## Final Activity Report Summary - MORIMAR (The spectrum of geometric operators on manifolds with singularities)

One of the research subjects, related to magnetic operators, was motivated by objects arising from theoretical physics. More specifically, magnetic Schroedinger operators were investigated on certain manifolds with thin ends. These manifolds admitted compactifications to smooth manifolds with boundary and the metric was asymptotically conformally cylindrical. In summary, we showed that the magnetic Laplacian on a conformally cusp manifold had purely discrete spectrum unless the magnetic field satisfied some integrality condition on the manifold. We obtained examples of compact perturbations of the magnetic field with long-range effect, which, in other words, affected the essential spectrum of the operator. In Euclidean space this was known to be impossible. The result was submitted for publication by the time of the project completion and could be consulted online at arxiv.org/abs/math.DG/0507443.

Regarding Dirac operators on Riemannian manifolds, Dr Sergiu Moroianu continued a scientific collaboration with Dr Mihai Visinescu from the National Institute of Physics and Nuclear Engineering in Bucharest and Dr Ion Cotaescu from the West University of Timisoara on 'Quantum anomalies for generalised Taub-NUT metrics'. The results of this collaboration were published in the Journal of Physics A: Mathematical and General, issue 38, 2005, pages 7005 to 7019, and the work continued this year with the study of the axial anomaly on R^4. A preprint was elaborated and was accessible in http://arxiv.org/pdf/math-ph/0511025.

Regarding quantum groups, the quantum permutation group of the X_n set corresponded to a Hopf algebra, constructed with generators and relations, known to be infinite dimensional for n greater than 3. We found an explicit representation of this algebra related to Clifford algebras. For n = 4 the representation was faithful in the discrete quantum group sense. Our goal for further research was to extend such results to infinite permutation groups. This paper, by T. Banica and S. Moroianu, was anticipated to appear in the proceedings of the Americal Mathematical Society.

A post-doctoral fellow coming from Cergy University was recruited for 12 months for research activity on some of the above problems. Moreover, five scientists from Europe and the United States of America were invited for short scientific visits and cooperation. A four days workshop was organised at the 'Institutul de Matematica al Academiei Romane' (IMAR) for investigating possible applications of the results to the study of some physical models of interest in nano-technologies.

Furtermore, Dr S. Moroianu was co-editor at the proceedings of the Seventh International Conference on Differential Geometry and its Applications. Some of the project results were presented by Dr S. Moroianu in four invited talks and by Dr S. Golenia in eight communications. Finally, two courses were delivered by Dr S. Moroianu for undergraduate students at Bucharest University and 'Scoala Normala Superioara - Bucharest'.

Regarding Dirac operators on Riemannian manifolds, Dr Sergiu Moroianu continued a scientific collaboration with Dr Mihai Visinescu from the National Institute of Physics and Nuclear Engineering in Bucharest and Dr Ion Cotaescu from the West University of Timisoara on 'Quantum anomalies for generalised Taub-NUT metrics'. The results of this collaboration were published in the Journal of Physics A: Mathematical and General, issue 38, 2005, pages 7005 to 7019, and the work continued this year with the study of the axial anomaly on R^4. A preprint was elaborated and was accessible in http://arxiv.org/pdf/math-ph/0511025.

Regarding quantum groups, the quantum permutation group of the X_n set corresponded to a Hopf algebra, constructed with generators and relations, known to be infinite dimensional for n greater than 3. We found an explicit representation of this algebra related to Clifford algebras. For n = 4 the representation was faithful in the discrete quantum group sense. Our goal for further research was to extend such results to infinite permutation groups. This paper, by T. Banica and S. Moroianu, was anticipated to appear in the proceedings of the Americal Mathematical Society.

A post-doctoral fellow coming from Cergy University was recruited for 12 months for research activity on some of the above problems. Moreover, five scientists from Europe and the United States of America were invited for short scientific visits and cooperation. A four days workshop was organised at the 'Institutul de Matematica al Academiei Romane' (IMAR) for investigating possible applications of the results to the study of some physical models of interest in nano-technologies.

Furtermore, Dr S. Moroianu was co-editor at the proceedings of the Seventh International Conference on Differential Geometry and its Applications. Some of the project results were presented by Dr S. Moroianu in four invited talks and by Dr S. Golenia in eight communications. Finally, two courses were delivered by Dr S. Moroianu for undergraduate students at Bucharest University and 'Scoala Normala Superioara - Bucharest'.