# Seminar of the PhD Program in Mathematics Porto-Coimbra

The seminar will take place on Fridays at 14:30 in room 0.07 of the Mathematics Department in Porto

### 07/10/11

- Speaker: Christian Lomp (University of Porto)
- Title: Aspects of non-commutative algebra
- Abstract: Introdution to basic examples and ideas from non-commutative algebra.

### 14/10/11

- Speaker:
**Pedro Silva**(University of Porto) - Title: Finite automata for Schreier graphs of virtually free groups
- Abstract:
*Finite automata became over the years the standard representation of finitely generated subgroups H of a free group F_A. The Stallings construction constitutes a simple and efficient algorithm for building an automaton S(H) which can be used for solving the membership problem of H in F_A and many other applications. This automaton S(H) is nothing more than the core automaton of the Schreier graph (automaton) of H in G, whose structure can be described as S(H) with finitely many infinite trees adjoined. Such an approach invites naturally generalizations for further classes of groups. In joint work with Xaro Soler-Escrivá and Enric Ventura, we present a Stallings type approach with some generality which is robust enough to exhibit several prized algorithmic and geometric features, namely in connection with Schreier graphs. Moreover, using Bass-Serre theory, we succeed on identifying those groups G for which it can be carried on: (finitely generated) virtually free groups.*

- Speaker:
**Andrea Solotar**(University of Buenos Aires) - Title: Gradings of matrix algebras and intrinsic fundamental group
- Abstract:
*In this talk I will comment on the definition of the fundamental group of a k-algebra and I will compute explicitly the fundamental group of several algebras. For this purpose, given a k-algebra A, we will consider the category of all connected gradings of A by a group G the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of A, which allows its computation using complete lists of gradings. This is a joint work with Claude Cibils and María Julia Redondo.*- Y. Bahturin, M. Zaicev, Group gradings on matrix algebras, Canad. Math. Bull. 45 (2002), no. 4, 499–508.
- C. Boboc, S. Dascalescu, Gradings of matrix algebras by cyclic groups, Comm. Algebra 29 (2001), 5013–5021.
- C. Cibils, E. Marcos, Skew category, Galois covering and smash product of a k-category, Proc. Amer. Math. Soc. 134 (2006), no. 1, 39–50.
- C. Cibils, M. J. Redondo, A. Solotar, The intrinsic fundamental group of a linear category, arXiv:0706.2491
- S. Dascalescu, Group gradings on diagonal algebras, Arch. Math. (Basel) 91 (2008), 212–217.
- A. Valenti, M. Zaicev, Group gradings on upper triangular matrices, Arch. Math. (Basel) 89 (2007), 33–40.

### 21/10/11

- Speaker:
**Ana Paula Santana**(University of Coimbra) - Title: Schur Algebras
- Abstract:
*Schur algebras are fundamental tools in the representation theory of the general linear group. In this seminar I introduce Schur algebras and show how they are related with the general linear and the symmetric groups. The problem of the construction of projective resolutions of Weyl modules is also considered.*

### 28/10/11

- Speaker:
**Ana Paula Dias**(University of Porto) - Title: Coupled cell networks
- Abstract:
*It is known that the dynamics of networks relates to the network architecture. In this talk we introduce the formalism of coupled cell networks of Golubistky, Stewart and co-workers. We consider then some of the questions raised in the field towards the understanding of the relationship between the dynamics of networks and the network topology.*

### 04/11/11

- Speaker:
**António Leal**(University of Coimbra) - Title: Eigenvalue multiplicities and graphs.
- Abstract:
*The zero pattern of a matrix (that is its graphs) impose several restriction on the multiplicities of its eigenvalues. The problem of describing the list of eigenvalues that can occur on symmetric (Hermitian) matrix with a given graph seems to be very hard and is far from being solved even for when the graph is a tree. the solution of some related problems (e.g. minimal number of distinct eigenvalues, minimal number of simple eigenvalues) is also also unknown. We will discuss those as well as some structural properties of matrix with multiple eigenvalues with respect with its graph.*

### 11/11/11

- Speaker:
**Maria Manuel Clementino**(University of Coimbra) - Title: From sets to categories: exponentials and subobject classifiers
- Abstract:
*We will introduce categorical notions and techniques that allow a unified study of function objects and power objects, having as guiding example the notion of topos. If time permits, we will analyse these constructions in algebraic and topological settings.*

### 18/11/11

- Speaker:
**Lars Kadison**(University of Porto) - Title: Subalgebra Invariants
- Abstract:
*Depth and H-depth are bimodule H-equivalence class invariants of a subring B in A taking values in the natural numbers or infinity. Depth d(B,A) is defined as the least n for which the (n+1)-fold tensor product of A over B is isomorphic as bimodules to a direct summand of a multiple of the n-fold tensor product. For semisimple complex algebras A and B the depth d(B,A) is a certain width of a bipartite graph; equivalently, a certain power of the induction-restriction matrix at which it becomes stably positive. For a complex group algebra extension A = kG and B = kH the depth d(H,G) seems limited to the odd numbers as well as {2,4,6}. The permutation groups have depth d(S*_{n},S_{n+1})=2n-1, but one may ask the combinatorial problem a la Littlewood-Richardson, what is d(S_{n}x S_{m},S_{n+m})? For general classes of subalgebras it is interesting to ask if d(B,A) is finite. If B tensor the opposite algebra of B has finite representation type in a Krull-Schmidt category, then it is finite for an easy module-theoretic reason. Boltje, Danz and Kuelshammer have shown d(B,A) is finite for completely general finite group algebra extension by establishing an upper bound with a group- and G-set theoretic depth, which in turn is bounded by twice the index of the normalizer of the subgroup. One may ask if something similar is true for finite semigroup semigroup algebras or finite dimensional Hopf algebras? Like cohomology and K-theory, low-level depth is the best understood and applied. A ring extension having depth two is essentially a Galois extension, where Hopf algebroids that act and co-act can be found. Subgroups satisfying d(H,G) < 3 are normal subgroups; something similar is true for Hopf algebras where normal means invariant under the adjoint actions. In the alloted time I will just cover some basics of depth and H-depth.

### 25/11/11

** Open Day of the Center of Mathematics of the University of Porto **

### 02/12/11

- Speaker:
**Alberto Adrego Pinto**(University of Porto) - Title: Explosion of diferentiability for conjugacies between multimodal maps.
- Abstract:
*Let f and g be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy h between f and g is differentiable at a point in the nearby expanding set of f, then h is a smooth diffeomorphism in the basin of attraction of a renormalization interval of f. In particular, if f and g are smooth unimodal maps and h is differentiable at a boundary of the domain I of f then h is smooth in I.*

### 09/12/11

- Speaker:
**Peter Gothen**(UPorto) - Title: Surface group representations - algebra and geometry
- Abstract:
*We study representations of the fundamental group of a surface in the isometry group of the hyperbolic plane, up to deformation equivalence. There is an integer invariant, called the Toledo invariant, associated to such a representation. We state a theorem due to W. Goldman which characterizes Fuchsian (i.e. uniformizing) representations in terms of the Toledo invariant. Finally we indicate how methods from harmonic map theory and holomorphic geometry can be used to prove Goldman's Theorem and some generalizations.*

### 16/12/11

- Speaker:
**Pedro Lima**(CEMAT, Instituto Superior Técnico, Technical University Lisbon) - Title: Analysis and numerical modelling of singular boundary value problems for nonlinear ordinary differential equations
- Abstract:
*We present the analysis and numerical treatment of a nonlinear singular second order boundary value problem in ordinary differential equations, posed on an unbounded domain. The differential operator is the degenerate p-laplacian. In the case p=2, this equation represents the density profile equation for the description of the formation of microscopic bubbles in a non-homogeneous fluid; this case was analyzed in P.M.Lima et al. J.ComputApplMath (2006) and Kitzhofer et al. J.Sci Comput (2007). By considering p different from 2, the model analyzed in this paper describes a more general class of mixtures of fluids. First we give an asymptotic analysis of the considered equation and provide asymptotic expansions for one-parameter families of solutions satisfying the boundary conditions at the singular points. Then the problem is solved numerically using algorithms that are robust with respect to singularities. We illustrate the theoretical results by numerical experiments. This is a joint work with***Luisa Morgado**,from Universidade de Tras-os-Montes e Alto Douro, Vila Real, Portugal,**E. Weinmuller**and**G. Hastermann**, from Institute for Analysis and Scientific Computing, Vienna University of Technology,Austria.

### HAPPY NEW YEAR 2012

### 17/02/12

- Speaker:
**Azizeh Nozad**(PhD student of the program) - Title: The fundamental group and Hurewicz' theorem for first cohomology
- Supervised by: Peter Gothen (University of Porto)
- Abstract:
*We give an introduction to the fundamental group of a topological space and explain how it can be calculated using covering spaces. We then use integration of 1-forms along closed loops to prove that the first de Rham cohomology group of a connected smooth manifold is isomorphic to the space of homomorphisms from its fundamental group into the real numbers.*

### 24/02/12

- Speaker:
**Olga Azenhas**(Universidade de Coimbra) - Title: Schur positivity order on skew shapes and full support of ribbon shapes
- Abstract:
*Schur functions are considered to be the most important basis for the ring of symmetric functions. They are indexed by partitions and have a combinatorial expansion as a sum over the monomials associated to all Young tableaux of a given partition shape. Skew Schur functions come as a generalization of Schur functions to skew shapes. As symmetric functions they have an integral linear expansion over Schur functions. Interestingly, the set of partitions which appear in that expansion fit, in a certain sense, that skew shape and run over a subposet of the dominance lattice of partitions with bottom and top elements determined by the skew shape. We shall discuss some recent results, in collaboration with A. Conflitti and R. Mamede, on criteria for full support of a special class of ribbon shapes and their relationship with the classification of maximal connected skew shapes in the Schur positivity order due to P. McNamara and S. van Willingenburg.*

### 02/03/12

**change of time: this seminar will start at 16h00**

- Speaker:
**Jorge Freitas**(University of Porto) - Title: Rare events for chaotic dynamical systems
- Abstract:
*This lecture is about the study of extreme 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 a fixed random variable along the obits of the system. The other has to do with the phenomenon of recurrence to arbitrarily small (hence rare) events, which is commonly known as Hitting Time Statistics (HTS) and Return Time Statistics (RTS). 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 have a tendency to appear concentrated in time. The strength of the clustering is quantified by the Extremal Index (EI), that 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/RTS we will interpret the existence of an EI less than 1 as due to the presence of underlying periodic phenomena.*

### 9/03/12

**change of time: this seminar will start at 16h00**

- Speaker:
**Sofia Castro**(Faculty of Economics, U.Porto) - Title: Stability of heteroclinic networks
- Abstract:
*Invariant sets are important objects in the study of dynamical systems since they allow us to describe families of trajectories/solutions. In fact, a solution starting in an invariant set will remain in this set for all time. Whether this solution is complex or simple depends on the complexity, or simplicity, of the invariant set. Another important concept in the study of dynamical systems is that of stability. This goes back to Lyapunov at the end of the 19th century. When an invariant set is stable, it can be used to retrieve information not only about solutions belonging to it but also to study nearby solutions (which, because of stability, will tend to the invariant set in the future). In this talk the objects of interest are heteroclinic networks, that is, a collection of equilibria and the connections amongst them, as well as the study of solutions near the heteroclinic connection. The behaviour of these nearby solutions is especially interesting when the heteroclinic network is somewhat, but not asymptotically, stable. I shall discuss various notions of stability for a heteroclinic network and present examples of the behaviour of nearby solutions in each case.*

### 16/03/12

**change of time: this seminar will start at 15h30**

- Speaker:
**Maryam Khaksar**(PhD student of the program) - Title: Image Enhancement and Denoising by Diffusion Processes
- Supervised by: Sílvia Barbeiro (University of Coimbra)
- Abstract:
*Diffusion processes are common procedures used in image processing to remove noise from digital images in order to improve their quality. In this talk, we focus on anisotropic diffusion, called Perona-Malik diffusion, which is a nonlinear diffusion method used to remove noise from digital images without blurring edges.*

### 23/03/12

- Speaker:
**Adérito Araújo**(CMUC, U.Coimbra) - Title: Complex diffusion in image denoising
- Abstract:
*Optical coherence tomography (OCT) is a non-invasive imaging modality with an increasing number of applications and it is becoming an essential tool in ophthalmology allowing in vivo high-resolution cross-sectional imaging of the retinal tissue. However, as any imaging technique that bases its image formation on coherent waves, OCT images suffer from speckle noise, which reduces its quality. The aim of the work herein is present an adaptive complex diffusion filter that improve the process of speckle noise reduction and to improve the preservation of edge and image features. Joint work with: Rui Bernardes, Cristina Maduro, Pedro Serranho, Sílvia Barbeiro, José Cunha-Vaz*

- Speaker:
**Sílvia Barbeiro**(CMUC, U.Coimbra) - Title: Stability of finite difference schemes for complex diffusion processes
- Abstract:
*This talk focuses on the proof of the stability of a class of finite difference schemes applied to nonlinear complex diffusion equations. Complex diffusion is a common and broadly used denoising procedure in image processing. To illustrate the theoretical results we present some numerical examples based on an explicit scheme applied to a nonlinear equation in context of image denoising. This talk is based on joint work with A. Araújo and P. Serranho.*

### 13/04/12

- Speaker:
**Filipa Alexandra Cardoso da Silva**(PhD Student of the program) - Title:
*Integer-valued conditionally heteroskedastic time series models* - Supervisors: Maria de Nazaré Mendes Lopes and Maria Esmeralda Gonçalves (U.Coimbra)
- Abstract:
*The analysis of time series of counts is an emerging field of science. In fact, there are several real situations in which the results are discrete, and in particular of integer values. Many examples are known from epidemiology (e.g., count of cases of a certain disease) or economics (e.g., count of price changes). In this talk we present a new operation called thinning operation that replaces the scalar multiplication used in the models for real data. We also presente the Integer-valued GARCH model whose definition is motivated by the GARCH model, introduced in 1986 by Bollerslev. We also develop a distributional study on stationarity for this model.*

### 20/04/12

- Speaker:
**Alberto Hernandez Alvarado**(PhD student of the program) - Title: The Grigorchuk Group
- Supervised by: Pedro Silva (U.Porto)
- Abstract: In 1902 Burnside proposed the question of whether or not there exists a finitely generated group, for which all it’s elements are of finite order, while the group itself is infinite. In a seminal paper d’Adyan-Novikov proved the existence of such a group, yet it is our concern to introduce the theory involving self similar groups and the tools that will allow us to make computations on them, so that, after defining the Grigorchuk Group, we could be able to prove in a simple and elegant way that it is an example of such a group.

### 27/04/12

**change of time: the first talk will start at 14h00, while the second talk will start at 18h00**

- Speaker:
**Célia Mariana Rabaçal Borlido**(PhD student of the program) - Title: From Tarski' Problem to Universal Algebraic Geometry
- Supervised by: Jorge Almeida (U.Porto)
- Abstract:
*Around 1945, a Polish matematician named Alfred Tarski proposed the problem of deciding whether a first order sentence was fulfilled at a given free group. Since then, a lot of well-known matematicians have been putting their effort at solving this problem. A particular case is solving systems equations over a free group. Then, the same question over free semigroups arises naturally. In 1982, Makanin exhibited an algorithm for solving such kind of equations. Nevertheless, it's still an open problem to have a good description of the set of solutions (even for quadratic equations). In this talk, we will see how these questions can be formulated via universal algebraic geometry and why the techniques already developed to deal with richer algebraic structures rather than free semigroups fail when one tries to apply them at the latter.*

- Speaker:
**Juliane Fonseca de Oliveira**(PhD student of the program) - Title: Spatial Hidden Symmetries in Pattern Formation
- Supervised by: Isabel S. Labouriau and Sofia de Castro (U.Porto)
- Abstract:
*Partial differential equations such that are invariant under Euclidian transformations are traditionally used as models in pattern formation. These models are often posed on finite domains (in particular, multidimensional rectangles). Defining the correct boundary condition is often a very subtle problem. On the other hand, there is pressure to choose boundary conditions which are attractive to mathematical treatment. Geometrical shape and mathematically friendly boundary conditions usually imply spatial symmetry. The presence of symmetry effects that are “hidden” in the boundary conditions was noticed and carefully investigated by several researchers during the past 15-20 years. Here we review developments in this subject and introduce a unifying formalism to recover spatial hidden symmetries (hidden translations and hidden rotations) in multidimensional rectangular domains with Neumann boundary conditions.*

### 04/05/12

- Speaker:
**Diogo Judice**(PhD student of the program) - Title: Global Convergence and Worst Case Complexity of Trust-Region Methods
- Supervised by: Luís Nunes Vicente, University of Coimbra
- Abstract:
*Trust-Region Methods (TRM) are iterative methods for the solution of Nonlinear Optimization problems, where a quadratic model of the objective function is minimized (and trusted) in a region of pre-specified size. It is well known that these methods attain convergence from arbitrary starting points (global convergence) to first and second order stationary points, for smooth unconstrained and constrained optimization. In this talk, we will describe TRM in general for unconstrained optimization and focus on the case where there is no gradient and Hessian information available, describing the corresponding global convergence properties in the smooth case. We will then show how many iterations are needed to reach stationarity in the worst case. Finally, we will explain how these results can be extended to the nonsmooth case, where the objective function is Lipschitz continuous, but not necessarily differentiable.*

- Speaker:
**Helena Albuquerque**(University of Coimbra) - Title: Cayley algebras and exceptional non associative simple algebras
- Abstract:
*In this talk we will show the importance of the class of Cayley algebras for the discription of exceptional Lie, Jordan and Malcev algebras.*

### 18/05/12

- Speaker:
**Evelina Shamarova**(researcher, U.Porto) - Title: Forward-Backward SDEs driven by Lévy Processes and Application to Option Pricing
- Abstract:
*Recent developments on ﬁnancial markets have revealed the limits : Christian Loof Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Diﬀerential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing.*

### 01/06/12

- Speaker:
**Muhammad Ali Khan**(PhD student of the program) - Title: Coupled Coincidence and Common Coupled Fixed Points of Hybrid Pair of Mappings
- Supervised by: Christian Lomp (University of Porto)
- Abstract:
*Bhaskar and Lakshmikantham proved the existence of coupled fixed point for a single valued mapping under weak contractive conditions and as an application they proved the existence of a unique solution of a boundary value problem associated with a first order ordinary differential equation. Later on, Lakshmikantham and Ćirić obtained a coupled coincidence and coupled common fixed point of two single valued maps. We extend these concepts to multi-valued mappings and obtain coupled coincidence points and common coupled fixed point theorems involving hybrid pair of single valued and multi-valued maps satisfying generalized contractive conditions in the frame work of a complete metric space. We also present two examples to support our results.*