### Melissa Lee

#### Thesis

The m-covers of generalised quadrangles and their generalisations

#### Summary

This project aims to extend the concept of relative hemisystems in various ways that have not been explored in the literature.

Relative hemisystems are inspired by the concept of hemisystems. A hemisystem of a generalised quadrangle R of order (q^2,q) is a set of lines L such that every point of the generalised quadrangle meets L in half of its lines.

There has been much exploration in the literature about hemisystems of classical and nonclassical generalised quadrangles, the non-existence of nontrivial covers of the points of R that are not hemisystems and so on. In particular, it has been shown that hemisystems only exist for odd q.

Relative hemisystems were defined as an analogous version of hemisystems, but for q even. A handful of infinite families have been discovered, but the concept has not been extended in the same way as it has for hemisystems.

#### Why my research is important

My research is important because relative hemisystems give rise to nice structures called association schemes, which are of interest to mathematicians in a variety of areas.

Finding new examples of relative hemisystems and extending the concept in different ways may allow us to find new examples of these association schemes.

#### Funding

- This higher degree by research is funded by a Hackett Postgraduate Scholarship.