**Classificação**: *Very Good*

CMAT is a research unit of the University of Minho. This Unit is a merger of two previous research units of Mathematics integrated in the Pluriannual Funding Programme for Portuguese research units: CMAT and Officina Mathematica.

This research team has 68 members who, following a criteria defined in the regulations referred to above, are classified as PhD-qualified integrated members (FTE), PhD-qualified collaborators (C) and members who are not PhD-qualified (S). In accordance with scientific interests and affinities, the members of the Unit are organized in 5 Research Groups in the following areas: Algebra and Logic (10FTE, 1C, 1S); Analysis and Geometry (10FTE, 3C, 4S); Statistics and Probabilities (5FTE, 1C, 3S); Computational Mathematics (7FTE, 3C, 10S); Mathematical Physics (5FTE, 1C, 4S). Before the merger there were 15 different research groups. In general the merger proved to be a success in all categories: research, training and international visibility.

A description of the main multidisciplinary research activity by area is presented below.

Probability and Statistics: The members in this area carry out research typically focused at the interface between methodology and applications such as Biostatistics, Geostatistics, Point Processes, Branching Processes, Extreme Value Theory, Nonparametric Estimation and Trend Analysis Methods for Environmental Data. This research promotes strong interdisciplinary connections. The members in this area are collaborating with other groups within the UM like GESTAP (Grupo de Estatística Aplicada) and outside with researchers from Universities in various countries (Santiago de Compostela, Vigo, Copenhagen, Louvain-la-Neuve, Leiden and Chalmers Technology). The scientific output of this area is still low but, as a result of the site visit, we see a potential for improvement in the near future.

There is no evidence concerning training of young researchers.

Mathematical Physics: The group works on relativity and kinetic theory that involves difficult problems. One of their objectives is to study stability of steady states for Vlasov-Einstein. This seems like a good idea; at least for the symmetric case it may be possible to get something interesting. This is well motivated physically, too.

External funding is very modest, an attempt should be made to secure European funding.

The number of publications is a bit thin (5 senior members with 5 publications during the period 2003-2006) but the quality is high. The journals seem to be mostly Physics journals thus it is not evident that there is an impact of this research on the mathematics community.

There is no information concerning training of young researchers.

Logic and Algebra: The research topics match those areas of algebra strong in Portugal generally. The work in Proof Systems and Programming is multidisciplinary in nature, aiming at obtaining application to programming of the study of proof systems. Specifically, the task is concerned with fundamental issues in reasoning about (the implementation of) functional programs, like types, termination, continuations, and explicit substitution. These concerns have led to collaborative work with computer scientists, for example at the Informatics Department of UM or INRIA Sophia Antipolis, resulting in published papers and participation in European networks like TYPES and APPSEM II. The scientific output of the area is very good. That said, it is hoped that in the future the area will diversify its efforts to new research topics in Algebra.

Computational Mathematics and applications: There are two distinct research areas (1) Dynamic field approach in the domain of robotics, bio-mechanics and neuro-biology (2) development of computational algorithms for nonlinear optimization, matrix decomposition, and various problems in mechanics.

Some of the goals in (2) are somewhat too general .The first area is associated with the Robotics project. The publication output is good but it is mainly that of the 2 PIs.

Geometry and Analysis: This is the strongest group in the unit. Dynamical systems is a strong component of Portuguese mathematics with an international impact. This group illustrates this statement. The Analysis part too seems to be solid

Most of the researchers in the group appear to be internationally recognized.

Some of the research on efficient algorithms for the computation of eigensystems of large matrices has been motivated by mathematical models of problems studied by colleagues from Centro de Física of UM. Examples of these problems are the calculation of energy transfer in certain media, electronic spectra and the vibration of nanoparticles.

Topology: Joint work of three members of the Geometry and Topology group and P. Ghienne (Université d’Artois) have led to results concerning the motion planning problem in robotics, which have been presented at the interdisciplinary conference Topology and Robotics 2006, ETH Zürich.

The total research output of the unit in the period of evaluation is 86 journal publication and 85 papers in Int'l conferences, 50% of these publications are internationally co-authored. The quality of journals is mixed. In 2007 the output increases by almost 30%.

Three int'l meetings and two national ones were organized by members of the unit.

All in all the international exposure of the unit is very good.

A weakness of the unit is that low number of members lead externally funded research projects or supervise student. In some of the younger groups leadership by a senior member is lacking.

- Algebra, Logic and Computation [RG-MATH-Norte-Braga-13-651]
- Four research areas are covered by the group:

Matrix completion Problems

Decidability questions on finite semigroups

Structures and ordered products

Proof systems

The research areas match those areas of algebra strong in Portugal generally, linear algebra, semigroup theory and algebraic logic. Strong links exist between the researchers here and related ones elsewhere in Portugal, and with foreign institutes. Efforts were made to engage with experts in other areas and diversify the research profile.

PRODUCTIVITY: In the three year period 2003/6, 23 papers were published in refereed journals, as well as good participation by the group in delivering lectures at conferences and publishing in conference proceedings (the latter is the main venue for the two logicians in the group). Worthy to note is that the applied logic

members became partners in the TYPES European network in type theory.

INTERNATIONAL VISIBILITY: A core of the more senior members has international recognition. It would help if the younger members were able to participate in foreign conferences a bit more.

RESOURCE NEEDS: The program of bringing in foreign experts is paying dividends and should be supported, if possible. Such a program is really important in reducing the isolation of the group.

TRAINING RESEARCHERS: The group has been quite active in this area, mostly at the master’s level, as is appropriate with such a predominantly young group.

- Computational and Theoretical Mechanics (Mechanics) [RG-X-MATH-Norte-Braga-13-776]
- Computational Mathematics and Applications (COMAPP) [RG-MATH-Norte-Braga-13-733]
- This is a large group of 10 Ph.Ds and 10 doctoral/master students. There are two distinct research areas (1) Dynamic field approach in the domain of robotics, bio-mechanics and neuro-biology (2) development of computational algorithms for nonlinear optimization, matrix decomposition, and various problems in mechanics.

Some of the goals are far too general "The mission of the group is to develop new techniques of mathematical modelling, mathematical analysis, and numerical algorithms for solving problems in applied areas", that boils down to continuing previous work on a very narrow area "We will continue previous work on the design and analysis of algorithms for the computation of eigenvalues and singular values of large matrices, very much in the spirit of the LAPACK and ScALAPACK libraries."

The first topic is certainly attracting students and seems to be an activity not found elsewhere in Portugal. As such it should be given "the benefit of doubt" even if the mathematical content of it is currently questionable.

The two senior (assoc. prof.) members are reasonably productive and with collaborations and funded research, but exclusively in Europe. Some of the papers are highly cited. However, it is not clear how novel the mathematics in these publications is. The junior staff (assist. prof) productivity is low. Significant "self breeding" is evident.

It is not easy to see how the two subgroup plans to work as a group.

- Computational Mathematics and Applications (COMAPP) [RG-X-MATH-Norte-Braga-13-746]
- This is a small group: 2 PhD internals (seniors), 2 PhD externals (seniors) plus 2 juniors without PhD.

The group works on three Topics. Hypercomplex function theory: applications to differential equations. Robust algorithms of linear algebra: applications to the reduction of matrices and the efficient computation of eigenvalues. Iterative refinement methods: application to Fredholm equations with weakly singular kernels. Solution methods: An important measure of success will be whether they can develop software packages that are comparable with the LAPACK and ScaLAPACK packages they cite.

Fairly good productivity: Two publications in Linear Algebra and its Applications, one publication in J. of Comp. and Applied Math. (several citations) plus several papers in proceedings of International Conferences.

Good visibility at international level: Participation to the Scientific Committee of two conferences, Visits of good American and Spanish scientists. Long-term and short-term invitations to good foreign centers. Participation to one Integrated Action Portugal-Germany. Three co-directed PhD theses by foreign supervisors.

Good training of students: 2 PhD theses defended, 2 Master theses defended.

- Exact solutions of Einstein's equations in General Relativity (GR Solutions) [RG-X-MATH-Norte-Braga-13-752]
- This is a small group: 3 PhD internals (seniors), 2 PhD externals (seniors) plus 1 junior without PhD. Its research topics are: Double warped space-time. Relativistic elasticity. Relativistic kinetic theory. Invariant Differential operators for solving Einstein's equations.

Very good productivity. Publications in high quality international journals, high level of citations: J. of Math. Physics, Classical and Quantum Gravity, Indiana Univ. Math. J., Annales Inst. Henri Poincaré, J. Diff. Equations, Math. Proc. Camb. Phil. Soc., Comm. in Math. Physics, General Relativity and Gravitation. Participation to conferences (conference proceedings, oral communications, posters).

Poor training of students: 3 Master theses defended.

Little funding for the period: app. 15.000 Euros.

- Geometry and Analysis: Theory, History and Applications (GATHA) [RG-MATH-Norte-Braga-13-2405]
- This is an excellent sizable and very varied group, with researchers in varied subfield of mathematics, the history of mathematics and mathematical education.

Dynamical systems are a strong components of Portuguese mathematics with an international impact. This group illustrates this statement. The Analysis part too seems to be solid

Most of the researchers in the group appear to be internationally recognized, with one or several recent publications in high-level peer-reviewed mathematical journals. The group should benefit greatly from the arrival in 2007 of A. Pinto as a "Catedratico", this appears as an excellent hiring choice which could bring many interesting developments.

- Geometry and Topology (GEOTOP) [RG-X-MATH-Norte-Braga-13-750]
- History and Epistemology of Mathematics and Calculus Education (HISEPCALC) [RG-X-MATH-Norte-Braga-13-747]
- Mathematical Physics and Modelling (MathPhys). [RG-MATH-Norte-Braga-13-655]
- The group works on relativity and kinetic theory. These are very difficult problems – non linear Einstein-Boltzmann equation, Vlasov-Einstein system, Vlasov-Maxwell system, global regularity in 3D, etc. One of their objectives is to study stability of steady states for Vlasov-Einstein. This seems like a good idea. A lot has been done on stability for Vlasov-Poisson, but basically there are two papers on stability for

Vlasov-Einstein (due to Wolansky, 2001 and Rein&Andreasson, 2007). There is some resemblance between Vlasov-Poisson and Vlasov-Einstein for symmetric solutions, so at least for the symmetric case it may be possible to get something interesting. This is well motivated physically, too.

External funding is very modest, an attempt should be made to secure European funding.

The number of publications is a bit thin (5 senior members with 5 publications during the period 2003-2006) but the quality is high. The journals seem to be mostly Physics journals thus it is not evident that there is an impact of this research on the mathematics community.

There is no information concerning training of young researchers.

- Non-linear evolution, Modelling and Stability problems in Mathematical Physics (MathPhys) [RG-X-MATH-Norte-Braga-13-748]
- This is a small group: 2 PhD internals (1 senior, 1 junior), 2 PhD externals (juniors) plus 2 juniors without PhD. Young group. They cover a wide range of topics. Non-linear and stability problems in Math. Physics: controllability, Fredholm theory, functional differential equations, general relativity, Kinetic theory, magneto-hydrodynamics.

Good productivity. Publications in high quality international journals, good level of citations: J. of Math. Anal. and Applications, Physical Reviews, Classical and Quantum Gravity, Applicable Analysis, Phys. Fluids, Math. Meth. in Applied Sciences, J. of Physics, Discrete Continuous Dynamical Systems. Participation to conferences (conference proceedings, oral communications, posters).

Good visibility at international level: Organization of seminars. Organization of short courses. Common papers with good foreign scientists.

Fair training of students: 4 Master theses defended. No Ph.D.

Good funding for the period: Participation to 4 research projects plus FCT, totalling app. 108.000 Euros.

- Probability and Dynamical Systems (DYNPRO) [RG-X-MATH-Norte-Braga-13-749]
- This is a small group; its publication output is thin (only 3 papers by a group of 3).

Some Local Organization of seminars and conferences.

No indication of depth or international collaboration Topics are classical but not necessarily very exciting. These are well studied problems and they need to be very strong to make a significant contribution.

They produced 1 PhD and 1 Masters Student.

- Project Optimization and Control (POpCon) [RG-X-MATH-Norte-Braga-13-858]
- Statistics and Applied Probability (SAAP) [RG-MATH-Norte-Braga-13-652]
- This group is good based partially on potential for future achievements rather then on past performance.

The group is a union of elements from the "Statistics in the North" group and DynPro group. It has 5 Ph.Ds, 3 Ph.D students and one collaborator

Main research efforts are on applying statistical methods to medical applications and evolution of populations. The work is decent and somehow current but not very productive and the journals are not of high quality. Connection to life sciences is a good sign. The oncology study is undoubtedly important, though if it were evaluated for NIH in the USA, one would want to see a lot of details on the study design before evaluating it against competing alternatives.

This is a small group with no student output yet, the use of money is not clear.

- Statistics in the North (SIN) [RG-X-MATH-Norte-Braga-13-751]
- This is a small and a very young group (academically) most got a PH.D in 2005-6. Publication output is still very low and the few published papers are in low ranking journals, but some of which seems to be on what could be regarded as interesting and important topics The group should be assimilated into the SAAP group (three members are in fact co-participate in this group).
- The Dynamical Field Approach to Cognition (DyFACo) [RG-X-MATH-Norte-Braga-13-859]
- PI and C. Igel are productive with good publication record. Group is successful in obtaining research funds, also outside Portugal. Research goals are focused even though the topics are interdisciplinary. There is a sizable intersection with the COMAAP group and the planned merger is recommended.

In relation to the Evaluation Panel’s Report, the Unit would like to call the attention of both the Panel and FCT to the aspects presented below.

The Unit is a merger of two previous units. As a result of this, the application process consisted of a unit form and an annexe with a

further 15 forms, one for each of the Groups of the previous units and of the present unit.

The Panel made General Comments about the Unit and Specific Group Comments for each of the 5 new groups and for 6 of the groups of the

period 2003-2006, leaving 4 groups from this period with no comments. It is not clear if and how these groups contributed to the

overall assessment of the Unit. It should also be noted that in the General Comments on the Unit it is indicated that there were 15

groups before the merger, when, in fact, there were 11 groups. This aspect is evident from the Unit application form and was explained

during the visit of the Panel to the Unit.

The Unit considers that, in spite of the detailed discussions during the Panel’s visit, some aspects of how the old groups (of the period

2003-2006) relate to the new ones are not appropriately reflected in the Panel’s Report. This is particularly evident in the area of

Mathematical Physics, where the links between the new group (ref. 655) and the two old groups (refs. 748 and 752) do not appear to be

appropriately established:

1. In relation to the new group, it is reported that “The number of publications is a bit thin (5 senior members with 5 publications during the period 2003-2006)”. In relation to the old groups, it is reported that they have “Good productivity” and “Very good

productivity”. In fact, the 5 senior members mentioned published 27 international publications (14 in journals and 13 in conference

proceedings) in 2003-2006, as listed in the forms of the two groups of this period. It should also be noted that the new groups only

filled in the Future Research part in the FCT form and this specifically requested the indication of only 5 publications from

the past.

2. The Panel reports on the new group that “There is no information concerning training of young researchers”. Again, the Future

Research part does not include information on research training. The information for the period 2003-2006 can be found in the forms of

the previous groups. It should be noted that the 4 non-PhD members of the group are all supervised by group members.

The Unit is a merger of two previous units. As a result of this, the application process consisted of a unit form and an annexe with a

further 15 forms, one for each of the Groups of the previous units and of the present unit.

The Panel made General Comments about the Unit and Specific Group Comments for each of the 5 new groups and for 6 of the groups of the

period 2003-2006, leaving 4 groups from this period with no comments. It is not clear if and how these groups contributed to the

overall assessment of the Unit. It should also be noted that in the General Comments on the Unit it is indicated that there were 15

groups before the merger, when, in fact, there were 11 groups. This aspect is evident from the Unit application form and was explained

during the visit of the Panel to the Unit.

The Unit considers that, in spite of the detailed discussions during the Panel’s visit, some aspects of how the old groups (of the period

2003-2006) relate to the new ones are not appropriately reflected in the Panel’s Report. This is particularly evident in the area of

Mathematical Physics, where the links between the new group (ref. 655) and the two old groups (refs. 748 and 752) do not appear to be

appropriately established:

1. In relation to the new group, it is reported that “The number of publications is a bit thin (5 senior members with 5 publications during the period 2003-2006)”. In relation to the old groups, it is reported that they have “Good productivity” and “Very good

productivity”. In fact, the 5 senior members mentioned published 27 international publications (14 in journals and 13 in conference

proceedings) in 2003-2006, as listed in the forms of the two groups of this period. It should also be noted that the new groups only

filled in the Future Research part in the FCT form and this specifically requested the indication of only 5 publications from

the past.

2. The Panel reports on the new group that “There is no information concerning training of young researchers”. Again, the Future

Research part does not include information on research training. The information for the period 2003-2006 can be found in the forms of

the previous groups. It should be noted that the 4 non-PhD members of the group are all supervised by group members.