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 research line Computational Mathematics has developed from the recognition by CMUP members of the intense and important computational effort that was being undertaken independently in different areas of Mathematics.

In what concerns the experience in this field, which includes implementing algorithms, one should mention work in automata theory, numerical analysis, computational semigroup theory and contribution to the development of computer algebra systems.

Various programming languages are used, including the ones provided by computer algebra system themselves, such as Mathematica, Magma, Matlab, PARI or GAP.

On the other hand, there are cases where the usage of the computer plays a fundamental role in research, either by intensive use such as in signal and data analysis or in other contexts such as the direct use of tools provided by the available computer algebra systems.

An important example is our involvement with GAP (Groups, Algorithms, Programming; http://www.gap-system.org/), a powerful and flexible system for computation in group theory, algebra and discrete mathematics. User contributed programs, called packages, are distributed with GAP. Both the system and the packages are open-source, commonly distributed under the GPL license. M. Delgado is co-author of various GAP packages and responsible for the implementation of GAP objects such as automata and numerical semigroups, as well as for starting the production of routines for automatic visualization of GAP objects. One of M. Delgado's co-authors is S. Linton who is director of the Centre for Interdisciplinary Research in Computational Algebra, of the University of St Andrews, one of the four Centres that together lead the international consortium developing GAP. M. Delgado's experience with GAP involves also contacts with various members of The GAP Group, namely through visits to St Andrews and Braunschweig, another centre of the referred consortium, with participation in workshops for GAP package authors.

The packages in which M. Delgado has been involved have been motivated by concrete problems and influenced his research.

We refer 'numericalsgps' as an example of a successful package whose maintenance and recent development is done jointly with P. A. García-Sánchez (University of Granada and collaborator of CMUP). It is dedicated to numerical semigroups and the algorithms implemented reflect the state of the art in the area. The package is successfully used not only by the authors (it has greatly influenced recent works published in Mathematics of Computation and IEEE Trans. on Information Theory; on the other hand, the package benefited from the new algorithms provided in these papers), but also by other authors, which is reflected inhttp://www.swmath.org/software/640, a page given by an information service for mathematical software provided by Springer (which is still a prototype).

Besides the case of the computer algebra system just presented, we present here an overview of research currently done at CMUP that can be broadly considered Computational Mathematics. Most of the work is done in connection with research in specific areas of Mathematics, and often involves applications to technology or other sciences.

That aspect of the work is described in more detail in the texts about research groups and other research lines. Here the focus is on the development and intensive use of computational tools.

Generally, the computational part will help in the formulation of conjectures or the obtention of counter-examples; the theorems are proved by the traditional means.

As an example, we can mention that Carlos Rito uses computer algebra, mainly the software Magma, to support his research with algebraic varieties, in particular surfaces. He has had success in the computation of new examples of complex algebraic surfaces and intends to continue research in this line.

Research done in CMUP also deals with providing computational resources for the scientific community and the general public. This aspect is better described in the justification for the laboratory intensity level of the unit (6.4).

Another aspect deals with developing code for work on numerical simulations of models and analysis of large sets of data. This is applied to problems arising in Bioengeneering, Medicine, Biology and Economics and is best described elsewhere in this proposal.