PPS · Projections of Polar Spaces
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2006-11-01 → 2008-10-31
- EU contribution
- €139,484
- Participants
- 1
- Scheme
- IIF
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Results in brief
Final Activity Report Summary - PPS (Projections of Polar Spaces)
The work of this project was on the interaction between semipartial geometries and polar spaces, and in particular, the construction and characterisation of new semipartial geometries and the study of those that are obtained via projection. Finite polar spaces are one of the central objects of study in finite geometry, in the sense that they are closely related to the finite simple groups of Lie type, and that they exhibit configurations which are connected with other areas of finite geometry. The semipartial geometries include partial quadrangles, and the discovery of a new partial quadrangle by the investigators and their colleague Dr Durante also gave rise to a new strongly regular graph and a cometric association scheme (with 4 classes). Bamberg, Penttila and Schneider proved that an elation generalised quadrangle (which is both a semipartial geometry and polar space) for which the number of lines on a point is one more than a prime, is classical or a flock quadrangle. A new geometric construction of the Mathon perp-system was found, and hence by projection, we obtained a new geometric construction of the partial geometry that arises. We investigated certain equitable partitions of partial quadrangles, known as intriguing sets, and in the case where the partial quadrangle arises from projection of a polar space (generalised quadrangle), we were able to prove a characterization of those intriguing sets that arise from hemisystems. The development of a finite geometry software package for the computer algebra system GAP, is nearing completion and was used heavily in the research leading to the results of this project.
Data: CORDIS, © European Union
Project objective
In algebraic graph theory, the use of linear algebraic techniques applied to graphs consisting of vertices and edges, has been a major component in the study of strongly regular graphs; an area of modern mathematics which interests a wide variety of researchers.Finite geometry is the study of incidence structures with a finite number of points, lines, planes, etc. The most important ambient objects are projective spaces, affine spaces, and polar spaces.One of the dominant interests in recent times in finite geometry is in finding geometric models which yield strongly regular graphs and other important and related structures such as generalized quadrangles, (semi)partial geometries, and projective 2-weight codes.Since the 1970's, geometers have not only succeeded in finding such rich models from polar spaces, but have also derived characterization and classification results, which give greater insight into this phenomena.One of the central themes of this project is to introduce new techniques in order to complete these results and in particular to study the geometries arising from projecting a polar space embedded in a projective space onto a hyperplane of that space. The goal is to prove theorems in the same spirit as the ones that are known in the literature regarding a similar projection of generalized quadrangles.Development of new techniques will be necessary, and in particular the use of group theory, the mathematical study of symmetry, will be utilised. Also included in this study is an investigation into embeddings of (semi)partial geometries into affine spaces, begun in the works of De Clerck, Delanote, De Winter, and De Feyter.
Original text from CORDIS.
Participants
- UNIVERSITEIT GENT · GENTCoordinatorBelgium
Links
Data: CORDIS, © European Union