Sesión número 98 del Seminario Iberoamericano de Matemáticas (SIM98)
Lugar: Excepcionalmente tendrá lugar en el Aula A118 de la Facultad de Ciencias de la Universidad de Valladolid.
Fecha: Jueves, 23 de Febrero de 2017.
17:45 Armin Rainer (University of Vienna)
Titulo: Recognizing (ultra)differentiable functions on closed sets
Resumen: A classical theorem of Boman states that a function on an open subset of R^d is smooth if and only if it maps smooth curves to smooth curves. There is also an ultradifferentiable version of Boman’s theorem. In this talk I will show that these results persist on closed subsets X of R^d whose interior is dense in X and has the uniform cone property. I shall also discuss the case that X is a closed subanalytic set with dense interior.
18:45 André Belotto (Université Paul Sabatier, Toulouse)
Título: Solutions of quasianalytic equations.
Resumen: I will present new techniques to solve equations G(x,y)=0, where G(x,y)=G(x_1,...,x_n,y) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class). Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related). No previous knowledge on quasianalytic functions is necessary.
In the first part of the talk, I will give a brief overview on quasianlytic functions, focusing on the difference with analytic functions. Next, I will present a technique of "quasianalytic extension" (based on resolution of singularities) and the following result: if G(x,y)=0 has a formal power series solution y=H(x) at some point a, then H is the Taylor expansion at a of a quasianalytic solution y=h(x), where h(x) is allowed to have a certain controlled loss of regularity, depending on G.