Abordarea noţiunilor de simetrie şi spaţiu din perspectiva grupoizilor (A groupoid approach to notions of symmetry and space)

Grant CNCSIS (Romanian National University Research Council)

Code At 127/2004 ( for young researchers)

2004-contract no. 33346/29.06.2004, 2005-contract no. 34682/24.06.2005.

Project manager: Mădălina Roxana Buneci

Research team: Constantin Cercel, Alina Dincă (students)

Abstract

ReportsThe notion of symmetry is of fundamental importance in all of the sciences. At present it is generally recognized that groups are not sufficient to characterize the symmetry. For example, objects such as crystals containing only a few unit cells, or crystals with extensive voids and inclusions like nanotechnology crystals exhibit "symmetry" but admit few automorphisms. A possibility to describe such symmetries is to use groupoids.

The main theme of this proposal is the study of the convolution algebras associated to groupoids (more precisely, we focus on two open problems concerning the C*-algebras associated to a locally compact groupoid). They play a major part in the field of noncommutative geometry, which is a rapidly growing new area of mathematics with links to many branches in mathematics and physics. It introduce a completely new concept of space (noncommutative space), by unifying methods of classical geometry with noncommutative C*-algebras.

Another goal of this proposal is to acquaint the students in the project with the way in which groupoids measure symmetry and to encourage them to apply this concept to computer graphics.

List of papers that are (partially) supported by this grant

- 2004-report (in Romanian)
- 2005-report (in Romanian)
- final report (in Romanian)

Other papers (in connection with the theme of this project)

M. Buneci, C. Cercel and A. Dincă, A program for obtaining the symmetry groupoid associated to a labeled graph, Proceedings of the International Scientific Conference "Economy and Globalization", June 4-5, 2004, Targu-Jiu, Romania, 289-294. [ISBN 973-8436-97-4].M. Buneci, Morphisms that are continuous on a neighborhood of the base of a groupoid, Studia Sci. Math. Hungar.42(3) (2005), 281-292.M. Buneci, Isomorphic groupoid C*-algebras associated with different Haar systems, New York J. Math.11, 225-245 (2005).M. Buneci, An application of Mackey's selection lemma, Studia Universitatis Babes-Bolyai, Matematica vol. L(4) (2005), 23-32.M. Buneci, Amenable equivariant maps defined on a groupoid, to appear in Proceedings of Operator algebras and mathematical physics conference (Sinaia, June 26-July 4, 2003)(eds. O. Bratteli, R. Longo, H. Siedentop), Theta Foundation.M. Buneci, C*-algebras associated to groupoids with proper orbit space, to appear in Proceedings of OT20 (20th International Conference on Operator Theory), Timişoara (Romania), June 30 -July 5 2004.M. Buneci, Amenable map and semi-direct products, Proceedings of the 9th National conference of the Romanian Mathematical Society, Lugoj, May 6-7 2005, Editura Universitatii de Vest, Seria Alef, Timisoara 2005, 69-80.M. Buneci, A notion of open generalized morphism that carries amenability , Acta Univ. Apulensis Math. Inform No.10(2005), 177-187 - Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics (ICTAMI 2005).

Other results in this project

M. Buneci, Necessary and sufficient conditions for the continuity of a pre-Haar system at a unit with singleton orbit, Quasigroups and Related Systems12(2004), 29-38.M. Buneci, The equality of the reduced and the full C*-algebras and the amenability of a topological groupoid, Recent advances in operator theory, operator algebras, and their applications, Oper. Theory Adv. Appl. (Proceedings of the 19th International Conference on Operator Theory -Timişoara, June 27-July 2, 2002-eds. D. Gaspar, I. Gohberg, D. Timotin, L. Zsido), Birkhauser Verlag, Basel/Switzerland, Vol.153(2005), 61-78.M. Buneci and V. Ungureanu, Symmetry groupoids associated to a labeled graph, Proceedings of 2nd the International Scientific Conference Economy and Globalization, June 3-4, 2005, Targu-Jiu (Romania), Ed. Universitaria, Craiova, 2005, 373-376. [ISBN 973-742-094-2].M. Buneci and V. Ungureanu, Coloring of graphs-an approach using symmetry, Proceedings of the 2nd International Scientific Conference Economy and Globalization, June 3-4, 2005, Targu-Jiu (Romania), Ed. Universitaria, Craiova, 2005, 393-398. [ISBN 973-742-094-2].

Contantin Cercel and Alina Dincă (students at Automatics) have been encouraged to attend national and international conferences. They have obtained second prize to student meeting StudIT 2004 Timisoara. Also they have been part of team that obtained first prize at 13th Summer school for image processing (June 30-July 8 2005, Szeged, Hungary).

But the main achievement is given by the fact that they learnt new mathematical concepts and applied these concepts in their field of interest.