The first call for applications for the new edition of the PhD Program in Mathematics UC|UP (starting September 2018) will be open during January 02, 2018 - February 02, 2018
Events
Higgs bundles and the beauty of unexpected connections in Mathematics André Oliveira (CMUP) 15h30m, Room 108, UP Math. Dept. (Seminar)
November 28, 2017
The theory of vector bundles over algebraic curves is a topic usually classified as a subarea of Algebraic Geometry, hence standing at the crossroads of several key areas of Mathematics and Modern Physics. D. Mumford constructed their moduli spaces more than 50 years ago, and since then much progress has been achieved in the geometrical description of these spaces. A new chapter has begun in the late 80's with the introduction of Higgs bundles over curves by N. Hitchin. On one hand, these moduli spaces have a very rich geometric and topological structure which is far from being fully understood. On the other hand, they play a crucial role in many different, apparently unrelated, areas, including representation theory, hyperkähler geometry, Langlands duality, mirror symmetry and more. In this talk I will briefly review some of these aspects, and present some results and open problems.
Past Events
Nonlinear cross-diffusion models and applications Adérito Araújo (CMUC) 15h30m, Room 108, UP Math. Dept. (Seminar)
November 21, 2017
After the pioneering work of Keller and Segel in the 1970s, cross-diffusion models became very popular in biology, chemistry, ecology, population dynamics, economy and physics to emulate systems with multiple species. Meanwhile, the underlying mathematical theory has been developed in a synergistic way with applications and, in recent years, this topic became the focus of an intensive research within the mathematics community. From a mathematical point of view, cross-diffusion models are described by time-dependent partial differential equations of diffusion or reaction-diffusion type, where the diffusive part involves a nonlinear non-diagonal diffusion matrix. This leads to a strongly coupled system where the evolution of each dependent variable depends on itself and on the others in a way governed by the diffusion matrix. Cross-diffusion terms are nowadays widely used in reaction-diffusion equations encountered in models from mathematical biology and in various engineering applications. In this talk we review the basic model equations of such systems and give an overview of their mathematical analysis with an emphasis on pattern formation. Finally we present numerical simulations in the context of relevant applications, namely, the dynamics of crime behaviour and medical image processing.
Nonlinear diffusion for image processing Diogo de Castro Lobo (student) 15h30m, Room 108, UP Math. Dept. (Seminar)
November 14, 2017
The heat equation is probably the oldest and most investigated equation in image processing. We are going to discuss some approaches based on nonlinear diffusion for images denoising, namely the Perona-Malik and Complex Diffusion models. Numerical implementations and practical examples will be presented.
Evolutionary Game Dynamics Ricard Trinchet (student) 16h15m, Room 108, UP Math. Dept. (Seminar)
November 14, 2017
Evolutionary game dynamics is the application of population dynamical methods to game theory and has been introduced by evolutionary biologists in the 1970s. In this talk, we will review some of the basic definitions and results of this field and motivate one of the possible dynamics which can be associated to a game: the replicator dynamics. Namely, we will see what is the relation between the Nash Equilibria and Evolutionarily Stable Strategies of a given game and the stable equilibria of its associated dynamics. We will try to illustrate all of these concepts with some examples.
Coupling ultrasound propagation and drug transport: a numerical approach Daniela Jordão (student) 15h30m, Room 108, UP Math. Dept. (Seminar)
November 7, 2017
In this presentation we study a system of partial differential equations defined by a hyperbolic equation and a parabolic equation. The convective term of the parabolic equation depends on the solution and eventually on the gradient of the solution of the hyperbolic equation. This system arises in the mathematical modeling of several physical processes as for instance ultrasound enhanced drug delivery. In this case the propagation of the acoustic wave, which is described by a hyperbolic equation, induces an active drug transport that depends on the acoustic pressure. Consequently the drug diffusion process is governed by a hyperbolic and a convection-diffusion equation. Here, we propose a numerical method that allows us to compute second-order accurate approximations to the solution of the hyperbolic and the parabolic equation. The method can be seen as a fully discrete piecewise linear finite element method or as a finite difference method. The convergence rates for both approximations are unexpected. In fact we prove that the error for the approximation of the pressure and concentration is of second-order with respect to discrete versions of the H^1-norm and L^2-norm, respectively.
(student under supervision of Professor José Augusto Ferreira from UC)
Torsion Theories in Category Theory Leonardo Larizza (student) 16h15m, Room 108, UP Math. Dept. (Seminar)
November 7, 2017
In the category of finitely generated Abelian Groups we have particular objects called torsion-free groups and torsion groups and any other objects can be decomposed as a direct sum of those particular groups. This behavior gives rise to an interesting classification for the objects of this category. Starting from this particular known example, I will present the categorical generalization of this situation. We will introduce Torsion Theories for Abelian Categories, outlining some basic properties, and I will briefly describe the case of Hereditary Torsion Theories. If time permits I will conclude describing the construction of Derived Torsion Theories for arrow categories.
Highlights in Dynamical Systems Maria Carvalho (UPorto) 15h30m, Room 108, UP Math. Dept. (Seminar)
October 31, 2017
The aim of this talk is to present some relevant concepts and methods used in Dynamical Systems through a few interesting examples.
The spectral expansion approach to index transforms and connections with generalized convolutions and Lévy-type processes Rúben Sousa (student) 15h30m, Room 108, UP Math. Dept. (Seminar)
October 24, 2017
Index transforms are integral transforms whose kernel depends on the parameters of well-known special functions. In this talk we will explain how these transformations can be systematically constructed via the classical spectral theory for Sturm-Liouville differential operators, and we will see how these spectral methods yield a general connection between the index transforms and the associated parabolic partial differential equations. Furthermore, we will show that this theory can be used to construct Lévy processes with respect to generalized convolution operators. Applications to pricing problems in mathematical finance will also be discussed.
(student under supervision of Professors Semyon Yakubovich from UP and Manuel Guerra from UL)
Drug release enhanced by temperature: mathematical perspective Elisa Silveira (student) 16h15m, Room 108, UP Math. Dept. (Seminar)
October 24, 2017
The stimuli responsive polymers are long chain molecules that are sensitive to certain external stimuli - as temperature, chemical, light, electrical or magnetic field - and respond with observable or detectable changes in its properties.
In this talk, we study a quasilinear system of parabolic equations that can be used to simulate the drug transport enhanced by the temperature. In this case the first equation describes the temperature evolution and the drug transport is described by the second one. We observe that the diffusion coefficient of the second equation depends on the temperature.
We present fully discrete piecewise linear finite element methods that allow us to obtain numerical approximations for the temperature and concentration that present second order convergence with respect to a discrete L2 norm.
(student under supervision of Professors José Augusto Ferreira and Paula Oliveira, both from UC)
-------------------------------------------------------------------------------
11h00 - Peter Lombaers
-------------------------------------------------------------------------------
Integers and Ideals: There and Back Again
-------------------------------------------------------------------------------
Abstract: In number theory, when you try to solve an equation in a number field, it is often more convenient to work with ideals than with integers. This stems from the fact that ideals have unique factorization, but integers may not. I will explain the advantages and difficulties of this method using concrete examples.
Peter Lombaers is a PhD student of the Joint PhD Program UC—UP, working at the University of Porto, in Number Theory, under the supervision of Professor António José Machiavelo.
-------------------------------------------------------------------------------
13h30 - Willian Silva
-------------------------------------------------------------------------------
Introducing (T, V)-categories
-------------------------------------------------------------------------------
Abstract: In this seminar we introduce the concept of (T, V)-categories through its fundamental examples. In order to do so, we explore the concepts of monads and quantales, also with examples. We finish relating to the work on cartesian closed categories.
Willian Silva 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.
-------------------------------------------------------------------------------
14h30 -Mina Saee Bostanabad
-------------------------------------------------------------------------------
SOS versus SDSOS polynomial optimization
-------------------------------------------------------------------------------
Abstract: It is NP-hard to decide whether a polynomial is nonnegative, however, semidefinite programming can be used to decide whether a polynomial is a sum of squares of polynomials (SOS) in a practically efficient manner. In the context of polynomial optimization, it has become usual to substitute testing for nonnegativity with testing for SOS. Since there are much fewer sums of squares than nonnegative polynomials, we get only a relaxation and one that does not scale very well with the number of variables and degree of the polynomial. Recently, Ahmadi and Majumdar introduced a more scalable alternative to SOS optimization that they refer
to as scaled diagonally dominant sums of squares (SDSOS). The idea is searching for sums of squares of binomials, instead of general polynomials, which leads to a more scalable SOCP problem. In this presentation, we investigate the quantitative relationship between sums of squares of polynomials and scaled diagonally dominant polynomials. More specifically, we use techniques established by Blekherman to bound the ratio between the volume of the cones of these two classes of polynomials, showing that there are significantly less SDSOS polynomials than SOS polynomials. This drawback can be circumvented by using a recently introduced basis pursuit procedure of Ahmadi and Hall that iteratively changes the polynomial basis to a more suitable relaxation. We illustrate this by presenting a new application of this technique to an optimization problem.
Mina Saee Bostanabad is a PhD student of the Joint PhD Program UC—UP, working at the University of Coimbra, in Optimization, under the supervision of Professor João Eduardo Gouveia.
On Takens' Last Problem: times averages for heteroclinic attractors Alexandre Rodrigues (U. Porto) 15h30m, Room 108, UP Math. Dept.
October 17, 2017
In this talk, after introducing some technical preliminaries about the topic, I will discuss some properties of a persistent family of smooth ordinary differential equations exhibiting tangencies for a dense subset of parameters.
We use this to find dense subsets of parameter values such that the set of solutions with historic behaviour contains an open set. This provides an partial affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure.
The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. In addition, further open questions will be discussed.
This is a joint work with I. Labouriau (University of Porto).
Reference: I.S. Labouriau, A. A. P. Rodrigues, On Takens' Last Problem: tangencies and time averages near heteroclinic networks, Nonlinearity 30(5), 1876-1910, 2017
ADI Methods for heat equation Carla Jesus (student) 15h30m, Room 108, UP Math. Dept. (Seminar)
October 10, 2017
The production of metallic objects in three dimensions using selectivelaser melting (SLM) has become increasingly important in the manufacturing of small and complex molds. This technique uses digital information to produce a 3D metallic object through the action of a laser passing over a metal powder. In this talk, we will present a model that describes the heat transfer process during SLM. Due to not having access to the exact solution to most of these problems, we explore a numerical method to obtain an approximated solution to the problem. In particular, we will consider the alternating direction implicit (ADI) method that will be adjusted according to the different boundary conditions under consideration. We will present a convergence analysis of two bidimensional cases and conclude with an example of an ADI method in three dimensions.
Schur-Weyl duality: From the symmetric group to the general linear group João Santos (student) 16h15m, Room 108, UP Math. Dept. (Seminar)
October 10, 2017
The Schur-Weyl duality is a way of relating irreducible representations of algebras, if these algebras are under some special conditions. In this talk, we will start with the definition of a representation with a goal to talk about these conditions and the relations given by the Schur-Weyl duality. In the end, we will see the most common example of this duality, relating the representations of two very well-known groups, the symmetric group and the general linear group.
In this talk, we will introduce a new and a weaker version of the famous Faddeev-Takhtajan-Volkov algebra in Sl_2 case and a complete calculation towards it by getting help of a new defined Poisson bracket just by using the Cartan matrix A_2, and then by employing this structure we will extend it to the Sl_3 case and so on to Sl_n case by using the Cartan matrix A_n.
Network dynamical systems - synchrony and graph fibrations Oskar Weinberger (student) 16h15m, Room 108, UP Math. Dept. (Seminar)
October 3, 2017
Flows of vector fields (or any dynamical systems) associated to graphs/networks are often called network dynamical systems. These are systems that in some sense respect a graph structure, much like equivariant (symmetric) dynamical systems are systems that respect a group structure. Among various ways to formalise such systems, coupled cell systems is arguably the formalism which adopts the most algebraic and combinatorial perspective on the connection between networks and dynamics. In my talk I will try to motivate and give some basic definitions and results in this area. In particular, I want to present the notions of synchrony subspaces (invariant subspaces analogous to fixed point spaces for equivariant systems) and certain morphisms between graphs called graph fibrations. The main take away is that graph fibrations induce conjugacies on the level of vector fields. I will briefly sketch how this relates to ``hidden symmetries" of coupled cell systems, in the sense of the dynamics embedding as a synchrony subspace in the dynamics on another network with non-trivial self-fibrations. If time permits, I will also say a few words on some categorical aspects of the above.
PhD Defense Pier Giorgio Basile - Descent theory of (T,V)-categories: global-descent and étale-descent 15:00 - Sala dos Capelos, Univ. Coimbra
This Short Course is part of the Course on PDEs, Dynamical Systems and Geometry of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor José Miguel Urbano (jmurb@mat.uc.pt). It will be 10 hours long.
This Short Course is part of the Course on Statistics and Stochastic Processes of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Jorge Milhazes de Freitas (jmfreita@fc.up.pt). It will be 10 hours long.
This Short Course is part of the Course on Statistics and Stochastic Processes of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Carlos Tenreiro (tenreiro@mat.uc.pt). It will be 10 hours long.
This Short Course is part of the Course on PDEs, Dynamical Systems and Geometry of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Helena Reis (hreis@fep.up.pt). It will be 10 hours long.
PhD Defense Deividi Ricardo Pansera - On semisimple Hopf actions 11h00m - Room 031, Dep. Mathematics, Univ. Porto
June 22, 2017
PhD Defense Artur de Araújo - Representations of Generalized Quivers Room 107, Dep. Mathematics, Univ. Porto
June 2, 2017
Research Seminar Program (RSP) 2016/17 second session Room 2.4, DMat UC
June 1, 2017
-----------------------------------------------------------------------------------------
11h15 - Maryam Khaksar Ghalati
Two mathematical approaches in ophthalmology
-----------------------------------------------------------------------------------------
Lunch Break
-----------------------------------------------------------------------------------------
14h00 - Azizeh Nozad
Reducibility of nilpotent cone for G-Higgs bundle moduli space
-----------------------------------------------------------------------------------------
14h50 - Muhammad Ali Khan
Statistical instability for the contracting Lorenz flow
-----------------------------------------------------------------------------------------
Maryam Khaksar Ghalati is a former PhD student of the Joint PhD Program UC|UP, in the area of "Numerical Analysis and Optimization" under the supervision of professors Adérito Araújo and Sílvia Barbeiro.
Azizeh Nozad is a Postdoctoral fellow at Faculty of Science, University of Lisbon.
Muhammad Ali Khan is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in the area of "Dynamical Systems" under the supervision of professor José Ferreira Alves.
Orthogonal polynomials, generalized coherent pairs of measures, and Sobolev orthogonal polynomials Dieudonne Mbouna (student) 15h00m, UC Math. Dept. (Seminar)
May 31, 2017
In this talk we present results involving extensions of the concept of coherent pair of measures, introduced by A. Iserles, P. E. Koch, S. P. Norsett, and J. M. Sanz-Serna. In the first part of the talk we give a survey about basic results from the theory of orthogonal polynomials, (the inductive limit topology in the space of the polynomials, Favard’s theorem,the spectral theorem for orthogonal polynomials, Markov’s theorem, classical and semiclassical orthogonal polynomials, etc.). In the second part we present recent results involving the so-called (M,N)-coherent pairs of measures, and we give an application in the framework of Approximation Theory. We will use the above mentioned results to compute the Fourier-Sobolev coefficients appearing in the approximation of functions on appropriate Sobolev spaces.
The work for this presentation is concerned with the theory of estimation in manifolds. Here, we will not focus on the estimation of the set itself but on some of its topological properties. Based on a random sample of points from a manifold M, the goal is to achieve the exact description of topological properties of M, almost surely when the sample size increases to infinity. We will discuss two different models, depending on whether or not the observed points include noise.
An introduction to convexity properties and optimization methods for data analysis Lili Song (student) 15h40m, UC Math. Dept. (Seminar)
May 24, 2017
First-order methods have been studied for convex optimization for many decades. They typically converge slowly, at rates that are only sublinear without strong convexity. However, the fact they demand little at each iteration have made them highly popular in the last decade for problems involving heavy data, such as those in information processing. Second-order methods are also becoming popular these days but for nonconvex problems such as those arising in deep learning.
In this talk we will cover how convexity is exploited in first-order methods, what are the main classes of algorithms and their
convergence rates. We will only cover the fully deterministic case where problems and algorithms are both deterministic.
An introduction to Clarke nonsmooth calculus and its application in optimization Ali Moghanni Dehkhargani (student) 15h, UC Math. Dept. (Seminar)
May 17, 2017
In this seminar, we provide an overview of the main aspects of the Clarke nonsmooth calculus and cover a few of its applications to the type of optimization problems we have in mind.
The Clarke nonsmooth analysis has been widely applied due to the convenience of its generalized derivatives and to the convexity of its subdifferentials. We will cover the main concepts and results involved assuming that the functions are Lipschitz continuous.
Then we will briefly describe two applications to optimization algorithms, in one case by showing that Clarke generalized derivatives are nonnegative along limit directions of directional type methods.
In this seminar, our main purpose is to study the Mathematical Model of Glioma Growth and its Invasion. Cancer is a complex disease which leads to uncontrolled growth of abnormal cells, destruction of normal tissues and invasion of vital organs and gliomas are diffusive and highly invasive brain tumors. In our study, we will focus on the different aspects of cell tumor growth. We also describe the effect of chemotherapy on tumour growth. It is based on quantitative image analysis of histological sections of a human brain glioma and especially on cross-sectional area or volume measurements of serial CT images while the patient was undergoing chemotherapy. We also provide the Modelling Tumour Polyclonality and Cell Mutation.
Given an nxn nonnegative matrix M, its phaseless (signless) rank is the minimum rank of a complex (real) matrix whose entry-wise absolute value matrix is M.
We conjecture that, for any n, the boundary of nxn nonnegative matrices of phaseless rank less than n is contained in the union of coordinate planes and the set of nxn nonnegative matrices of signless rank less than n. In this session, we prove the n=3 case using two approaches: an analytical one and an algebraic one.
The spectral expansion approach to index transforms and connections with the theory of diffusion processes Ruben Sousa (student) 15h40m, UC Math. Dept. (Seminar)
May 3, 2017
Index transforms are integral transforms whose kernel depends on the parameters ofwell-known special functions. In this talk we will explain how these transformations can be systematically constructed via the classical spectral theory for Sturm-Liouville differential operators, and we will see how these spectral methods yield a general connection between the index transforms, the associated parabolic partial differential equations and the corresponding diffusion processes. Applications of these results to pricing problems in mathematical finance will also be discussed.
Second order approximations for kinetic and potential energies in Maxwell's wave equations Daniela Jordão (student) 17h10m, UC Math. Dept. (Seminar)
April 19, 2017
In this presentation we study a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. The numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L^{2}-norm.
Geodesic flow on manifolds of non-positive curvature Ahmed Elshafei (student) 16h30m, UC Math. Dept. (Seminar)
April 19, 2017
The study of geodesic flows on manifolds of non-positive curvature has been around for almost a century, it combines methods from geometry, dynamics, analysis and ergodic theory. This made the subject both more interesting and more difficult. In this seminar presentation the goal is to try to expose the subject using methods from Riemannian geometry and hyperbolic dynamics to show the interplay between these fields.
Various nonlinear evolutionary partial differential equations (coming from physics, fluid and solid mechanics, biology, chemistry etc.) can be viewed as geodesic equations or gradient flows on infinite-dimensional Riemannian structures.Understanding the underlying geometry of a PDE immediately provides certainconserved (in the case of geodesics) or dissipating (in the case of gradient flows) quantities which can be used, for instance, to get a priori bounds, to study asymptotic behaviour of solutions, or to develop numerical schemes. Further insights can be gained by observing that many of those Riemannian structures are related to the theory of optimal transport, which is enjoying a tremendous recent analytic progress. In my talk, I will try to introduce the students into this subject.
On the construction of complex algebraic surfaces Carlos Rito (UP) 15h40m, UC Math. Dept. (Seminar)
March 8, 2017
Surfaces of general type are far from being classified. Frequently the construction of a single example is a challenge, even for the ones with low values of the invariants. The most efficient methods of construction are classical: quotients by the action of a group and coverings. In this talk I will review these and I will explain some of my recent constructions.
Extremal behavior of chaotic dynamical systems Ana Cristina Moreira Freitas (UP) 15h, UC Math. Dept. (Seminar)
March 8, 2017
This talk is about the study of rare events for chaotic dynamical systems.We will address this issue by two approaches. One regards the existence of Extreme Value Laws (EVL) for stochastic processes obtained from dynamical systems, by evaluating an observable function (which achieves a global maximum at a single point of the phase space) along the orbits of the system. The other has to do with the phenomenon of recurrence to arbitrarily small sets, which is commonly known as Hitting Time Statistics (HTS). We will show the connection between the two approaches both in the absence and presence of clustering. Clustering means that the occurrence of rare events has a tendency to appear concentrated in time. The strength of the clustering is quantified by the Extremal Index (EI), which takes values between 0 and 1. The stronger the clustering, the closer the EI is to 0. No clustering means that the EI equals 1. Using the connection between EVL and HTS we associate the existence of an EI less than 1 to the occurrence of periodic phenomena.
PhD Defense Maryam Khaksar Ghalati - Numerical Analysis and Simulation of Discontinuous Galerkin Methods for Time-Domain Maxwell’s Equations Sala dos Capelos, Universidade de Coimbra
February 24, 2017
In this thesis we present a detailed analysis of a fully explicit leap-frog type discontinuous Galerkin (DG) method for the numerical discretization of the time-dependent Maxwell’s equations. The study comprehends models capable to deal with anisotropic materials and different types of boundary conditions. Despite the practical relevance of the anisotropic case, most of the numerical analysis present in the literature is restricted to isotropic materials. Motivated by a real application, in the present dissertation we consider a model which encompasses heterogeneous anisotropy, extending the existing theoretical results.
The DG formulation for the spatial discretization is developed in a general framework which unifies the study for different flux evaluation schemes. The leap-frog time integrator is applied to the semi-discrete DG formulation yielding to a fully explicit scheme. The main contribution of this thesis is to provide a rigorous proof of conditional stability and convergence of the scheme taking into account typical boundary conditions, either perfect electric, perfect magnetic or first order Silver- Müller absorbing boundary conditions and for different choices of numerical fluxes. The bounds
of the stability region point out not only the influence of the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. Under the stability condition, we prove that the scheme is convergent being of arbitrary high-order in space and second order in time. When Silver-Müller boundary conditions are considered we observe only first order convergence in time. To overcome this order reduction we propose a predictor-corrector time integrator which is also analysed in this dissertation.
We illustrate the stability and convergence properties of the various schemes with numerical tests. The numerical results of our simulations support the theoretical analysis developed along the thesis.
On a regularity conjecture for degenerate elliptic pdes José Miguel Urbano (UC) 15h, UC Math. Dept. (Seminar)
February 22, 2017
We establish a new oscillation estimate for solutions of nonlinear partial differential equations of degenerate elliptic type, which yields a precise control on the growth rate of solutions near their set of critical points. We then apply this new tool in the investigation of a longstanding conjecture which inquires whether solutions of the degenerate p-Poisson equation with a bounded source are locally of class C^{1,1/p-1}. This is a joint work with Eduardo Teixeira and Damião
Araújo.
Non Fickian diffusion in porous media José Augusto Ferreira (UC) 15h40m, UC Math. Dept. (Seminar)
February 22, 2017
Transport processes in porous media have being described by the classical convection-diffusion equation for the concentration coupled
with an elliptic equation for the pressure and Darcy’s law for the velocity. Despite the popularity of this model, gaps between
experimental data and simulation results were observed in different scenarios. To overcome the limitation of the traditional diffusion
models, integro-differential models involving an integral in time were proposed. In this case, Fick’s law that defines a relation between the mass flux and the gradient of the concentration is replaced by an equation where the mass flux is given by the past in time of the
gradient of the concentration. In this talk, we present some numerical analysis results for IBVPs defined by integro-differential equations for the concentration and elliptic equations for the pressure.
The combinatorics of noncrossing and nonnesting partitions Ricardo Mamede (UC) 15h, UC Math. Dept. (Seminar)
February 8, 2017
After a brief introduction to abstract Coxeter systems using a combinatorial approach based on words, we discuss recent developments on two combinatorial objects associated to the classical Coxeter groups: noncrossing and nonnesting partitions.
On the representation theory of some Noetherian algebras Paula Carvalho (UP) 15h40m, UC Math. Dept. (Seminar)
February 8, 2017
I will present some classes of Noetherian algebras that I have been studying regarding their representation theory; down-up algebras and skew polynomial rings, presenting past and present results. If time allows, another class of algebras will be introduced for which it is unknown when they are Noetherian.
PhD Defense Juliane Fonseca de Oliveira - Bifurcation of projected patterns UP Math. Dept.
January 21, 2017
This thesis is related to the study of pattern formation in symmetric physical systems. The purpose of this thesis is to discuss a possible model, namely the projection model, to explain the appearance and evolution of regular patterns in symmetric systems of equations.
Results found in Crystallography and Equivariant Bifurcation Theory are used extensively in our work. In particular, we provide a formalism of how the model of projection can be used and interpreted to understand experiments of reaction-diffusion systems.
We construct a scenario where systems of symmetric PDEs posed in different dimensions can be compared as projection. In particular, we show how we can overcome the boundary conditions imposed by the problems.
We prove a correspondence between irreducible representations and fixed points subspaces, given by the action of a (n + 1)-dimensional crystallographic group, with the action of its projection on lower dimension. Such results are the first step to compare typical structures in dimension (n + 1), after projection, and the typical solutions of the posed problem in
dimension n.
We show that complex structures, as the black-eye pattern, obtained both as projection and as an experimental observation in CIMA reactions are the same. In particular, we believe that the projection model provides extra information to the study of pattern forming system, since it allows us to embed the original problem into one with more symmetry.
Castelnuovo-Mumford regularity and graph invariants Jorge Neves (UC) 15h40m, UC Math. Dept. (Seminar)
December 16, 2016
The work of R. Stanley exposed a strong link between the theory of ideals of a polynomial ring generated by monomials and the theory of simplicial complexes. Recently, other bridges have been established between combinatorial structures and classes of ideals not necessarily generated by monomials. The Castelnuovo-Mumford regularity of a module over the polynomial ring is a basic invariant of the module. It is related with its free resolution and is, indeed, a measure of its complexity. It is natural to expect that, whenever a link between a class of ideals of a polynomial ring and a certain type of combinatorial structure exists, the Castelnuovo-Mumford regularity will translate into a meaningful combinatorial invariant. The focus of this talk will be on a recent joint work with A. Macchia (CMUC), Maria Vaz Pinto (CAMGSD, Lisbon) and Rafael Villarreal (Cinvestav, Mexico) in which the Castelnuovo-Mumford regularity of a particular class of ideals associated to graphs is equated with the combinatorics of the graph.
On Taken's Last Problem: times averages for heteroclinic attractors Alexandre A. Rodrigues (UP) 15h, UC Math. Dept. (Seminar)
December 2, 2016
In this talk, after giving a technical overview about the topic, I will discuss some properties of a robust family of smooth ordinary differential equations exhibiting tangencies for a dense subset of parameters. We use this to find dense subsets of parameter values such that the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. This is a joint work with I. Labouriau (University of Porto).
Singularities of vector fields and the dimension of groups of holomorphic diffeomorphisms Helena Reis (UP) 15h40m, UC Math. Dept. (Seminar)
December 2, 2016
The group of holomorphic diffeomorphisms, Aut(M), of a compact complex manifold M is a Lie group of finite dimension. To provide bounds for the dimension of these groups is a classical problem in complex analysis. It is well known that the dimension of Aut(M) cannot be bounded in terms of the dimension of M solely. However several important problems arise once specific constraints are imposed on the manifold M. For example, the case of homogeneous manifolds has been intensively studied in connection with which is sometimes called Remmert conjecture. Another interesting situation corresponds to the case of algebraic manifolds whose Picard group is Z (the Hwang-Mok problem). There is an evident relation between bounds for the order of the zeros of holomorphic vector fields on M and bounds for the dimension of Aut(M). In this sense, results on singularities of holomorphic vector fields (particularly on specific questions concerning the extent to which a singularity of a vector field can be degenerate) have implications to problem mentioned above. We will discuss some concepts and recent results in this direction.
Research Seminar Program (RSP) 2016/17 first session 11h, Room M005, UP Math. Dept. (Research Seminar Program)
November 25, 2016
MORNING SESSION - Algebra, Combinatorics and Number Theory
-------------------------------------------------------------------------------
11:00h Antonio Macchia
-------------------------------------------------------------------------------
Title Proper divisibility as a partially ordered set (*)
-------------------------------------------------------------------------------
Abstract We define the order relation given by the proper divisibility of monomials, inspired by the definition of the Buchberger graph of a monomial ideal. From this order relation we obtain a new class of posets. Surprisingly, the order complexes of these posets are homologically non-trivial. We prove that these posets are dual CL-shellable, we completely describe their homology (with integer coefficients) and we compute their Euler characteristic. Moreover this order relation gives the first example of a dual CL-shellable poset that is not CL-shellable.
(*) joint work with Davide Bolognini, Emanuele Ventura and Volkmar Welker
------------------------------------------------------------------------------------
12:00h Alberto José Hernandez Alvarado
-------------------------------------------------------------------------------------
Title The Quotient Module, Coring Depth and Factorisation Algebras
-------------------------------------------------------------------------------------
Abstract In this conference, I will be reviewing the main aspects of my thesis dissertation. I will introduce the notion of depth of a ring extension $B\subseteqA$ and give several examples as well as important results of recent years. I will then consider a finite dimensional Hopf algebra extension $R \subseteq H$ and its quotient module $Q := H/R^+H$ and show that the depth of such an extension is intrinsically connected to the representation ring of $H, A(H)$. In particular, we will see that finite depth of the extension is equivalent to the quotient module $Q$ being algebraic in $A(H)$. Next, I will introduce entwining structures and use them to show that a certain extension of crossed product algebras is a Galois coring and use that to give a theoretical explanation for a result of S. Danz (2011). Finally, I will discuss factorisation algebras and their roll in depth, in particular a result on the depth of a Hopf algebra $H$ in its generalised factorised smash product with $Q^{*op}$.
----------------------------------------------------------------------------
***LUNCH BREAK***
----------------------------------------------------------------------------
AFTERNOON SESSION - Algebra, Logic and Topology
----------------------------------------------------------------------------
14:30h Pier Giorgio Basile
----------------------------------------------------------------------------
Title A lax version of the Eilenberg-Moore adjunction (*)
---------------------------------------------------------------------------
Abstract In Category Theory there is a well developed theory of monads, proved to be very useful for 1-dimensional universal algebra and beyond. The relation between adjunctions and monads was first noticed by Huber (Homotopy Theory in General Categories): every adjunction gives rise to a monad. Then, Eilenberg, Moore and Kleisli realized that every monad comes from an adjunction. In particular, Eilenberg and Moore (Adjoint Functors and Triples) realized that, for every monad T, there is a terminal adjunction (called Eilenberg-Moore adjunction) which gives rise to T. Category Theory can be also developed in a 2-dimensional case, that is, considering not only morphisms between objects but also morphisms (usually called 2-cells) between morphisms themselves. Thereby, one can study lax versions of the theory of monads. In the pseudo version, that is when we replace commutative diagrams by coherent invertible 2-cells, the relation between biadjunctions and pseudomonads has been investigated by F. Lucatelli Nunes in the paper On Biadjoint Triangles as a consequence of the coherent approach to pseudomonads of S. Lack. The next step consists of studying the lax notion of monads, in which the associativity and identity works only up to coherent (not necessarily invertible) 2-cells. In this talk we present a work in progress where we try to generalize to the lax-context the classical result of Eilenberg-Moore. For this purpose, having in mind the notion of lax extension of monads introduced and studied in the context of Monoidal Topology (Metric, topology and multicategory: a common approach - M.M. Clementino and W. Tholen), we use a generalization of Gray's lax-adjunction (see the monograph Formal Category Theory). Then, we show some steps of the construction leading to the positive answer.
(*) joint work with Fernando Lucatelli Nunes
--------------------------------------------------------------------------------------
15:30h Fernando Lucatelli Nunes
---------------------------------------------------------------------------------------
Title Kan construction of adjunctions
----------------------------------------------------------------------------------------
Abstract I will talk about a basic procedure of constructing adjunctions, sometimes called Kan construction/adjunction. In the first part of the talk, I will construct abstractly such adjunctions via colimits. In the second part, we give some elementary examples: fundamental groupoid, sheaves, etc. We assume elementary knowledge of basic category theory (definition of categories, colimits and Yoneda embedding).
---------------------------------------------------------------------------------------
ABOUT THE SPEAKERS
--------------------------------------------------------------------------------------
1. Antonio Macchia is a PosDoc researcher working at the University of Coimbra in the area of "Algebra and Combinatorics".
2. Alberto José Hernandez Alvarado is a former PhD student of the Joint PhD Program UC|UP working at the University of Porto in the area of "Algebra, Combinatorics and Number Theory" under the supervision of professor Lars Kadison.
3. Pier Giorgio Basile is a PhD student of the Joint PhD Program UC|UP working at the University of Coimbra in the area of "Algebra, Logic and Topology" under the supervision of professor Maria Manuel Clementino.
4. Fernando Lucatelli Nunes is a PhD student of the Joint PhD Program UC|UP working at the University of Coimbra in the area of "Algebra, Logic and Topology" under the supervision of professor Maria Manuel Clementino.
A meeting between Sobolev, Bessel, Hölder, Lorentz and Karamata in order to discuss optimal embeddings and some open problems Júlio Neves (UC) 15h, UC Math. Dept. (Seminar)
November 18, 2016
In this talk we give a short survey on the results of embeddings of Sobolev type spaces into Hölder type spaces, including as well the famous result of Brézis and Wainger about almost Lipschitz continuity of elements of the Sobolev space with super-critical exponent of smoothness. Afterwards, we discuss different sharp inequalities that will allow to derive necessary and sufficient conditions for embeddings of Sobolev type spaces modelled upon rearrangement invariant Banach function spaces X into generalized Hölder spaces. Some open problems will be presented as well.
Smoothness Morrey spaces and their envelopes Susana Moura (UC) 15h40m, UC Math. Dept. (Seminar)
November 18, 2016
The classical Morrey space {\cal M}_{u,p}(R^n), 0 < p \leq u < \infty, is defined to be the set of all locally p-integrable functions f such that \|f \mid {\cal M}_{u,p}(R^n)\| :=\, \sup_{x\in R^n, R>0} R^{\frac{n}{u}-\frac{n}{p}} \left(\int_{B(x,R)} |f(y)|^p \;\mathrm{d} y \right)^{\frac{1}{p}} is finite, where B(x,R) denotes the ball centered at x\in R^n with radius R>0. They are part of the wider class of Morrey-Campanato spaces and can be considered as an extension of the scale of L_p spaces. Built upon these basic spaces Besov-Morrey spaces \cal N^{s}_{u,p,q} and Triebel-Lizorkin-Morrey spaces \cal E^s_{u,p,q} attracted some attention in the last years, in particular in connection with Navier-Stokes equations. Closely related to theses scales are the spaces of Besov type B_{p,q}^{s,\tau} and Triebel-Lizorkin type F_{p,q}^{s,\tau}, \tau\geq 0, which coincide with their classical counterparts when \tau=0. We present a survey on such different scales of smoothness spaces of Morrey type. We also introduce the general concept of growth and continuity envelope of a function space and determine the envelopes of the above mentioned spaces. In some cases a specific behaviour appears which is different from the "classical" situation in Besov or Triebel-Lizorkin spaces. This talk is based on joint work with D.D. Haroske (Jena), L. Skrzypczak (Poznan), D. Yang (Beijing) and W. Yuan (Beijing).
Ring theoretical and combinatorial aspects of representation theory Samuel Lopes (UP) 15h, UC Math. Dept. (Seminar)
November 4, 2016
I will discuss some of my research interests in relation to representation theory of infinite-dimensional algebras, highlighting the interplay with topics in ring theory and combinatorics.
A successful pair: symbolic dynamics and (pro)finite semigroups Alfredo Costa (UC) 15h40m, UC Math. Dept. (Seminar)
November 4, 2016
My research has been mostly about exploring natural links between the field of symbolic dynamics and the theory of finite and profinite semigroups, with applications in both directions. The aim of this talk is to give examples of relevant results and open problems stemming from this line of research.
Diverse paths in network optimisation Marta Pascoal (UC) 15h, UC Math. Dept. (Seminar)
October 28, 2016
A path in a graph is a sequence of vertices linked by graph edges. The determination of an optimal path with respect to a given objective function has led to classical network optimization problems. However, for many applications, it is useful to know an alternative backup solution,which can replace the optimal path if needed and, in some cases, which is reasonably diverse (or "different") from the latter. In this talk we will discuss applications which justify finding diverse
paths, approaches that have been used to model and solve such problems, as well as open questions within this line of research.
Positive semidefinite rank and representations of polytopes João Gouveia (UC) 15h40m, UC Math. Dept. (Seminar)
October 28, 2016
Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices A_i,B_j of size k×k such that M_{ij}=trace(A_i,B_j). The psd rank was recently introduced, and has has many appealing interpretations, capturing some geometrical aspects of the power and limitations of semidefinite programming. In this talk, we will briefly cover basic properties and open questions on this quantity, and proceed to use some of these results to provide a complete characterization of polytopes in R^4 that can be represented as economically as possible by means of a semidefinite program.
A fractional diffusion model with resetting Ercília Sousa (UC) 15h, UC Math. Dept. (Seminar)
October 14, 2016
We consider a fractional partial differential equation that describes the diffusive motion of a particle, performing a random walk with Lévy distributed jump lengths, on one dimension with an initial position x0. The particle is additionally subject to a resetting dynamics, whereby its diffusive motion is interrupted at random times and is reset to x0. A numerical method is presented for this diffusive problem with resetting. The influence of resetting on the solutions is analysed and physical quantities such as pseudo second order moments will be discussed. Some comments about what happens in the presence of boundaries will be also included. This talk is based on joint work with Amal K. Das from Dalhousie University (Canada).
When assigned with the task of image reconstruction, the first challenge one faces is the derivation of a truthful model for both the information we want to extract and the data. The natural question arises: how can we make our model adaptive to the given data? Diffusion processes are commonly used in image processing in order to remove noise. The main idea is that if one pixel is affected by noise, than the noise should be diffused among the neighboring pixels in order to smooth the region. In this way, proper diffusion partial differential equations (PDE) have been considered to achieve this end. The choice of the diffusion parameter plays a very important role for the purpose of denoising. Roughly speaking, one wants to allow diffusion on homogeneous areas affected only by noise and to forbid diffusion on edges to preserve features of the original denoised image. Consequently, the efficient models exhibit solution-dependent adaptivities in form of nonlinearities or non-smooth terms in the PDE. After a critical discussion of models based on nonlinear diffusion, we will turn towards the second modelling strategybased on nonlinear complex diffusion, which is suggested taking into account its advantages with regard to edge preservation, speckle filtering capabilities and potential to recover the original (uncorrupted) signal. The models will be compared by means of illustrative practical examples. Some applications of the complex diffusion filter, namely for despeckling optical coherence tomograms from the human retina, will be highlighted.
PhD Defense Alberto Hernandez - The Quotient Module, Coring Depth and Factorisation Algebras Room 107, Dep. Mathematics, Univ. Porto
September 30, 2016
In Boltje-Danz-Kulshammer, J. of Algebra, 03-019, (2011) it was shown that for a finite group algebra extension over any commutative ring the depth is always finite. Later, in Kadison, J. Pure and App Algebra, 218: 367-380, (2014) depth of such a subgroup pair was obtained by computing on the permutation module of the left or right cosets. This holds more generally for finite dimensional Hopf algebra extensions. We show that the depth of the Hopf subalgebra pair R H is related to the depth of its generalised permutation module Q := H=R+H in its module category. Furthermore we establish that the pair is finite depth if and only if Q is an algebraic module in the representation ring of either H or R. A necessary condition for finite depth is provided as the stabilisation of the descending chain of annihilators of the tensor powers of Q. In Danz, Comm of Alg, 39:5, 1635- 1645, (2011) the author provides a formula for the depth of a complex twisted group extension of the symmetric groups of order n and n + 1. We provide a general setting in which the depth of the complex crossed product algebra extension D#H D#G of a group pair H < G is always less or equal than the depth of the algebra extension kH < kG. For this we use the entwining of a left H-module algebra A with an H-module coalgebra C, over a Hopf algebra H. We show that such a structure is a Galois coring when A = H and C = Q for a finite dimensional Hopf algebra extension R H, and that this extends to the crossed product algebra extension of a finite group extension. We also provide a setting for the depth of factorisation algebras and provide a formula for the value of depth of a subalgebra of a factorisation algebra in terms of its module depth.
The Short Course is part of the Course on Algebra and Categories of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Manuel Delgado (mdelgado@fc.up.pt). It will be 7.5 hours long (1.5h per day).
Short Course on Categorical Algebra Lecturer: Professor Marino Gran, Université Catholique de Louvain Department of Mathematics, University of Coimbra
September 12, 2016 - September 16, 2016
The Short Course is part of the Course on Algebra and Categories of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Maria Manuel Clementino (mmc@mat.uc.pt). It will be 7.5 hours long (1.5h per day).
The Short Course is part of the Course on Dynamics and Optimization of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Manuela Aguiar (maguiar@fep.up.pt). It will be 7.5 hours long (1.5h per day).
The Short Course is part of the Course on Dynamics and Optimization of the Summer School of the UC|UP Joint PhD Program in Mathematics. It is organized by Professor Joao Gouveia (jgouveia@mat.uc.pt). It will be 7.5 hours long (1.5h per day).
PhD Defense Fatemeh Esmaeili Taheri - Numerical ranges of linear pencils Sala dos Capelos, Universidade de Coimbra
July 27, 2016
In recent years, the numerical range of finite matrices and linear operators has been intensively investigated. In this thesis, the concept of numerical range of a linear pencil is discussed, and the geometry of the numerical range is investigated by using techniques of plane algebraic geometry. The classification of all possible boundary generating curves of the numerical range of pencils of two-by-two and three-by-three matrices is explicitly given, when one of the matrices is hermitian. The numerical range of linear pencils with hermitian coefficients has been studied by some authors. We have characterized the numerical range of self-adjoint linear pencils, pointing out and correcting an error reproduced in the literature. For the case n=2 , the boundary generating curves of numerical range are conics. Geometrical proofs of the Elliptical Range Theorem, Parabolical Range Theorem and Hyperbolical Range Theorem, have been obtained in an unified way. We remark that the two-by-two case is particularly important, since for a pencil of arbitrary dimension the compression to the bidimensional case gives us information on the general n by n case. For n = 3, we obtained the classification of all possible boundary generating curves of the numerical range, distinguishing the case of one of the matrices being positive (negative) definite, semidefinite and indefinite. All the possible boundary generating curves of the numerical range of three-by-three linear pencils can be completely described by using Newton’s classification of cubic curves. The obtaining results are illustrated by numerical examples.
Count time series modeling has drawn much attention and considerable development in the recent decades since many of the observed stochastic systems in various contexts and scientific fields are driven by such kind of data. The first modelings, with linear character and essentially inspired by the classic ARMA models, are proved to be insufficient to give an adequate answer for some empirical characteristics, also observed in this type of data, such as the conditional heteroscedasticity. In order
to capture such kind of characteristics several models for nonnegative integer-valued time series arise in literature inspired by the classic GARCH model of Bollerslev [10], among which is highlighted the integer-valued GARCH model with conditional Poisson distribution (briefly INGARCH model), proposed in 2006 by Ferland, Latour and Oraichi [25].
The aim of this thesis is to introduce and analyze a new class of integer-valued models having an analogous evolution as considered in [25] for the conditional mean, but with an associated comprehensive family of conditional distributions, namely the family of infinitely divisible discrete laws with support in N0, inflated (or not) in zero. So, we consider a family of conditional distributions that in its more general form can be interpreted as a mixture of a Dirac law at zero with any discrete
infinitely divisible law, whose specification is made by means of the corresponding characteristic function. Taking into account the equivalence, in the set of the discrete laws with support N0, between infinitely divisible and compound Poisson distributions, this new model is designated as zero-inflated compound Poisson integer-valued GARCH model (briefly ZICP-INGARCH model).
We point out that the model is not limited to a specific conditional distribution; moreover, this model has as main advantage to unify and enlarge substantially the family of integer-valued stochastic processes. It is stressed that it is possible to present new models with conditional distributions with interest in practical applications as, in particular, the zero-inflated geometric Poisson INGARCH and the zero-inflated Neyman type-A INGARCH models, and also recover recent contributions such as the
(zero-inflated) negative binomial INGARCH [81, 84], (zero-inflated) INGARCH [25, 84] and (zeroinflated) generalized Poisson INGARCH [52, 82] models. In addition to having the ability to describe different distributional behaviors and consequently, different kinds of conditional heteroscedasticity, the ZICP-INGARCH model is able to incorporate simultaneously other stylized facts that have been recorded in real count data, in particular overdispersion and high occurrence of zeros.
The probabilistic analysis of these models, concerning in particular the development of necessary and sufficient conditions of different kinds of stationarity (first-order, weak and strict) as well as the property of ergodicity and also the existence of higher order moments, is the main goal of this study. It is still derived estimates for the parameters of the model using a two-step approach which is based on the conditional least squares and moments methods.
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 pre spectrums 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.
(*) Fernando Lucatelli Nunes is a PhD student for the Joint PhD Program in Mathematics UC|UP working at the University of Coimbra in the area of Algebra, Logic and Topology under the supervision of professor Maria Manuel Clementino.
Homological Algebra Artur de Araujo (UP student) 15h, Room 5.5, UC Math. Dept. (Research Seminar Program)
June 28, 2016
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.
(*) Artur de Araujo is a PhD student for the Joint PhD Program in Mathematics UC|UP working at the University of Porto in the area of Geometry and Topology under the supervision of professor Peter Gothen.
Topological spaces as algebras Pier Giorgio Basile (UC student) 14h, Room M004, UP Math. Dept. (Research Seminar Program)
May 19, 2016
Monads, or triples, and the algebras they define, proved to be very important in several fields of mathematics. For example, they allow us to see the algebraic nature owned by topological spaces. This fact, for what concerns compact Hausdorff spaces, is known since 1969 (see [4]). By suitably weakening the axioms of the algebras, M. Barr in 1970 (see [1]) generalized the result to all topological spaces. This step represented a starting point of the new theory of lax (Eilenberg-Moore) algebras, introduced about thirty years later by M.M. Clementino, D. Hofmann and W. Tholen (see [2] and [3]).
[1] M. Barr, Relational Algebras, in: Reports of the Midwest Category Seminar, IV, pp 39-55, Lecture Notes in Mathematics 137, Springer, Berlin (1970).
[2] M.M. Clementino and D. Hofmann, Topological features of lax algebras, Appl. Categ. Structures 11 (2003), 267-286.
[3] M.M. Clementino and W. Tholen, Metric, topology and multicategory - a common approach, J. Pure Appl. Algebra 179 (2003), 13-47.
[4] E. Manes, A triple theoretic construction of compact algebras, 1969 Sem. on Triples and Categorical Homology Theory (ETH Zurich 1966/67), 91118, Lecture Notes in Mathematics, Springer, Berlin.
(*) Pier Giorgio Basile is a student for the Joint PhD Program in Mathematics UC|UP working at University of Coimbra in the area of "Algebra, Logic and Topology" under the supervision of Prof. Maria M. Clementino.
PhD Defense Célia Borlido - The word problem and some reducibility properties for pseudo-varieties of the form DRH Univ. Porto
April 27, 2016
On topological semi-abelian algebras Mathieu Duckerts-Antoine (CMUC, Portugal) 14h, Room 5.5, Department of Mathematics, University of Coimbra
April 7, 2016
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.
The k-word problem over DRG Célia Borlido (UP student) 14h, Room 5.5, UC Math. Dept. (Research Seminar Program)
March 10, 2016
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 ?-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.
PhD Defense Azizeh Nozad - Hitchin Pairs for indefinite unitary Group Room 107, Dep. Mathematics, Univ. Porto
February 26, 2016
Besov and Triebel-Lizorkin Spaces with Variable Exponents Helena Gonçalves (Technische Univ. Chemnitz, Germany) 14h, Room M030, UP Math. Dept. (Research Seminar Program)
February 18, 2016
After an introduction on classical function spaces, we introduce spaces of Besov and Triebel-Lizorkin type B^{s}_{p,q}(R^{n}) and F^{s}_{p,q}(R^{n}) 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 L_{p(.)}(R^{n}). With this space in mind, we introduce two generalizations of B^{s}_{p,q}(R^{n}) and F^{s}_{p,q}(R^{n}): Besov and Triebel-Lizorkin spaces with variable smoothness and integrability B^{s(.)}_{p(.),q(.)}(R^{n}) and F^{s(.)}_{p(.),q(.)}(R^{n}), and 2-microlocal Besov and Triebel-Lizorkin spaces B^{w(.)}_{p(.),q(.)}(R^{n}) and F^{w(.)}_{p(.),q(.)}(R^{n}). 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.
The spectral inclusion regions of linear pencils and numerical range Fatemeh Esmaeili Taheri (UC student) 14h, Room M030, UP Math. Dept. (Research Seminar Program)
January 28, 2016
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 form A-?B, 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 \in Cn, |x| = 1, λ \in 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 \in 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 of 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. Natália Bebiano.
An Overview on the Dimension of Projected Patterns in Reaction-Diffusion Systems Juliane Fonseca de Oliveira (UP student) 12h, Room 5.5, UC Math. Dept. (Research Seminar Program)
December 17, 2015
In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modeled 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 Diffusion 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 defined on a plane.
Tuning Polymeric and Drug Properties in a Drug-Eluting Stent: A Numerical Study Jahed Naghipoor (Institute of Structural Mechanics (ISM), Bauhaus-Universität Weimar) 14h30, Room 5.5, UC Math. Dept. (Research Seminar Program)
December 17, 2015
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 profiles on these properties are included.
The Electrostatic Limit for the Zakharov System Luigi Forcella (Scuola Normale Superiore di Pisa, Italy) 14h, Room M031, UP Math. Dept. (Research Seminar Program)
November 19, 2015
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.
The speaker 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.
Analogies between Optimal Transport and Minimal Entropy Luigia Ripani (Univ. Claude Bernard Lyon 1, France) 14h30, Room M031, UP Math. Dept. (Research Seminar Program)
November 19, 2015
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.
The speaker 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.
On Pseudovarieties of Forest Algebras Saeid Alirezazadeh (UP student) 15h, Room 5.5, UC Math. Dept. (Research Seminar Program)
October 27, 2015
Forest algebras are used in the theory of formal languages. They consist of two monoids, the horizontal one H and the vertical one V , with an action of V on H, and a complementary axiom of faithfulness. The main example is the forest algebras of plane forests and contexts, that is to say plane forests with a deleted leaf, which is the free object of the theory. In the study of forest algebras one of the main difficulty is how to handle the faithfulness property. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products.
A profinite algebra is defined to be a projective limit of a projective system of finite algebras. We tried to adapt in this context some of the results in the theory of semigroups, specially the results on relatively free profinite semigroups which have been shown to be an important tool in the study of pseudovarieties of semigroups.
We recall definition of forest algebras and state several basic results concerning definition and properties of forest algebras and the free forest algebra which are used later on.
Bojanczyk and Walukiewicz in ["Forest algebras", Logic and Automata, 2008, pp. 107-131] defined the syntactic forest algebra over a forest language. We define a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language, which is used in the proof of an analog of Hunter's Lemma ["Certain finitely generated compact zero-dimensional semigroups", 1988, pp. 265-270]. The new version of syntactic congruence is the natural extension of the syntactic congruence for monoids in case of forest algebras. We show that for an inverse zero action subset and a forest language which is the intersection of the inverse zero action subset with the horizontal monoid, the two versions of syntactic congruences coincide.
Almeida in ["Profinite semigroups and applications", Structural Theory of Automata, Semigroups, and Universal Algebra, 2005, pp 1-45] established some results on metric semigroups. We adapted some of his results to the context of forest algebras. We define on the free forest algebra a pseudo-ultrametric associated with a pseudovariety of forest algebras. We show that the analog of Hunter's Lemma holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite. We show an analog of Reiterman's Theorem ["The Birkhoff theorem for finite algebras", 1982, pp. 1-10], which is based on a study of the structure profinite forest algebras.
Rolling Maps and Applications Maria de Fátima Alves de Pina (UC student) 14h, Room M031, UP Math. Dept. (Research Seminar Program)
October 22, 2015
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.
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.
Numerical Solution of Time-Dependent Maxwells Equations in Anisotropic Materials for Modelling Light Scattering in Human Eyes Structure Maryam Khaksar Ghalati (UC student) 14h, Room M031, UP Math. Dept. (Research Seminar Program)
September 24, 2015
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.
Research and Events
Defended Theses
Descent theory of (T,V)-categories: global-descent and étale-descent