**Classificação**: *Very Good*

The unit's internal structure remains the same as in the last evaluation, counting six groups:

1. Algebra and Geometry

2. Complex and Hypercomplex Analysis

3. Functional Analysis and Applications

4. History of Mathematics

5. Mathematical Systems Theory

6. Probability and Statistics

Currently the unit has 34 PhD members and 22 PhD students. The large number of graduate students is interesting in view of the fact that most units, including those that are considered to be doing outstanding work, have serious difficulties in attracting students. This is also observed in the newly proposed joint PhD program with the U. Minho, that has secured already 5 (Portuguese) PhD students, and 10 more will be enrolled next year. It is possible that this discrepancy may be resolved with a nationwide basic criteria of recruitment hinged on a standardized GRE-type (Graduate Record Examination).

The Panel recommends that the unit encourages recent PhDs to pursue postdoctoral training abroad.

A few areas of interdisciplinary work have been identified, including genome analysis, environment, and molecular genetics. It is not clear how the unit plans to implement this strategic vision, given the recruitment constraints at the faculty level. It is possible that its integration in the Bioinformatics group, created at the University of Aveiro, and including Computer Science, Statistics, Biology, will facilitate this process.

The unit has a remarkable record in outreach activities, mostly targeted to basic and secondary schools, including the project Geometrix of the Algebra and Geometry for the development of educational software.

The unit has been very active in the organization of conferences, mini-courses and summer schools, and the level of internationalization is reasonable, mostly in what concerns the group on Complex and Hypercomplex Analysis.

The panel recommends that this unit and the other unit at the University of Aveiro (CEOC- Centro de Estudos em Optimização e Controlo) should merge. The two units should make an effort to reflect on their respective strengths and identify strategic areas. This merging should streamline both units, and should benefit groups in both units by increasing the visibility and impact of mathematics research carried out at the University of Aveiro.

- Algebra and Geometry Group [RG-MATH-Centro-Aveiro-255-1252]
- This is a very young team, with 6 out of 10 PhD members having obtained their degree during the period 2003-05. Just as in the previous evaluation, the Panel observes that one focus of activity of this group is the development of the teaching/computer learning packages, the Geometrix project.

This is a group which works in a variety of areas including combinatorics, category theory, geometry, semigroups, applied logic, and linear algebra. The research done on the topology of surfaces is close enough to mainstream to attract outside attention, but other research subjects are perhaps removed from current interests, or very abstract areas where progress is difficult (e.g. category theory).

Several members of the group have international collaborators, and publications appear in a large number of well-known international journals.

The group supervised 5 PhD students who are currently members.

A new area, number theory, is being introduced, and a potential member has been identified already. It is not clear what the plans are to build critical mass around in this area so that it may make an impact in the field.

- Complex and Hypercomplex Analysis Group [RG-MATH-Centro-Aveiro-255-1253]
- This group works on Clifford analysis, and its applications to PDEs, function spaces, inverse problems, wavelets (adaptive, frames on the sphere, discretization), polynomials (orthogonal, monogenic), and numerical analysis. Very few groups worldwide work in the area of Clifford Analysis; the Panel questions its current interest and impact on the frontier of mathematics research. This is reflected in the rating of the group.

The group has been strongly engaged in the training of graduate students. The productivity is very good, both in terms on number of paper as well as the quality of international journals.

The group has good visibility at the international level, including contacts with the Konrad-Zuse Institute in Berlin, and some of its members are on Editorial Boards of two journals. The group is very active in the organization of conferences.

The group has been successful in securing FCT and external to FCT funding, including European, of research projects.

- Functional Analysis and Applications Group [RG-MATH-Centro-Aveiro-255-1254]
- This group seems to lack cohesion, with work ranging from theory of function spaces (factorization of matrix functions), operator theory, differential equations, differential inclusions, calculus of variations, including the study of the p-Laplacian, Lavrentiev phenomena, regularity of minimizers in 2D, differential inclusions of gradient type. None of this work is applied or seems to be applicable in the short run, nor are these researchers interacting with researchers from other disciplines in search of motivations outside mathematics. For this reason, the Panel recommends that the group reconsiders its name.

Some of these topics are very contemporary and major research centers in Europe and in the USA (e.g. SISSA, Max-Planck Institute in Leipzig, UC Berkeley, U. Naples, U. Parma, etc.) are tackling them. The group does not seem to be in contact with these. This is also reflected on the quality of the journals that may be considered to be of moderate level. There are many conference proceedings papers. The group is encouraged to invest less on these and more on peer reviewed high level publications.

Training of PhD students is limited, with only one PhD thesis supervised during the period of this evaluation.

Involvement in the organization of conferences and workshops is adequate.

- History of Mathematics Group [RG-MATH-Centro-Aveiro-255-1255]
- Mathematical Systems Theory Group [RG-MATH-Centro-Aveiro-255-1256]
- This group is working on important topics including linear systems theory (periodic behavioral systems, convolutional codes, code decomposition) with applications in biomedical engineering (control for biomedical applications, drug administration, anesthesia). There is substantial interaction with physicians. The new research work on image processing, with a view towards image segmentation in biology, involves collaborations with the department of electrical engineering at IST.

The group publishes in good journals in the area, including journals in the biomedical area. There may be too many conference proceedings.

The training record is very strong, and this team has secured considerable external (to FCT) funding, including an European funded project.

- Probability and Statistics Group [RG-MATH-Centro-Aveiro-255-1257]
- The group strength is in applied statistics (environmental modeling, biology and genetics). Major topics here are extreme value theory, nonlinear time series, and genetics. There has been considerably less progress on the genetics component than the other two, and this raises the question of whether the genetics part of the future research proposal is viable in the international context. The extreme value and time series components have resulted in good publications on statistical estimation properties in respected journals.

The record for graduate student advising is strong for a group of this size.

This group has several interdisciplinary projects although this is not reflected in the origin of funding that it is mostly FCT.

Comments to the evaluation report of the R&D Unit “Matemática e Aplicações” (due to space limitations, we display here only an abridged version of the document sent to FCT):

1. About the “unusual” large number of PhD students and implicit doubts about our admission policy: a considerable number of such PhD students have a grant from FCT, therefore were submitted also to an external evaluation; it is also common that they publish papers in international journals during the PhD and have international experts in their vivas.

2. Two memberships in editorial boards of international journals were overseen.

3. The Complex and Hypercomplex Analysis group apparently received a lower ranking because of the allegedly lower current interest and impact of Clifford Analysis on the frontier of mathematics research. Nevertheless, the previous evaluation panel considered that our work was “serious and worthwhile”. Moreover, the number of mathematicians working in the field is significantly increasing worldwide, as shows the recent creation of two Birkhäuser journals ("Adv. in Appl. Clifford Algebras" and "Compl. Anal. Oper. Theory") where this area has considerable weight.

4. The report on the Functional Analysis and Applications group (FAAg) is superficial, unfair and inaccurate:

FAAg covers a broad research area, with several main topics of interest, but the “Functional Analysis” is the main tool for the group research and the cohesion factor. The word “Applications” is mainly used in the sense of applications to other parts of Mathematics, like ODEs and PDEs.

Factorization of matrix functions is wrongly included inside the area of Functions Spaces, instead of Operator Theory.

FAAg is in contact with several researchers working in its areas of interest from Belarus, Czech Rep., Georgia, Greece, Italy, Germany, Romania, UK, USA, Venezuela.

FAAg has a large list of publications, and publishes four times more papers in journals than in proceedings. Thus, the observations made about the proceedings publications cannot be understood. Moreover, we would like to express our reservation on a judgment about “moderate level” journals, with due respect for and without questioning the general competence of the evaluation panel.

The statement about the training of PhD students is wrong. FAAg supervised eight PhD students between 2003 and 2006.

1. About the “unusual” large number of PhD students and implicit doubts about our admission policy: a considerable number of such PhD students have a grant from FCT, therefore were submitted also to an external evaluation; it is also common that they publish papers in international journals during the PhD and have international experts in their vivas.

2. Two memberships in editorial boards of international journals were overseen.

3. The Complex and Hypercomplex Analysis group apparently received a lower ranking because of the allegedly lower current interest and impact of Clifford Analysis on the frontier of mathematics research. Nevertheless, the previous evaluation panel considered that our work was “serious and worthwhile”. Moreover, the number of mathematicians working in the field is significantly increasing worldwide, as shows the recent creation of two Birkhäuser journals ("Adv. in Appl. Clifford Algebras" and "Compl. Anal. Oper. Theory") where this area has considerable weight.

4. The report on the Functional Analysis and Applications group (FAAg) is superficial, unfair and inaccurate:

FAAg covers a broad research area, with several main topics of interest, but the “Functional Analysis” is the main tool for the group research and the cohesion factor. The word “Applications” is mainly used in the sense of applications to other parts of Mathematics, like ODEs and PDEs.

Factorization of matrix functions is wrongly included inside the area of Functions Spaces, instead of Operator Theory.

FAAg is in contact with several researchers working in its areas of interest from Belarus, Czech Rep., Georgia, Greece, Italy, Germany, Romania, UK, USA, Venezuela.

FAAg has a large list of publications, and publishes four times more papers in journals than in proceedings. Thus, the observations made about the proceedings publications cannot be understood. Moreover, we would like to express our reservation on a judgment about “moderate level” journals, with due respect for and without questioning the general competence of the evaluation panel.

The statement about the training of PhD students is wrong. FAAg supervised eight PhD students between 2003 and 2006.