FCT

R&D Institutions

Resultado da avaliação 2007 na área de Matemática

Unidade de I&D

Centro de Álgebra da Universidade de Lisboa [MATH-LVT-Lisboa-143] visitada em 20/02/2008

Classificação: Very Good

Comentários do painel de avaliação
Sobre a unidade
This unit is divided into three groups:

I. Algebras, Modules and Rings
II. Lattices, Universal Algebra and Algebraic Logic
III. Semigroups, Automata and Languages.

In Group I, the principal areas being investigated are algebras and rings, with particular emphasis on radicals and preradicals, injectivity properties of modules, and algebraic analysis, especially D-modules, sheaves and stacks, and also aspects of string theory and K-theory. The last couple of topics represents a new direction for this group and is designed to increase its interaction with other researchers in Lisbon, especially with those in the area of Mathematical Physics.
Group II studies subvarieties of lattice-ordered algebras, completions of finite orders and the algebras of logic. They investigate the strong inter-relationship between universal algebra, lattice theory and algebraic logic. A particular focus has been the application of these subjects to the constraint satisfaction problem in computer science.
Group III. This is the biggest group and contains researchers with a broad variety of interests, under the general umbrella of semigroup theory and related topics. The relationship between finite semigroups, finite automata and rational languages is a major topic of concentration. Following recommendations of previous evaluation panels, the group has diversified to include a large computational discrete algebra component, especially, the development and application of algorithms within the language GAP.

All three groups have been active in the supervision of Masters and PhD students. They have also made many contributions to the organization of conferences, both within Portugal and outside, and to the dissemination of their work at international meetings. The unit has organized several advanced courses and seminars, some of which were designed to provide up-to-date minute information on advances in the principal areas being investigated, while others were of a more instructional nature and delivered knowledge on more general areas of current research in algebra. The unit also delivered a substantial outreach program to schools and the general public, and engaged in the debate on mathematical education issues generally.
Substantial funding has been sought and obtained from the FCT for several projects, and, in addition, the Unit has obtained support from the Portuguese-Hungarian cooperation, the Anglo-Portuguese Cooperation and the Fundaçao Luso-Americana. However, the panel encourages the Unit to intensify its efforts to secure more funding from EU network programs and other international sources.
The panel views this as one of the leading research units in algebra in this country. Semigroup theory has a long tradition in Portugal and currently this unit is one of the principal contributors to the country’s reputation in this subject. Algebraic logic is also a strong feature of research throughout the country and this unit makes a good contribution to it. The research area of Group I, while related to the other groups, has a different focus from that done elsewhere in Portugal.
The quality of the research in the Unit is generally high, with the findings published in well-recognized, and highly regarded, journals. The level of productivity, while being satisfactory overall, varies somewhat over the groups. The group PIs and several members of the Unit are well-known internationally for their work.
Sobre os grupos de investigação
Algebras, Modules and Rings [RG-MATH-LVT-Lisboa-143-431]
The study of the microsupport of the complex of tempered D-modules and algebraic analysis, with applications to PDE, is a major part of the research effort. A new area of investigation has been self-extensions of simple modules over semi-simple Lie algebras. Progress was also made on developing a Goldie theory in lattices. Further diversification was achieved through the addition of research in related parts of string theory and geometric topology.
While the papers published by the group are of high quality and appear in leading journals, the panel views the overall output as low. The group continues to have good international visibility, and an early book by the PI with Kashiwara and Schapira is still important in this area. However, there is room for improvement in its level of interaction with other researchers locally. The new topics recently introduced in its research should lead to more collaboration with colleagues in geometry and mathematical physics.
Lattices, Universal Algebra and Algebraic Logic [RG-MATH-LVT-Lisboa-143-433]
The quality of its research is high and there is a good measure of international collaboration and visibility. However, its productivity seems a little low. The panel does recognize that the work of the group is relevant to the constraint problem in theoretical computer science. There is a general sense that the future of the area of universal algebra/lattice theory is uncertain (see, for example, the recent controversy over Wehrung's solution to a famous problem in this area). Most of the recent research on the algebraic side of logic involves category theory, and there has been recent progress on some longstanding problems in lattice theory, using ideas from set theory. We recommend that the group should consider broadening its perspective.
Semigroups, Automata and Languages [RG-MATH-LVT-Lisboa-143-434]
This is a sizeable group of researchers with a broad variety of interests, under the general umbrella of semigroup theory. The group leader, G. Gomes, has established a strong international reputation. Two others, J. Araujo and V. Fernandes, have made a good start to developing international reputations and contacts. The other members of the group are still relatively early in their careers. The group has been highly productive. We especially commend it for its contribution to GAP. Strong collaboration continues with colleagues in St Andrews, the current home of GAP, and we look forward to further significant progress in this area and to the strengthening of links with the Centre for Interdisciplinary Research in Computational Algebra in the UK. Within Portugal, the members play a major role in collaboration with colleagues in the related group in CMUP in organizing and directing research activity in this subject.