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.

# Past Projects

FCT Project PTDC/MAT-GEO/0675/2012

FAPERJ Edital nº 20/2008

Processo: E-26/111.416 /2010

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.

O objectivo geral deste projecto é desenvolver a teoria e os métodos de refinamento iterativo bem como a sua paralelização, no caso do problema espectral, a partir dos resultados recentes, obtidos pela equipe, para a equação integral de Fredholm de segunda espécie com núcleo possuindo uma singularidade fraca, e com importantes aplicações à modelação da transferência em atmosferas estelares. Os objectivos científicos específicos são detalhados em seguida na descrição das tarefas a desenvolver por cada estabelecimento do projecto.

Spectral problems are present in many mathematical problems in engineering, economy and other areas. These problems arise either as intermediate computations in the solution of many problems, for instance in image processing, or autonomously for studying dynamic properties of mathematical models, for instance in stability analysis.

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.

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.

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.

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].

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:

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.