July 15, 2011

14:30-15:30, Marisa Resende, The Conjugate Gradient Method and some applications

Abstract: Conjugate-Gradient Methods are a class of numerical optimization procedures developed with the aim of minimizing a given objective function. The methods in this class have had a widespread use with powerful, efficient and robust performances, since their introduction in 1952 till the end of the last century with the establishment of important global convergence results. In this talk we present some of their relevant aspects concerning the algorithmic structure, properties and applications, both for the linear and the non-linear cases, focusing also their relation towards the original method of Steepest Descent.
In particular we study the Polak-Ribière Method, a non-linear variant of the Conjugate-Gradient Methods, and its application to machine learning on a recent algorithm developed by Rasmussen aiming at estimate parameters associated to a Gaussian Process Model.
[Seminar supervised by Marta Pascoal.]

15:30-16:30, Serkan Karaçuha, The Yang-Baxter Equation and Its Solutions By Hopf Algebras
Abstract: The Yang-Baxter Equation (YBE) was first introduced in the field of statistical mechanics independently by C.N. Yang (1967) and R.J. Baxter (1971). The term YBE has become a common name  to denote a principle of integrability, i.e. exact solvability, in several fields of physics and mathematics. The solutions of the YBE are called the R-matrices. In this talk, it is shown that braided (quasi-triangular) Hopf algebras provides solutions of the YBE. To do this, the important concept of braided bialgebras is introduced. These bialgebras are the ones with a universal R-matrix inducing a solution of the YBE on any of their modules. At the end we explain how the R-matrix of a braided bialgebra produces solutions of the YBE.
[Seminar supervised by Christian Lomp.]

July 1, 2011

14:30-15:30, Nasim Karimi, Parking functions and labeled trees
Abstract: Suppose that n drivers want to park their cars in a one way street with n empty parking places and they have a preference for a special place: the i'th driver that enters the street wants to park his car in place f(i). Now suppose drivers enter to the street one by one and the i'th driver does park in place f(i) in case this place is empty; otherwise he probes the first next empty place and parks there. We say that f is a parking function if finally all drivers can park their cars in the street. It is well-known that the set of parking functions of size n is in bijection with the set of labeled forests with n vertices.
In this talk we define a bijection between these two sets. This bijection is not recursive.

[Seminar supervised by António Guedes de Oliveira.]

June 3, 2011

14:30-15:30, Mohammad Ahmadi, Parallel splitting-up methods for elliptic boundary value problems
Partial differential equations arise in the mathematical modelling of many physical, chemical and biological phenomena. Most of the time for solving these PDEs we need numerical methods, but the main problem is time limitation for calculation. So the alternating direction methods and a while later parallel splitting-up methods were proposed. With parallel methods we can reduced multidimensional problems into the fractional steps of independent one dimensional problems and therefore their computation can be carried out by parallel processors. 
This talk introduces parallel algorithms for solving elliptic problems and discusses their convergence and efficiency.
[Seminar supervised by Adérito Araújo.]

15:30-16:30, Mahdi Dodangeh, Degenerate elliptic problems in a class of free domains

Abstract: In this talk we study a mixed boundary value problem for an operator of p-Laplacian type. The main feature of the problem is the fact that the exact domain where it is considered is not known a priori and is to be determined so that a certain integral condition is satisfied. We establish the existence of a unique solution to the problem, by means of the analysis of the range of an appropriate real function, and we show the continuous dependence with respect to a family of operators.

[Seminar supervised by José Miguel Urbano.]

May 20, 2011

14:30-15:30Maria de Fátima Pina, Rolling Pseudo-Riemannian Manifolds
Abstract:We will present the concept of a rolling map for manifolds that are embedded in a pseudo-Riemannian manifold. This talk is based on recent work of F. Silva Leite and P. Crouch which, in turn, generalizes the notion of a rolling map given by Sharpe, for Euclidean manifolds. One particular case that will be studied in detail is the hyperbolic n-sphere rolling on the affine tangent space at a point, both embedded in the generalized Minkowski space R1n+1.
[Seminar supervised by Fátima Leite.]

15:30-16:30, Jahed Naghipoor, Studying a mathematical model for biodegradable polymeric drug delivery system
Abstract: Biodegradable polymeric coatings on cardiovascular stents can be used for local delivery of therapeutic agents to diseased coronary arteries after stenting procedures. In this seminar, a mathematical model will be presented to design and simulate of such coating drug delivery. This mathematical model can be used as a tool for predicting drug delivery from other coatings using the same polymer-drug combination. The studied model can be used to develop mathematical models for predicting the degradation and drug release kinetics for other polymeric drug delivery system. The linearization of general nonlinear mathematical models also will be discussed.
[Seminar supervised by José Augusto Ferreira.]

May 6, 2011

14:30-15:30, Manuela Sobral, Profinite structures and profinite completions

Abstract: Profinite algebras and profinite completions first appeared in Galois theory and algebraic number theory. Profinite topological spaces as well as profinite ordered topological spaces form a part (half of) famous dualities:  Stone duality and Priestley duality, respectively. In various other settings profinite structures and profinite completions played and play an important role.
In this talk we use results and problems in this area to exhibit the power and usefulness of some categorical ideas, tools and techniques.

April 29, 2011

14:30-15:30, Rui Sá Pereira, A tool for characterizing Vector bundles

Abstract: The classification of mathematical objects arises as a powerful tool for reducing the exhaustive study of the objects in general theories  to smaller classes where each object represents a bigger, possibly infinite, class of objects, each sharing the same “unifying” characteristic, where unifying means in almost every case “up to isomorphism”. The classification of finite simple groups in algebra, the classification of a surface by the genus in geometry, are both stark examples of the advantage  of reducing the study of a potential infinitude of different objects, to the study of a “handful” “interesting” ones . In algebraic topology full classification remains an elusive task, but nevertheless there are discrete algebraic invariants providing powerful tools for characterizing a central object naturally emerging in geometry,  the vector bundle which has a multitude of applications in mathematics and theoretical physics. In this seminar, we introduce the Chern class of a vector bundle, an algebraic invariant that can be assigned to every vector bundle, thus providing a way to see whether two given vector bundles are non-isomorphic.
[Seminar supervised by Peter Gothen.]

15:45-16:45, Maria Manuel Clementino, From sets to elementary toposes

Abstract: As a naive introduction to notions and techniques of Category Theory, we will try to guide the students, through a careful analysis of the categorical behaviour of sets and maps, to the definition of elementary topos.
[No Category Theory background is assumed.]

March 25, 2011

14:30-15:30, Ronald Alberto Zúñiga Rojas, The fundamental group and the van Kampen Theorem

Abstract: The main idea of the presentation is to talk about the importance of the fundamental group pi
1(X; x) of a topological space X and how it is related with coverings of X. To do that, it is necessary to introduce the concept of covering map, and some of their basic properties. Many coverings, including the particular case of the polar coordinate map, are examples of G-coverings, arising from an action of a group G on a topological space.
The study of closed path, and homotopic paths are also very important subjects to introduce and de fine, formally, the fundamental group, which is the group of equivalence classes of closed paths starting and ending at a fixed point, with the equivalence relation given by homotopy.
The Van Kampen Theorem is a very important result that describes the fundamental group of a union of two spaces in terms of the fundamental group of each and of their intersection, under suitable hypotheses.
At the end, if it is possible, there is an application of the Van Kampen Theorem that could be interesting to the audience.
[Seminar supervised by Peter Gothen.]

March 18, 2011

14:30-15:15, Maria de Fátima Carvalho, Generalized ergodic theorems

Abstract: Given a measurable space X with a probability measure which is invariant and ergodic by a dynamics T: X --> X, the classical Birkhoff's theorem states that, for any integrable test function, its sequence of Cesàro time averages converges almost everywhere to the space mean. We will discuss an extension of this result to a class of non-invariant sigma-finite measures.

March 11, 2011

14:30-15:15, Olga Azenhas, Littlewood-Richardson Miscellany

Abstract: Schur functions constitute one of the most important basis for the space of symmetric functions but their importance is due mainly to their ubiquitous nature. In fact, Schur has  identified these functions - they would later bear his name - as characters of certain irreducible polynomial representations. On other hand, in combinatorics, they are the generating functions for semistandard Young tableaux. A function is said to be Schur positive if it can be written as a linear combination of Schur functions with all coefficients non negative integers. Examples are the Schur function product and skew Schur functions where coefficients are the famous Littlewood-Richardson coefficients. For  combinatorialists Littlewood-Richardson coefficients are particularly interesting because they enumerate various combinatorial objects. I shall illustrate these ideas by showing the appearance of Littlewood-Richardson coefficients in several problems.

15:15-16:00, Alessandro Conflitti, How to study Coxeter systems and live to tell the tale

Abstract: An accessible introduction to Coxeter systems presented as combinatorics of words.
[No prerequisite required, all are welcome.]

March 07, 2011

16:45-17:30, Raquel Caseiro, Dirac structures

Abstract: Dirac structures were introduced by T. Courant and A. Weinstein as a unified  approach to Poisson and pre-symplectic geometry.  Instead of considering linear transformations on V or on V*, the key idea is to work on the direct sum V⊕V*.
The aim of this seminar is to introduce these structures and  review some of their basic properties. Then we will look to smooth Dirac structures and we will see some applications.

17:30-18:15, Camille Laurent-Gengoux, The Toda lattice: from theory to practice

Abstract: We will show how sophisticated tools of Poisson geometrie and Lie algebra theory can end up solving a relatively simple differential equation: the Toda lattice, and that abstract non-sense and concrete problems are not that far away one from the other.

February 25, 2011

14:30-15:15, Alexander Kovacec, Polynomial Inequalities and Minimization

Abstract: The problem to find good approximations to global minima of multivariate real polynomials defined on basic semialgebraic sets leads to profound algebraic questions and solutions have surprising applications, e.g. approximation schemes for NP-complete problems like

a) the partition problem: given a sequence a1,a2,..., an of positive integers, can these be partitioned into two sets whose sums of elements are equal? 
b) the stable set problem: what is the maximum cardinality of a set of vertices in a graph so that no two of them are neighbours?

In more precise terms the request is, given polynomials p, g1,..., gm in R[x]= R[x1,x2,..., xn], find good bounds for

p*:= infx in K p(x),       K ={x: g1(x)>= 0, ..., gm(x) >= 0}.

The talk concentrates on the algebraic questions arising: to find algebraic nonnegativity certificates; to decide whether a polynomial is a sum of squares of polynomials; to give good lower bounds for p* minimizing related polynomials; to find extremal elements in the cone of positive semidefinite polynomials, etc.

February 18, 2011

14:30-15:15, Carlos Tenreiro, Combining cross-validation and plug-in methods for kernel density bandwidth selection

Abstract:The cross-validation and multistage plug-in methods are two of the most widely used procedures for choosing the bandwidth in a kernel density estimation setting. In this talk we review the basic ideas and results about kernel density bandwidth selection and we propose a combination of these well-known procedures in order to obtain a data-based bandwidth selector that presents an overall good performance for a large set of underlying densities.

15:15-16:00, Paulo Eduardo Oliveira, Smoothing sparsely observed discrete distributions

Abstract: In categorical models it is often reasonable to assume some adjacency and contiguity relations between neighboring cells. In such cases it becomes justifiable the use of smoothing to improve upon simple histogram approximations of the probabilities. This is particularly convenient when in presence of a sparse number of observations. We will discuss approaches to this kind of problem, using kernel methods and local polynomials, with respect to usual least squares error criterium and a relative least squares that is inspired on the approaches suggested by chi-square tests.

February 11, 2011

14:30-15:15, Jorge Almeida, Profinite Algebra

What do the construction of reals (from the rationals), of p-adic integers (from the integers), of power series (from polynomials), of absolute Galois groups, of free profinite semigroups, and so on, have in common? They are all obtained by completion of more elementary structures. In all these cases, the completion is a topological operation, but there is a very strong interaction with an algebraic structure. In fact, in all but the first case, the completion can be realized as an inverse limit of quotients of the more elementary structure. Moreover, with the additional exception of power series with coefficients in infinite rings, the quotients are finite. Be it in Number Theory, in Field Theory, in Group Theory or in Semigroup Theory, the latter motivated by applications in Computer Science, finite quotients are of special interest simply because it is in principle easy to compute in finite structures. Inverse limits of finite algebras are called profinite algebras. The aim of this talk is to introduce such structures and to explain why they have been playing an increasingly important role in Algebra.

15:15-16:00, Manuel Delgado, An algorithm to compute generalised Feng-Rao numbers of a numerical semigroup

Abstract: A numerical semigroup is a co-finite submonoid of the non-negative integers under addition.
In the framework of the Theory of Error-Correcting Codes, Feng and Rao introduced a notion of distance for the Weierstrass semigroup at a rational point of an algebraic curve, with decoding purposes. It is a purely combinatorial concept that can be defined for any numerical semigroup. Later on, that notion has been generalised and is used not only in the theory of error correcting codes, but also in cryptography.
Let s be an element of a numerical semigroup S. An element a of S is said to divide s if there exists b in S such that s=a+b. The set of divisors of s is denoted by D(s).
The (classical) Feng-Rao distance is a function d from S into the non negative integers defined by d(m) = min{#(D(n)): n>=m, n in S}.
Replacing the element n in the preceding definition by a set of r elements of S greater than m, one obtains the definition of the rth Feng-Rao distance.
For a sufficiently large m, there exists a constant, the so-called rth Feng-Rao number, depending only on r and S, such that the rth Feng-Rao distance is the classical Feng-Rao distance plus that constant.
An algorithm to compute generalised Feng-Rao numbers will be presented. It can be used in practice and therefore can be extremely useful in the search for formulas for the generalised Feng-Rao numbers of numerical semigroups of certain classes.
(Joint work with J.I. Farrán, P.A. García-Sánchez and D. Llena.)

February 8, 2011

09.45-10:45, Ezgi Iraz Su, Doing Topology in the category of locales: Sublocale lattices

Abstract: The lattices of subobjects in Loc (sublocale lattices) are much more complicated than their counterparts in Top (which are nice complete atomic Boolean algebras).
Some of the main differences are that
(1) most sublocales are not complemented, and
(2) each locale has a smallest dense sublocale.
Even the lattice of sublocales of a topological space can be much larger than the Boolean algebra of its subspaces (e.g. Q has 2c many non-isomorphic sublocales [J. Isbell, Some problems in descriptive locale theory, Canad. Math. Soc. Conf. Proc. 13 (1992) 243-265]).
In this talk we describe the basic structure of sublocale lattices.

[Seminar supervised by Jorge Picado.]

17:00-18:00, Ebrahim Azhdari, Hausdorff Measures and Dimension on R

Abstract: In this seminar we study the Hausdorff measures Hs, 0<=s <∞, and the topology induced by them. We define the Hausdorff dimension on R
showing that the Lebesgue measure, defined on R by R. Baker, operates as a measure H . Then we will present some properties of these Hausdorff measures and dimension. Finally some examples will be given.
[Seminar supervised by Susana Moura.]

December 15, 2010

14:30-15:15, Júlio Neves, Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces

Abstract: Starting with the Theorem of Sobolev, 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 give necessary and sufficient conditions for embeddings of Bessel potential spaces  modelled upon rearrangement invariant Banach function spaces X into generalized Hölder spaces. We also apply our results to the case when X is a Lorentz-Karamata space and, in particular, we present better target spaces than the ones given by the Brézis and Wainger result.

15:15-16:00, Susana Moura, Envelopes and sharp embeddings in function spaces

Abstract: We describe the concept of (growth and continuity) envelopes in function spaces, present some basic properties and give a short survey on the results in the context of the  Besov and Triebel Lizorkin spaces of generalized smoothness. In general the knowledge of the envelope of a function space gives sharp results regarding embeddings, but we will show that this might not be the case in critical situations.

November 24, 2010

15:00-16:00, Colloquium: Richard Tsai (Univ. Texas at Austin, USA), Point source discovery in complicated domains

November 17, 2010

14:30-15:15, Marta Pascoal, Dealing with uncertainty in network optimisation

Abstract: Network optimisation is a branch of optimisation the problems of which are
modelled over a valued graph, that is, a network.
We briefly present introductory concepts in this field and discuss connections between network optimisation and related subjects.
Classical methods applied in this area assume that deterministic information is associated with the graph structure, however, in real problems these parameters are often incomplete, inaccurate or stochastic.
We describe some of the possible formulations of network optimisation problems when uncertainty is present.

15:15-16:00, Sílvia Barbeiro, Splitting methods in the design of coupled flow and mechanics simulators

Abstract: The dynamics of coupled flow and mechanics are of interest in many areas of science. Developments in this field are contributing to important achievements not only in soil mechanics but also in civil, petroleum and even biomedical engineering.
The interactions between flow and mechanics can be modeled using various coupling schemes. In this seminar we analyse different operator-split strategies which lead to iteratively coupled schemes. The resulting sequential procedures are iterated at each time step until the solution converges within an acceptable tolerance.  If the sequential solution strategies have stability and convergence properties that are closed to those of the fully coupled approach, they can be very competitive for solving problems of practical interest.
We will discuss the development of numerical solutions and give some insight into the theoretical basis of the underlying methods. Both the theory and the numerics will be illustrated via some examples.

November 10, 2010

14:30-15:15, Fátima Leite, Introduction to geometric control theory – controllability and Lie bracket

Abstract: We will present an introduction to the theory of nonlinear control systems, with emphasis on controllability properties of such systems. The basic tools in geometric control come from differential geometry. A control system can be seen as a family of vector fields and the most basic theoretical tool of the geometric view point is the Lie bracket. We first introduce the differential geometric language of vector fields, Lie bracket, distributions, integrability etc., and then analyze basic controllability problems and give criteria for complete controllability.

November 03, 2010

14:30-15:15, Adérito Araújo, Numerical Approximation of Mean Curvature Flow
Abstract: The mean curvature flow problem for graphs is closely related to the mean curvature of level sets. Taking into account this fact, we consider 
a level set algorithm in the context of finite differences together with a semi implicit time discretization. The main goal of this work is to study the qualitative
and quantitative properties of the numerical
solution as well as the efficiency of the algorithm.
We also provide
some numerical tests.

15:15-16:00, Ercília Sousa, Anomalous diffusion equations

Abstract: Fractional space derivatives are used to model anomalous diffusion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion.

In this talk, a one dimensional diffusion model is considered, where the usual second-order derivative gives place to a fractional derivative of order alpha, with 1< alpha <= 2. Different definitions for the fractional derivative are shown.
Some insights on how to solve fractional diffusion models numerically will be given, focusing on its additional difficulties when compared with the classical models. These difficulties are related to the fact that the fractional derivative of order alpha at a certain point, x, is not a local property anymore, except when alpha is an integer. Therefore it is expected that the theory involves information of the function further out of the region close to the point at which we are computing the derivative.

October 27, 2010

14:30-15:15, Amílcar Branquinho, Interpretation of some integrable systems via multiple orthogonal polynomials
Abstract: Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based on the analysis of the matrical moments 
for the main operator of the pair. The solutions of
these systems are studied in terms of properties of this operator, giving, under some conditions, explicit
expressions for the resolvent

15:15-16:00, Luís Daniel Abreu, The Uncertainty Principle in Time-Frequency Analysis
Abstract: Our departure point in this talk will be the Heisenberg Uncertainty Principle formulated as a proposition involving a real function f
(function of "time") and its Fourier transform (function of "frequencies").
Within this formulation, The Uncertainty Principle becomes quite intuitive: it provides a precise mathematical description of the impossibility
of finding a "instantaneous frequency".
Mathematically, this is rather intuitive, since to measure a frequency we need to look at a function over a certain period of time.
However, the pleasurable activity of listening to music requires our ears to constantly beat the uncertainty principle: in a given instant they are
able to regognize what frequency is being played!
How is this possible?-The answer to this question is at the heart of a modern branch of mathematics called "Time-Frequency Analysis", which had
a tremendous development in recent years, with an hitherto unseen collaboration between pure and applied mathematics which had a tremendous
impact in our lifes: from mobile communications and file compression to Banach Algebras and Noncomutative geometry.
I will give an idea of the outputs of this endeavour, like "Wavelets" and "Compressed Sensing" as well as possible entry points for research at the
Phd student level.

October 20, 2010

14:00-14:45, Inês Cruz, Symplectic Geometry versus Riemannian Geometry
Abstract: This talk will be an introduction to Symplectic Geometry. Basic notions in symplectic geometry (symplectic volume, symplectomorphism, Hamiltonian vector field, 
symplectic capacity) will
be introduced while comparing them with their Riemannian counterparts.

14:45-15:30, Antonio de Nicola, An introduction to contact geometry

Abstract: We will introduce the first elements of contact geometry and briefly mention some of its applications to physics as well

as its relation with symplectic geometry.

October 13, 2010

14:30-15:15, Dmitry Vorotnikov, Attractors in the absence of a dynamical system

An attractor of a dynamical system is a certain set to which every orbit eventually becomes close. When an autonomous differential equation (or boundary value problem) generates a dynamical system, the corresponding attractor characterizes the long-time behaviour of its solutions. However, if the differential equation (or BVP) is either non-autonomous, or does not have uniqueness of solutions, or lacks continuity properties, or is not dissipative, then the standard approach does not work. We will discuss the possible ways out of the situation. 

October 06, 2010

14:30-15:15, António Leal Duarte, Inverse Eigenvalue Problems for graphs

Abstract: The (real) Inverse Eigenvalue Problem IEP for a graph G (with vertices 1,  ... , n) consists in describing the set of all n-tuples
of real numbers that may occur
as eigenvalues of real symmetric matrices A with graph G, that is, with a fixed zero pattern: a non diagonal element of A in position (i, j) is nonzero if and only if
there is an edge between i and j in G; the set of such matrices is denoted by S(G).
This seems to be a very difficult problem even when the graph is a tree T; in this case any sequence of n distinct real numbers does occur as eigenvalues of one matrix in S(T),
so the problem is just to describe the possible multiple eigenvalues. But even the apparently simpler problem of  just describing the possible lists of multiplicities that may occur
between the eigenvalues of  matrices in S(T) seems very difficult, depending heavily on the graph). Some related questions are:
i) What is the maximum possible multiplicities of  eigenvalues for matrices in S(G);
ii) what are the minimum nunber of distinct eigenvalues for A in S(G);
iii) what is the minimum number of  multiplicity one eigenvalues for A in S(T).
Problem i) is solved when G is a tree but not much is known about ii) and iii).

15:15-16:00, Alfredo Costa, Semigroup invariants of symbolic dynamical systems
Abstract: Many natural phenomena are described in terms of dynamical systems arising from systems of differential equations. 
One successful method for studying these systems is to discretize them, thus producing "symbolic" dynamical systems.
One of the main driving forces in symbolic dynamics has been the classification of symbolic dynamical systems up to isomorphism. Linear Algebra plays a central role in the field.
In this talk focus is given to another algebraic perspective: sofic systems are closely related with finite automata, and via this perspective, with semigroup theory.
This is one of the aspects of the interplay between symbolic dynamics and semigroup theory.
The deduction of semigroup theoretic invariants of symbolic dynamical systems received attention in the past few years. Research in semigroup theory also benefited
from interplay with symbolic dynamics.

Some results in both directions will be presented in the talk.

Maria Manuel Clementino