Research
My research is mainly in algebraic combinatorics, in particular the theory of association schemes. My interests also include permutation groups, finite geometry, graph theory, designs and codes, and combinatorial computing. I have an Erdős number of 3.
Publications
See preprints listed on the arXiv. Note: published versions may differ slightly.
13. The synchronisation hierarchy via coherent configurations, submitted. With John Bamberg. (See arXiv or abstract.)
Abstract:
We describe the spreading property for finite transitive permutation groups in terms of properties of their associated coherent configurations, in much the same way that separating and synchronising groups can be described via properties of their orbital graphs. We also show how the other properties in the synchronisation hierarchy naturally fit inside this framework. This combinatorial description allows for more efficient computational tools, and we deduce that every spreading permutation group of degree at most
12. Rational Delsarte designs and Galois fusions of association schemes, submitted. With William J. Martin. (See arXiv or abstract.)
Abstract:
Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial
11. Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension, submitted. With Philipp Heering and Klaus Metsch. (See arXiv or abstract.)
Abstract: A chamber of the vector space
10. Tactical decompositions in finite polar spaces and non-spreading classical group actions, Designs, Codes and Cryptography, 2024. With John Bamberg, Michael Giudici, and Gordon F. Royle. (See arXiv or Journal or abstract.)
Abstract: For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of synchronising and separating are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of spreading permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an AB-Lemma) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.
9. On Bruen chains, Finite Fields and their Applications, 2024. With John Bamberg and Geertrui Van de Voorde, (See arXiv or Journal or abstract.)
Abstract: It is known that a Bruen chain of the three-dimensional projective space
8. A census of small Schurian association schemes, International Journal of Algebra and Computation, 2024. (See arXiv or Journal or abstract.)
Abstract: Using the classification of transitive groups of degree
7. Separating rank 3 graphs, European Journal of Combinatorics, 2023. With John Bamberg, Michael Giudici, and Gordon F. Royle. (See arXiv or Journal or abstract.)
Abstract: We classify, up to some notoriously hard cases, the vertex-primitive strongly regular graphs which meet both the Delsarte and Hoffman bounds. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give the first known examples of synchronising but not
6. Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry, Algebraic Combinatorics, 2023. With John Bamberg. (See arXiv or Journal or abstract.)
Abstract: In this paper we show that if
5. Synchronising primitive groups of diagonal type exist, Bulletin of the London Mathematical Society, 2022. With John Bamberg, Michael Giudici, and Gordon F. Royle. (See arXiv or Journal or abstract.)
Abstract: Every synchronising permutation group is primitive and of one of three types:
affine, almost simple, or diagonal.
We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that
4. The non-existence of block-transitive subspace designs, Combinatorial Theory, 2022. With Daniel R. Hawtin. (See arXiv or Journal or abstract.)
Abstract: Let
3. A family of hemisystems on the parabolic quadrics Journal of Combinatorial Theory, Series A, 2020. With Alice C. Niemeyer. (See arXiv or Journal or abstract.)
Abstract: We constuct a family of hemisystems of the parabolic quadric
2. On m-ovoids of regular near polygons, Designs, Codes and Cryptography, 2018. With John Bamberg and Melissa Lee. (See arXiv or Journal or abstract.)
Abstract: We generalise the work of Segre (1965), Cameron – Goethals – Seidel (1978), and Vanhove (2011) by showing that nontrivial
1. Bruck nets and partial Sherk planes, Journal of the Australian Mathematical Society, 2018. With John Bamberg and Joanna B. Fawcett. (See arXiv or Journal or abstract.)
Abstract: In Bachmann’s Aufbau der Geometrie aus dem Spiegelungsbegriff (1959), it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines, and conversely. Sherk (1967) generalised this result to charac- terise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow non-collinear points.
Doctoral thesis
Designs in Finite Geometry, 2020 (available via RWTH or UWA)
Refereeing
I have refereed for the journals
- European Journal of Combinatorics
- Combinatorial Theory
- Designs, Codes and Cryptography
- Electronic Journal of Combinatorics
- Discrete Mathematics
- The Australasian Journal of Combinatorics.