The first call for applications for the new edition of the PhD Program in Mathematics UC|UP (starting September 2019) will be open during January 02, 2019 - February 04, 2019
The prize Eng. António de Almeida for the best doctoral thesis defended at FCUP in the years 2015 to 2017 in the areas of Mathematics, Statistics and Computer Science was awarded to Célia Borlido for her thesis "The word problem and some reducibility properties for pseudo- varieties of the form DRH", supervised by Prof. Jorge Almeida within the scope of PIUDM.
Convergence in L^p, in measure and a.e.; integrability; Fatou Lemma, Monotone Convergence and Dominated Convergence; isomorphism in measure. Periodic orbit, recurrent, non-wandering, dense; conjugacy, stability. Invariant measures by a dynamic. Invariants by isomorphism. Existence of invariant probability measures for continuous dynamics on compact metric spaces. Ergodicity, mixing, uniquely ergodicity; dynamical and spectral interpretations of these metrical properties. Examples: homeomorphism of R; qx(mod1); Gauss map; shifts; powers of z and rotation in S^1; linear maps in R^n; Anosov diffeomorphisms; Smale horseshoe; pedal triangle; coupled systems; Markov chains; search engines. Poincaré, Birkhoff and Kac Theorems; connection with the second law of Thermodynamics. Applications: Borel Theorem on normal numbers; Multiple Recurrence Theorem; van der Waerden Theorem. Ergodic decomposition. Topological and measure-theoretic entropy. Variational principle and equilibrium states.