14h00-15h00
Speaker: Fernando Lucatelli Nunes*
Title: Homotopy Excision
Abstract: Eilenberg and Steenrod proved that ordinary homology is characterized by five axioms. Later, Atiyah, Hirzebruch and Whitehead observed that there are other families of functors that satisfy the four "most important" axioms. They defined the so called "generalized homology theories" (or "homology theories") which are examples of stable phenomena in homotopy theory. The concept of a prespectrum was first introduced by Elon Lages Lima in his PhD thesis to study some kinds of stable phenomena, such as Spanier-Whitehead duality and Stable Postnikov invariants. Later, Adams and Boardman proposed the first homotopy category of prespectrums. This was the starting point of the research field called stable homotopy theory. Nowadays stable homotopy categories are fundamental for studying all kind of stable phenomena in homotopy theory, including generalized homology theories and cohomology theories. The goal of the talk is to present some basic results of algebraic topology and give some elementary stable (and unstable) results of homotopy theory. To reach this goal, we shall introduce the concept of derived functors, homotopy colimits and assume two basic theorems: homotopy excision and long exact sequence of homotopy groups. At the end, we shall prove that every prespectrum represents a homology theory.
15h00-16h00
Speaker: Artur de Araujo**
Title: Homological Algebra
Abstract: We will explain the basic concepts of Homological Algebra (Cech cohomology, injective/projective resolutions, derived functors,) and show why, useful as they are, they have shortcomings for a general theory of cohomology. Time allowing, we'll give a hint of why derived categories make up for those defficiencies.
(*) Fernando Lucatelli Nunes is a PhD student of the UC|UP Joint PhD Program in Mathematics working in the University of Coimbra, under the supervision of professor Maria Manuel Clementino, in the area of Algebra, Logic and Topology.
(**) Artur de Araujo is a PhD student of the UC|UP Joint PhD Program in Mathematics working in the University of Porto, under the supervision of professor Peter Gothen, in the area of Geometry and Topology.
14h00 - 15h00
Speaker: Mathieu Duckerts-Antoine*
Title: On topological semi-abelian algebras
Abstract: In this talk, we will study some aspects of the categories of topological semi-abelian algebras. In particular, I will explain why these categories are homological. If the time allows it, I will also explain what is a torsion theory in a homological category and give several examples in the context under consideration.
References:
- F. Borceux and M. M. Clementino, Topological semi-abelian algebras, Advances in Mathematics, 190 (2005), 425-453
- D. Bourn and M. Gran, Torsion theories in homological categories, Journal of Algebra, 305 (2006), 18-47.
(*) Mathieu Duckerts-Antoine is a Post-doc of the University of Coimbra working in the area of “Algebra, Logic and Topology”.
14h00 - 15h00
Speaker: Célia Borlido*
Title: The k-word problem over DRG
Abstract. The study of finite semigroups has its roots in Theoretical Computer Science. In particular, in the mid nineteen seventies, Eilenberg [2] established the link between “varieties of rational languages”, which is an important object of study in Computer Science, and certain classes of finite semigroups, known as “pseudo varieties”. At the level of pseudovarieties, some problems arise naturally, one of them being the so-called “word problem”. Roughly speaking, it consists in deciding whether two expressions define the same element in every semigroup of a given pseudovariety.In this talk, we start by introducing some basic background on finite semigroups. Our first goal is to explain what the “k-word problem over DRG” is about. After that, based on some illustrative examples, we intend to give intuition on how to show that the referred problem is decidable. Our solution extends work of Almeida and Zeitoun [1] on the pseudovariety consisting of all “R-trivial semigroups”.
References
[1] J. Almeida and M. Zeitoun, An automata-theoretic approach to the word problem for ω-terms over R, Theoret. Comput. Sci. 370 (2007), no. 1-3, 131-169.
[2] S. Eilenberg, Automata, languages, and machines. Vol. B, Academic Press, New York - London, 1976.
(*) Célia Borlido is a student for the Joint PhD Program in Mathematics UC|UP working at the University of Porto in the area of “Semigroups, Automata and Languages” under the supervision of Prof. Jorge Almeida.
14h00 - 15h00
Speaker: Helena Gonçalves*
Title: Besov and Triebel-Lizorkin Spaces with Variable Exponents
Abstract. After an introduction on classical function spaces, we introduce spaces of Besov and Triebel-Lizorkin type Bsp,q(Rn) and Fsp,q(Rn) by Fourier analytical methods and present some properties of those spaces. Thereafter, we step up to the scale of function spaces with variable exponents, mainly the variable Lebesgue space Lp(·) (Rn). With this space in mind, we introduce two generalizations of Bsp,q(Rn) and Fsp,q(Rn): Besov and Triebel-Lizorkin spaces with variable smoothness and integrability Bs(·)p(·),q(·)(Rn) and Fs(·)p(·),q(·)(Rn), and 2-microlocal Besov and Triebel-Lizorkin spaces Bwp(·),q(·)(Rn) and Fwp(·),q(·)(Rn). We focus our attention on the last scale, where some properties will be considered
(*)Helena Gonçalves is working as a Research Assistant at Chemnitz University of Technology, Germany in the area of "Analysis" under the supervision of Prof. Henning Kempka.
14h00 - 15h00
Speaker: Fatemeh Esmaeili Taheri*
Title: The Spectral Inclusion Regions of Linear Pencils and Numerical Range
Abstract. Let A,B be n×n (complex) matrices. We are mainly interested in the study of the structure of the spectrum of a linear pencil, that is, a pencil of the formA−λB, where λ is a complex number. Our main purpose is to obtain spectral inclusion regions for the pencil based on numerical range. The numerical range of a linear pencil of a pair (A, B) is the set W(A,B) = {x∗(A−λB)x : x ∈ Cn,∥x∥ = 1,λ ∈ C}. The numerical range of linear pencils with hermitian coefficients was studied by some authors.
We are mainly interested in the study of the numerical range of a linear pencil, A − λB, when one of the matrices A or B is Hermitian and λ ∈ C. We characterize it for small dimensions in terms of certain algebraic curves. The results are illustrated by numerical examples.
(*) Fatemeh Esmaeili Taheri is a student for the Joint PhD Program in Mathematics UC|UP working at University of Coimbra in the area of "Algebra and Combinatorics" under the supervision of Prof. Natalia Bebiano.
12h00 - 13h00
Speaker: Juliane Fonseca de Oliveira*
Title: Dimensionality of pattern formation in Reaction Di usion systems
Abstract: In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. As a contrast, in this work we use the full 3-dimensionality of the problem to give a theoretical interpretation and possibly decide whether the pattern seen in Reaction Di usion systems naturally occur in either 2- or 3- dimension. For this purpose, we are concerned with functions in R3 that are invariant under the action of a crystallographic group and the symmetries of their projections into a function de ned on a plane.
14h30 - 15h30
Speaker: Jahed Naghipoor**
Title: Tuning polymeric and drug properties in a drug-eluting stent: a numerical study
Abstract: In recent years, mathematical modeling of cardiovascular drug delivery systems has become an effective tool to gain deeper insights in the cardiovascular diseases like atherosclerosis. In the case of the coronary biodegradable stent, it leads to a deeper understanding of drug release mechanisms from polymeric stent into the arterial wall. In this talk, a two-dimensional coupled nonlinear non-Fickian model for drug release from a biodegradable drug-eluting stent into the arterial wall is presented. The influence of porosity and degradation of the polymer as well as the dissolution rate of the drug are analyzed. Numerical simulations that illustrate the kind of dependence of drug pro les on these properties are included.
(*) Juliane Oliveira is a student for the Joint PhD Program in Mathematics UC|UP working at University of Porto in the area of "Bifurcation and Crystallography Theory" under the supervision of Prof. Isabel Labouriau and Prof. So a Castro.
(**) Jahed Naghipoor is a Post-doctoral fellow at Institute of Structural Mechanics (ISM), Bauhaus-Universitt Weimar, Germany and working in the area of "Numerical Analysis".
14h00 - 14h30
Speaker: Luigi Forcella*
Title: The electrostatic limit for the Zakharov system
Abstract. The Zakharov system describes the coupled dynamics of the electric field amplitude and the low frequency fluctuation of the ions in a unmagnetized or weakly magnetized plasma. This system couples Schrödinger-like and wave equations and in its physical derivation depends on a parameter $\alpha.$ Large value of $\alpha$ describes a plasma that is very hot, so it is meaningful to study the limit for the solutions to this system as $\alpha$ goes to infinity. In this talk we give rigorous mathematical result in this direction.
14h30 - 15h00
Speaker: Luigia Ripani **
Title: Analogies between optimal transport and minimal entropy
Abstract. The Schrödinger problem is an entropic minimization problem and it’s a regular approximation of the Monge-Kantorovich problem, at the core of the Optimal Transport theory.
In this talk I will first introduce the two problems, then I will describe some analogy between optimal transport and the Schrödinger problem such as a dual Kantorovich type formulation, the dynamical Benamou-Brenier type representation formula, as well as a characterization formula and some properties of the respective solutions.
Finally I will mention, as an application of these analogies, some contraction inequalities with respect to the entropic cost, instead of the classical Wasserstein distance.
(*) Luigi Forcella is a PhD student at Scuola Normale Superiore di Pisa, Italy and working in the area of "Analysis" under the supervision of Prof. Luigi Ambrosio.
(**) Luigia Ripani is a PhD student at Institut Camille Jordan - Université Claude Bernard Lyon 1, France and working in the area of "PDE, Analysis" under the supervision of Prof. Ivan Gentil and Prof. Christian Léonard.
15h00 - 16h00
Speaker: Saeid Alirezazadeh (Defended in May 2015)
Title: On Pseudovarieties of Forest Algebra
Area: Semigroups, Automata and Languages
Supervisor: Jorge Almeida
14h00 - 15h00
Speaker: Fátima Pina*
Title: Rolling Maps and Applications
Abstract. Rolling motions are rigid motions subject to holonomic and nonholonomic constraints. These motions appear associated to certain engineering areas, such as robotics and computer vision. Rolling maps are the mathematical tools to describe rolling motions.
In this talk, the concept of rolling map in a Riemannian framework will be presented together with some properties and applications. From the nonholonomic constraints of no-slip and no-twist the kinematic equations of motion can be derived. This will be done for the rolling of some particular manifolds that play an important role in applications. Explicit solutions of the kinematic equations will be derived when the manifolds roll along geodesics
(*) Fátima Pina is a student of the UC|UP Joint PhD Program in Mathematics working at University of Coimbra in the area of "Differential Geometry / Dynamical Systems" under the supervision of Prof. Fátima Silva Leite.
15h00 - 16h00
Speaker: Azizeh Nozad*
Title: Birationality of moduli spaces of twisted U(p,q)-Higgs bundles
Abstract. Let X be a Riemann surface of genus g greater or equal than 2. A twisted U(p,q)-Higgs bundle consists of a pair of holomorphic vector bundles on a Riemann surface, together with a pair of twisted maps between them. Here we study the variation with the parameter of the moduli space of twisted U(p,q)-Higgs bundles with a view to obtaining birationality results.
(*) Azizeh is a PhD student from the University of Porto. Her supervisor is Peter Gothen.
14h00 - 15h00
Speaker: Maryam Khaksar*
Title: Numerical Solution of Time-Dependent Maxwell’s equations in Anisotropic Materials for Modelling Light Scattering in Human Eye’s Structure
Abstract. Modelling light propagation in biological tissue has become an important research topic in biomedical optics with application in diverse fields as for example in ophthalmology. Waveguides with induced anisotropy may worth to be modeled as they could play a role in biological waveguides. For instance, there is a strong correlation between retinal nerve fiber layer thinning and reduction in tissue birefringence. Simulating the full complexity of the retina, in particular the variation of size and shape of each structure, distance between them and refractive indexes, re- quires a rigorous approach that can be achieved by solving Maxwell’s equations.
The finite difference time domain method (FDTD) was the first commonly used technique to find a time-domain solution of Maxwell’s curl equations on spatial grids. In the last decades, many papers in electromagnetic simula- tions have shown interest in discontinuous Galerkin time domain methods (DGTD) to solve Maxwell’s equations. The DGTD methods presents some well-known major advantages compared with more classical FDTD and finite element time domain methods. In opposition to the traditional FDTD methods, the DGTD method is a high-order accurate method that can easily handle complex geometries. Moreover, the method is suitable for parallel imple- mentation on modern multi-graphics processing units and local refinement strategies can be incorporated due to the ability of the method to deal with irregular meshes with hanging nodes and local spaces of different orders. This also represents an advantage when compared also to finite element methods. Despite the relevance of the anisotropic case, most of the formulation of the DGTD methods have been restricted to isotropic and dispersive materials.
In this work we consider models with anisotropic permittivity tensors which arise naturally in our application of interest. Here we combine the DGTD method (considering central and upwind fluxes) with a leap-frog type time integration, arriving at a fully-discrete explicit leap-frog DG scheme. We give a rigorous proof of the stability and the high-order convergency of the scheme. We provide some numerical tests which illustrate the theoretical results.
(*) Maryam Khaksar is a student for the Joint PhD Program in Mathematics UC | UP working at University of Coimbra in the area of "Numerical Analysis and Optimization" under the supervision of Prof. Adérito Araújo.