**Classificação**: *Very Good*

1. Ockham Algebras, Semigroups and Graphs:

2. Statistical Inference:

3. Distribution Theory:

4. Differential Equations and Numerical Analysis:

5. Operational Research:

6. Actuarial and Financial Mathematics.

This is a significant restructuring, only three of the four research groups of 2002 stayed. Since the last evaluation period the CMA unit has went through an explosive growth both in size, as well as the quality and magnitude of its research enterprise. The number of PhD members has grown from 12 in 2003 to 32 in 2006 and over 35 by 2008.

It is strongly advised that all members maintain a WEB page in English as it improves international visibility.

The explosive growth of the unit indicates a vision and dedication of the leaders and the members of the group to develop vital, excellent research team at UNL. The unit leader has demonstrated again his excellent administrative skills and clearly plays a critical role in nurturing the young members of the centre that enable them to conduct cutting edge research.

The Algebra, Differential Equations, and Operations Research groups are the strong legs of the centre. Their research is mostly timely, addressing research problems of interest to the broader international community.

The Centre delivered a well organized, refreshing presentation. The informative reviews were followed by very good scientific presentations, where the presenters demonstrated competence and high scholarly standards.

The panel recognizes that it is a challenge for the leadership of the centre, for UNL and FCT to ensure that the enthusiast talented young members have the resources and the chance to solidify their scientific status, to further build the momentum of the centre.

The three probability/statistics oriented research groups: Statistical Inference, Distribution Theory, Actuarial and Financial Mathematics have much in common and would benefit from closer collaboration.

Recommendations:

Consideration should be given to merging the three groups #2, #3 and #6. The strengths are complementary and the actuarial/financial component will benefit from closer contact with statisticians, while the more theoretical groups would benefit from the direct influence of applications.

- Actuarial and Financial Mathematics [RG-MATH-LVT-Almada-297-1852]
- This is a small group that is working on several problems of potential importance but the "ruin theory" component seems old-fashioned and the main emphasis of the financial component is some consulting work with banks, which does however appear to be high-quality work of contemporary importance. Just four publications in refereed journals is reasonable output given the size of the group, but obviously there is room to expand that. One researcher who is just joining the group has refereed publications in the past but not in the time period of the review. It is recommended to maintain the requirement that only scientifically active researchers are listed as members. The presence of inactive researchers might negatively impact the ranking of research groups.
- Differential Equations and Numerical Analysis [RG-MATH-LVT-Almada-297-1849]
- This is a large group. The pyramid of ages within the group seems good. This group, created in December 2005, gathers researchers with common interests in the above topics: differential equations, numerical analysis, continuum mechanics, and dynamical systems. The group has a strong training record.

The group has published papers in very good journals, has established excellent collaborations with top researchers in several other countries, and is working on problems of substantial mathematical and applications interests. Although they state their intent to collaborate with scientists and engineers, it seems most of their collaborations have been of the mathematical type. On the other hand, the group as it is currently structured is relatively new, so it is hoped that as they mature as a group, they will indeed enter into more interdisciplinary research. Likewise, it is expected that the group will build on the successes they have had in the training of students; in this respect, it seems they have produced students with high quality theses and it is hoped that the number of students produced will increase substantially.

As far as differential equations, continuum mechanics and dynamical systems are concerned, the productivity of the group is very good and publications have appeared in high level international journals. In numerical analysis, the study of the discrete problem may be too ambitious because of stability issues.

The objective: application of mathematics and numerical analysis to Physics and Engineering problems, such as kinetic models, evolutionary models, nonlinear PDE's, dimensional reduction, and fluid mechanics, is interesting and currently relevant.

At the international level, the group has organized one international summer school and most members of the group have good collaborations with international colleagues (France, USA, Switzerland, Austria, Iran, Spain, Italy,...). Two junior members of the group have obtained their PhD in foreign institutions (EPFL and UPMC). In particular, the collaborations Portugal--CMU and Portugal--UT Austin are very valuable and promising.

At the national level, the group has organized one meeting and members of the group are leaders of three research projects.

- Distribution Theory [RG-MATH-LVT-Almada-297-1848]
- The main emphasis of this group is on "exact" and "near-exact" probability approximations with applications to some classical problems in multivariate analysis. The focus is a little narrow and it is not clear just how far the group should be encouraged to pursue this particular theme, but there is a lot of energy and a reasonable record of publications. The written submission listed only four papers in refereed journals, but it appears that this record has improved if papers published during 2007 or currently accepted for publication are counted. There is also going to be a new journal, edited by the coordinator of this group, which represents a commendable level of energy and commitment (and evidence of international activity) though this should not supplant an effort to publish papers in the top journals.

There is a separate activity on extreme value theory, led by a recent PhD. This work is of good quality but should be diversified, at least into other areas of extreme value theory if not whole other areas of statistics. The proposal on Dependent Estimation in Multivariate Extremes seemed interesting, but further enquiries revealed this is not an ongoing area. The productivity of graduate students (three PhDs, one of whom is the young researcher working in extreme value theory) is reasonable given the size of the group.

- Ockham Algebras, Semigroups and Graphs [RG-MATH-LVT-Almada-297-1846]
- This group works in several areas, including algebraic logic and semigroup theory (which are traditional areas of strength in Portugal), and in graph theory. They have recently integrated some young researchers who work in the very active areas of additive number theory and non-commutative algebraic geometry. The group seems to have a clear strategy for broadening and improvement of their research.

Given the size of the group, it has been remarkably productive and published in highly regarded international mathematics journals, such as, the AMS Contemporary Math Series, Discrete Mathematics and the recent book Handbook in Linear Algebra.

In the period covered by the unit report two PhD theses were finished, and the graduates were integrated into the group. Much of the research performed by the group members was done in collaboration with well-known experts in Europe and the US.

The size of the group is still small. The caliber of the people concerned and their success so far deserves our recognition. They have set out an ambitious and worthwhile program for future research. The group carries out pure mathematical research of high quality and collaborates with leading experts. They are doing very well.

- Operational Research [RG-MATH-LVT-Almada-297-1850]
- This is a small young research group. Almost all members have gained their PhD in the last 5 years, five of them in 2007. The recent PhD’s give considerable strength to the group and represent unique areas of optimization in Portugal. Considering their start-up position, the group has made notable achievements in publications and international contacts. The publication in refereed journals is respectable given the group size, and there are a number of well known collaborators (e.g. John Dennis from Rice University and Hanif Sherali from Virgina Tech in the USA). The results on derivative free optimization and semi definite programming are at the level of the highest international standards

Not surprising, there is not much evidence of graduate student involvement other than two of the co-authors who obtained their PhD within the time period of the project. The members of the group are rather in the PDF age than ready to supervise PhD students of PDFs. The group needs time to grow to the point where they will start advising their own PhDs and PDFs. To keep in touch with cutting edge research they need to have regular, preferably longer term, visits of leaders of optimization theory and allocate significant time (1 to 2 month a year) to spend at leading research labs abroad.

Operations research, with its mathematical core is an applied science. It is not clear what are the links to practice. The involvement of one member is not clear in the research venture of the group as no publication is recorded since the year 2000.The involvement in applied projects might be a plus, however that should lead to appropriate publications too. The involvement of inactive members negatively impacts the ranking of a research group.

The Operations Research Group is well positioned to play a central, unifying role in the CMA unit. Their work reach to optimization problems in Differential Equations and Numerical Analysis theme, has overlap with the Jordan Algebra theme in the Algebra team. The historical relation to statistics and stochastics is obvious.

The list of future projects includes an enormous number of ideas – this is commendable, but the group needs to focus on a few hot areas of high promise. A more coherent description of the objectives for the group, not only as individual researchers would be beneficial for the group itself to develop smoothly. Derivative free optimization, semi definite programming, MPECs and global optimization are hot topics and provide a solid base for the group to define its vision and identity.

- Statistical Inference [RG-MATH-LVT-Almada-297-1847]
- The emphasis of this group is on mixed linear models, which is an active area of applied statistics that also poses some interesting theoretical problems. The written proposal contained an interesting discussion of the application of Jordan algebras to analyze the theoretical properties of estimators based on quadratic forms. During the presentation it emerged that the Jordan algebra theme is now less active but an alternative approach has been started based on fiducial inference. The topic of fiducial inference has long been regarded as a somewhat discredited backwater in statistics, but it has received new impetus in the last few years due to the work of researchers in the US, such as Jan Hannig. It would be good to make a more explicit link with the kind of work Hannig and the others have done. However as far as the present recommendation is concerned, it is meritorious that the group is working on theoretical problems and further development is encouraged.

There is also a substantial applied project on drought management, and further work on this and similar applications is also encouraged. The number of refereed publications is not quite as high as some other groups of comparable size, but the productivity of PhDs is high. There is also strong evidence of international collaboration, such as R. Zymslony from Poland who appears to have been a pioneer of the Jordan algebra technique.