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).
30th of November, 2018 - Room FC1 007, DMat of University of Porto
10h30 - 11h00
Speaker: Maria Elisa Silveira¹
Title: Drug release enhanced by temperature: an accurate discrete model for solutions in $H^3$.
Abstract: In this talk we consider the coupling between two quasilinear diffusion equations: the diffusion coefficient of the first equation depends on its solution and the diffusion and convective coefficient of the second equation depend on the solution of the first one. This system can be used to describe the drug evolution in a target tissue when the drug transport is enhanced by heat. We study, from an analytical and a numerical viewpoints, the coupling of the heat equation with the drug diffusion equation. A fully discrete piecewise linear finite method is proposed for this system and we establish its stability. Assuming that the heat and the concentration are in $H^3$ we prove that the introduced method is second order convergent. Numerical experiments illustrating the theoretical results are also included.
11h00 - 11h30
Speaker: Dieudonné Mbouna²
Title: (M;N)-Coherent pairs of order (m; k) and Sobolev Orthogonal Polynomials on the non-uniform lattice.
11h30 - 12h00
Speaker: Nuno Picado³
Title: Deciding about the emptiness of the interior of a manifold based on a sample of points.
Abstract: In this talk, I will present some results about a method to decide, based on a sample of points from a manifold $\mathcal{M}$, whether its interior is empty or not. This method was introduced in the article "Stochastic detection of some topological and geometric feature" by Cuevas et al. for independent random variables and sufficiently smooth manifolds. The decision process is based on an estimator constructed using balls centered at the points of the sample. We will show that the approach still holds if the sample satisfies a suitable dependence assumption.
After this, the talk will be focused on the construction of models with the type of dependence needed for the results.
¹ Maria Elisa Silveira 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 José Augusto Ferreira.
² Dieudonné Mbouna is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Analysis, under the supervision of Professor José Carlos Petronilho.
³ Nuno Picado is a PhD student of the Joint PhD Program UC|UP, working at the University of Coimbra, in Probability and Statistics, under the supervision of Professor Paulo Eduardo Oliveira.