Algebra is a central area of mathematics and some of its many features are also found in this research group. During 2008-2012 members of the algebra group carried out research in CMUP's research lines 'Semigroups, Automata and Languages', 'Algebra, Combinatorics and Number theory' and 'Dynamical Systems'.

# Research areas

Analysis is a key area of mathematics, having enormous applications and relations with other areas of mathematics, physics, engineering, etc. The main strands of the work developed within the analysis group which completely described by the keywords are creation and developments of methods of integral transforms and integral equations with special functions as the kernels, methods of special functions and orthogonal polynomials to solve different analysis problems, numerical analysis and computation.

Computers have fundamentally changed the relationship between mathematics, computing and other sciences. Apart from their invaluable role in numerical, symbolic and experimental applications, computation is per se an important object of mathematical study, constantly proposing new challenges for mathematics.

With the rapid growth of computational power, today's computers provide increasingly better simulation capabilities and require the adaptation of algorithms to new architectures. The importance of computational mathematics, has never been greater.

The area of Dynamical Systems has its beginning at the University of Porto in 1982 when José Basto-Gonçalves returned to the former Department of Applied Mathematics at the Faculty of Sciences, having obtained his PhD at the University of Warwick working on Hamiltonian systems. A year later, Isabel Labouriau was hired by the same department after obtaining her PhD at the University of Warwick under the guidance of Ian Stewart. I. Labouriau and J. Basto-Gonçalves started a research line on Geometrical Methods on Differential Equations.

Geometry is a central area of modern mathematics. The 3 main strands of our work (described by the keywords) are closely intertwined, through the use of dynamical (symplectic) methods in the study of geometric problems, on the one hand, and the use of geometric methods in the study of problems in dynamics, on the other. We also contribute with applications to Economics and Biology, in the line Mathematical Models and Applications.

In the following each of the 3 strands of our work is described in more detail.

Many of CMUP's researchers work on applications of Mathematics to other subjects in science and technology. For some, this is the main aspect of their research; for others, applications appear as interesting uses for their mathematical expertise.

The group of Probability and Statistics aggregates researchers from CMUP with the common denominator that they all use tools of probability and statistics to carry out research.

The research subjects covered include ergodic theory, extreme value theory, signal processing and statistical modelling and inference. Moreover, applications to biomedicine, engineering, insurance and meteorology are also studied.

Research in this area ranges from algebraic topics such as semigroups and groups (finite, profinite, or general) to mathematical models used in computer science, namely various flavors of automata and formal languages. Current work aims not only to contribute to the theories of each of the topics but also to explore the connections between them, with applications in both directions.