Seminario Iberoamericano de Matemáticas (SIM 112) organizado por el equipo ECSING se realizará de forma online el viernes 5 de marzo a las 16h (hora de Madrid, GTM +1). El horario es el siguiente:
16h-17h Javier Ribón Herguedas (Universidade Federal Fuminense, Brasil). Title: The role of Puiseux characteristics in the local Poincaré problem.
Abstract: The usual Poincaré problem consists in determining whether or not there exist upper bounds for the degree of an invariant algebraic curve of a foliation of the complex projective plane in terms of the degree of the foliation. We are interested in the local analogue, in which degrees are replaced with vanishing multiplicities at the origin. Since cusps of arbitrary degree can be invariant curves of foliations of degree one in the global setting and of multiplicity one in the local one, there are no universal upper bounds. In order to deal with this issue, extra hypotheses were considered, either on the invariant curve (Cerveau - Lins Neto,...) or on the desingularization of the foliation (Carnicer...). We consider an irreducible germ of invariant curve and no extra hypotheses. We provide a lower bound for the multiplicity of a germ of complex foliation in dimension 2 in terms of Puiseux characteristics of an irreducible invariant curve. Obviously, the lower bound is trivial for a cusp but it is valuable if the curve has at least two Puiseux characteristic exponents. This is a joint work with José Cano and Pedro Fortuny.
17h15-18h15. Alberto Lastra Sedano (Universidad de Alcalá de Henares). Title: On Gevrey asymptotics for linear singularly perturbed equations with linear fractional transforms (joint work with Guoting Chen and Stéphane Malek)
Abstract: A family of linear singularly perturbed Cauchy problems is studied. The equations defining the problem combine partial differential operators together with the action of linear fractional transforms. The exotic geometry of the problem in the Borel plane, involving both sectorial regions and striplike sets, gives rise to asymptotic results relating the analytic solution and the formal one through Gevrey asymptotic expansions. The main results lean on the appearance of domains in the complex plane which remain intimately related to Lambert W function, which turns out to be crucial in the construction of the analytic solutions
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