The emergence of chaotic systems and the erratic behavior that they enclose triggered a new approach in their analysis, more concerned with their statistical properties. In order to learn about the long-term behavior of these systems through a probabilistic perspective, one can just consider dynamically defined stochastic processes arising from these systems by simply evaluating an observable function along the time evolution of the orbits of the system. These processes will be our starting point.

# Projects

Algebraic Geometry is an old subject in mathematics and, at the same time, a vibrant area of current research with close connections to other areas. Its main objects of study are algebraic varieties which means, roughly speaking, zero sets of of polynomials. For example, an algebraic curve is a 1-dimensional algebraic variety, and an algebraic surface is a 2-dimensional algebraic variety.

In general terms the goal of this project is to study statistical properties of dynamical systems, both deterministic and stochastic (perturbed), with special emphasis on laws of rare events. The starting point of the analysis is a stochastic process. The dynamical system may appear in different ways. It can describe the time evolution simply by moving the process from one state to the succeeding one, acting, in this way, on the space of all realisations of the process.

Pretende-se obter novos resultados para fragmentos da lógica de predicados de primeira ordem, cujos modelos são palavras finitas e inifinitas. O projeto visa questões fundamentais motivadas por considerações práticas para a validação de software e circuitos. Desde o trabalho pioneiro de Büchi, Elgot e Trakthenbrot, as ligações entrem os mecanismos de descrição lógica e a teoria de autómatos é a principal ferramenta para a solução de tais problemas. Trabalho preliminar importante é devido ao responsável português, que foi o primeiro a considerar o problema da separação.

Smart Cyberphysical, Mathematical, Computational and Power Engineering Research for Disruptive Innovation in Production, Mobility, Health, and Ocean Systems and Technologies.

This is a research plan for a collaborative work involving colleagues from Brazil, France and Portugal. Namely, the members of this project are the following:

- Brazil: H. Movasati (IMPA) and Y. Nikdelan (UERJ).

- France: J. Rebelo (IMT) and D. de la Rosa (IMT - graduate student).

- Portugal: H. Reis (UP).

Exchange project with Instituto de Ciências Matemáticas e Computação, USP, under the CAPES/FCT (Brazil/Portugal) agreement. The list below only includes participants from Portugal.

Applications of singularity theory techniques to problems in dynamical systems, to the study of bifurcation with symmetry and to the geometry of low-dimensional manifolds. The problems to be treated correspond to the following topics:

The aim of this project is to study, implement and deploy an extension of the algebraic formulation for the tau method for the numerical solution of partial differential problems set on domains in $\mathbb{R}^{n}$, $n>2$. This extension is based on an appropriate choice of a basis for the space of polynomials in $\mathbb{R}^{n}$ and on the construction of the algebraic equivalent representation of the problem. An important feature of the required implementation is related to the solution procedure for the necessarily large dimensional linear systems involved.

Aims of the project:

(1) To validate an optimized algorithm for interpreting angiogenic factors to predict pregnancy complications and provide evidence of its general applicability;

(2) To provide evidence to translate the use of these markers into clinical practice by following up on a subsequent randomized trial by providing knowledge of the test results to clinicians, aiming at reducing the rate of premature births among women with normal sFlt1/PlGF ratios and more appropriately using resources.

Our aim in this project is to develop mobile math trails in Europe (MoMaTrE) which provides materials and methodology on one hand for teachers to create outdoor math activities easily for their classes and on the other hand for lecturers to create courses for teacher student to teach them how to enrich their future classes with mobile math activities.

Derivatives from the project are: