2018-2019 Sessions

5th of April, 2019- Room 17 de Abril, DMat of University of Coimbra

11h00 - Reception

11h15 - 12h00

Speaker: Willian Ribeiro¹

Title: Weak exponentiability in categories of lax algebras.

Abstract: We start this talk by recalling, motivating, and formalizing in categorical terms the concept of exponentials. Then moving to the particular case of the category Top of topological spaces and continuous functions, we comment on the alternatives to the non-existence of arbitrary exponentials. Weak exponentials are one of these alternatives, and using the (T,V) setting introduced by Clementino and Tholen, these objects are proven to exist not only in Top, but also in many other categories from Analysis and Topology, as (probabilistic) metric spaces, bitopological spaces, approach spaces, and multi-ordered spaces, which are all topological over the category Set of sets and maps.

12h00 - 12h30

Speaker: Carla Dias²

Title: Gibbs-Markov-Young structures and stochastic stability.

Abstract: In the 1960’s, Sinai and Bowen showed that all smooth uniformly hyperbolic dynamical systems admit a finite Markov partition. Sinai, Ruelle and Bowen then used this remarkable geometric structure, and the associated symbolic coding of the system, to study ergodic properties such as the rate of decay of correlations. Some years ago, L.-S. Young proposed an alternative geometric structure, which we call Gibbs-Markov-Young (GMY) structure, as a way of studying the ergodic properties of certain dynamical systems. In this talk, we discuss the relation between GMY structure, Lyapunov exponents and stochastic stability in the setting of random perturbations. This is a joint work with J.F. Alves and H. Vilarinho.

Lunch Break

14h15 - 15h00

Speaker: Diogo Lobo³

Title: Image Restoration Models through Backpropagation on Medical Imaging.

Abstract: The use of neural networks for image processing tasks in the previous decade has set the pace of current research in this area. However, the general intrinsic “black box” nature of such algorithms is a drawback in the context of medical applications. We propose the use of deep learning techniques on already established non-linear diffusion schemes in order to optimize the parameters for each specific task. Consequently, we obtain improved image restoration models with good mathematical foundations. The learning framework and resulting models are presented along with related numerical results and image comparisons.

15h00 - 15h45

Speaker: Ahmed Elshafei⁴

Title: Geodesic completeness for Pseudo-Riemannian manifolds.

Abstract: We study geodesic completeness of pseudo-Riemannian Lie groups by applying techniques from complex dynamics. We recall that, for a semi-simple Lie group, a geodesic corresponds to an integral curve in the Lie algebra of the Euler-Arnold vector field. These vector fields are algebraic and homogeneous of degree 2 thus amenable to be studied by techniques from complex dynamical systems. We present a complete study for SL(2,R) and give some indication on the further application of the techniques to SL(2,C).

¹ Willian Ribeiro is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Category Theory, under the supervision of Professor Maria Manuel Clementino.

² Carla Dias is a post-doc researcher of CMUP, working at the University of Porto, in Dynamical Systems.

³ Diogo Lobo is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Numerical Analysis, under the supervision of Professor Sílvia Barbeiro.

⁴ Ahmed Elshafei is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in Geometry and Dynamics, under the supervision of Professors Helena Reis (University of Porto) and Ana Cristina Ferreira (University of Minho).