# Projects

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:

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.

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.

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.

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 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.

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.

This is a project that was submitted within the Medicine area. I am a member (15%) of the project, responsible for the

experimental design and statistical analysis.

Recent international and national mathematics curriculum guidelines indicate the development of students’ statistical literacy, at

different levels of schooling, as major educational aim. In Portugal, the current mathematics syllabus for basic education assigns an

increased emphasis on statistics and presents more demanding learning objectives in statistics, since the elementary levels. This is a

challenging situation for practicing teachers, requiring them to develop new perspectives and professional knowledge. In this context,

Pretende-se obter novos resultados para fragmentos da lógica de predicados de primeira ordem, cujos modelos são palavras finitas e infinitas. 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 entre os mecanismos de descrição lógica e a teoria dos 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.

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.