# Past Projects

In general terms the goal of this project is to study statistical properties of Dynamical Systems (DS), both deterministic and stochastic, with special emphasis on laws of rare events.

Os sistemas de células acopladas (SCA) são formados por sistemas dinâmicos individuais (as células) que interactuam. Aqui, uma célula corresponde a um sistema de equações diferenciais ordinárias (EDO). A arquitectura dos SCA é codificada em termos de um grafo direccionado - a rede de células acopladas (RCA)- indicando quais as células que são idênticas, quais as células que interactuam e quais as interacções (acoplamentos) que são do mesmo tipo (SGP03), (GST05).

In the last decades uniformly hyperbolic systems have played an important role in the development of the Dynamical Systems Theory. These are systems for which the tangent bundle splits into two invariant sub-bundles, one of them contracting uniformly and the other one expanding uniformly. The study of uniformly hyperbolic systems, its connection with stability and the consequence to the dynamical behavior of systems reached unparallel development. The attention has now shifted beyond uniform hyperbolicity and the persistence of properties that prevail in this wider perspective.

The use of differential geometry to understand topological properties precedes the formal establishment of Algebraic Topology as a mathematical discipline and has been a central theme in mathematics since the early 20th Century.

The goals of this project fall in this tradition, in that we propose to study a variety of geometric objects and their topological properties using tools from Differential Geometry and Analysis. The main areas of study are the following:

Automata theory is a fundamental part of Computer Science that was intensively studied during the 1960 ́s, and has again, in the last years, become a matter of fruitful research, both at theoretical and applied level. This renewed interest is explained by the important role that new applications of automata theory play in fields such as computational linguistics, bioinformatics, speech and image recognition, software certification and computer networks, to mention only a few.

The project "Automata, Semigroups and Applications" aims to contribute to the development of the theories of automata and semigroups, and some of their applications. Besides connections between these areas of, say, Theoretical Informatics and Mathematics, they are also naturally connected with the theory of formal languages. On the other hand, the usage of sophisticated methods (as is the case of symbolic dynamics, representation theory, or geometric group theory) will also lead to the further exploration of natural connections with other branches of Mathematics.

The main goal of this project is to study the behavior of discrete chaotic dynamical systems and the new phenomena arising when these systems are perturbed or some bifurcation occurs.

We call a coupled cell network (CCN) a set of ordinary differential equations (ODEs) (the cells) that are coupled together.We study bifurcations (steady-state and Hopf) and dynamics of CCNs according to: the symmetry, the topology and the symmetry groupoid of the couplings network, and the cell´s internal symmetries.

The main idea behind this project is the bringing together of two different areas of mathematics (singularity theory and Poisson geometry) hopefully with profit to both. It is well known (since1983 when A. Weinstein stated the Splitting Theorem for Poisson manifolds) that the local structure of Poisson manifolds is not interesting but at singular points, i.e., points of rank zero. The most important works (by A. Weinstein, J. F. Conn and J. P. Dufour) on the local structure of Poisson manifolds concern the possibility of bringing to a linear normal form the original structure.

FCT Project PTDC/MAT-GEO/0675/2012

Proposed strategies to improve tuberculosis (TB) control include the targeting of interventions, such as active case finding in

specific high-risk groups; therefore, it is essential to identify and characterize the determinants of TB risk so that resources

and interventions can be directed at those most likely to develop and transmit TB. On this, epidemiological studies have

established an association between diabetes mellitus (DM) and active TB demonstrating that diabetic patients have an

The project, coordinated by Miguel Abreu, started in May 2013 and aims at fostering the interaction of research in Geometry and Mathematical Physics within the Department of Mathematics of IST and throughout the country, through the stimuli for interaction among researchers, the reinforcement of international connections, the attraction of post-docs and doctoral students, and the organization of seminars, short courses and international meetings.

The scope of the present project is the verification of properties of safety-critical software. Our approach will focus simultaneously on two techniques: software model checking and deductive reasoning, both advancing the state of the art at the theoretical level and producing tools that can be used by companies developing critical applications. We are particularly interested in interactions between both approaches, which have historically fed back into each other.

BREUDS is a research partnership between leading European and Brazilian research groups in dynamical systems, a prominent subject in mathematics. An extensive consortium of European and Brazilian institutions collaborates to provide world leading critical mass and support for research on the very forefront of the field. Work Packages reflect parallel priorities in the research. Transfer of knowledge is facilitated by conferences and five workshops.

Recent international and national mathematics curriculum guidelines indicate the development of students’ statistical literacy, at different levels of schooling, as a major educational aim. In Portugal, the mathematics syllabus for basic education, which began to be implemented in 2009, gives a greater emphasis on statistics, presenting more demanding learning goals, since the elementary levels. This represented a challenging situation for practicing teachers, requiring them to develop new perspectives about the teaching and learning of statistics.

This is a project in Mathematics with applications to Biology. There are three main goals in this project: a) To develop the understanding of hyperbolic dynamics; b) To develop the applications of the renormalization theory in hyperbolic dynamics, Physics and Biology; c) To analyse real data coming from Biology and to develop techniques of Dynamical Systems, Renormalization, Quantum Physics and Game Theory to study the data.

Studies of rings with additional structure form one of the main branches of studies in the theory of associative rings and are conducted in many research centers in the world. There are many open problems in this area. Some of them are of substantial importance not only for ring theory or, more generally, for algebra but also for other branches of mathematics and other scientific disciplines, e.g., physics.

We intend to study the following classes of rings:

Álgebras envolventes de Lie Álgebra, álgebras de grupo, esquemas de grupos afins e grupos quânticos, além dea estrutura de álgebra têm uma estrutura adicional: a estrutura de co-algebra. As classes de álgebras mencionadas são exemplos de álgebras de Hopf com acções e co-acções muito importantes. Uma álgebra A graduada por um grupo G é um co-módulo sobre a álgebra de grupo K[G] ou, usando outra terminologia, A é um álgebra na categoria de co-módulos da co-álgebra K[G].