## 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.

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.