involved in the characterization of the orbit space of a singular holomorphic (complex) foliation. It is known that this kind of foliations are sometimes uniquely defined by the holonomy of its separatrices (the leaves that are images of punctured complex disks): two foliations with holomorphically conjugate holonomies are holomorphically orbitally equivalent themselves. This principle was justified by R. Perez-Marco & J.-C. Yoccoz (1994) in the simplest singular case where the singularity is a nondegenerate saddle (with the ratio of the eigenvalues of the linear part being a negative number). In this case the holonomy map can also be suitably realized by a saddle foliation. Further, in the presence of extra parameters analytically deforming the singularity, the correspondence between holomorphic types of the foliation and its holonomy remains holomorphic.
In the case of phase portraits of germs of real analytic vector fields the holonomy map does not naturally represent the foliation. This is because there are additional symmetries (generated by an antiholomorphic involution of the complex phase space) that are not preserved by the holonomy. However, for the elliptic singularities (with the pair of nonreal eigenvalues) the germ of a self-map, called the Poincare monodromy, is well defined. The question whether the germ of the monodromy defines the analytic class of the real foliation under orbital equivalence has been positively answered by myself (J. Diff. Eqs., 2012).
These problems belong to a more general setting which is the study of equilibria of parameter-dependent analytic dynamical systems or unfoldings and the identification of a complete set of invariants under analytical equivalence. The corresponding moduli space is dramatically huge. Hence, it is an exception rather than the general rule that an ellitptic singularity be normalizable.
PhD Mathematics, Mathematical Engineer, Bachelor of Sciences