# Research

My research interests lie largely in finite incidence geometry, algebraic combinatorics, and computation.

My Erdős number is 3 via Lansdown → Bamberg → Cameron → Erdős or via Lansdown → Niemeyer → Seress → Erdős.

# Publications

See preprints listed on the arXiv. Note: published versions may differ slightly.

##
10.
Tactical decompositions in finite polar spaces and non-spreading classical group actions
*submitted*.
With John Bamberg, Michael Giudici, and Gordon F. Royle.
(*See arXiv 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,
*submitted*.
With John Bamberg and Geertrui Van de Voorde.
(*See arXiv or abstract.*)

**Abstract:** It is known that a Bruen chain of the three-dimensional projective space $\mathrm{PG}(3,q)$ exists for every odd prime power $q$ at most $37$, except for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not exist for $41\leqslant q\leqslant 49$. We develop a model, based on finite fields, which allows us to extend this result to $41\leqslant q \leqslant 97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$.
Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs.

##
8.
A census of small Schurian association schemes,
*International Journal of Algebra and Computation*, 2024.
(*See arXiv or Journal or abstract.*)

*International Journal of Algebra and Computation*

**Abstract:** Using the classification of transitive groups of degree $n$, for $2 \leqslant n \leqslant 48$,
we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 \leqslant n \leqslant 48$. The results of this classification are made available as a supplementary database.

##
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.*)

*European Journal of Combinatorics*

**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 $\mathbb{Q}\mathrm{I}$ groups of affine type.

##
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.*)

*Algebraic Combinatorics*

**Abstract:** In this paper we show that if $\theta$ is a $T$-design of an association scheme $(\Omega, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $\theta$ consists of precisely half of the vertices of $(\Omega, \mathcal{R})$ or it is a $T'$-design, where $|T'|>|T|$.
We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s, s^2)$ do not exist. We give short proofs of similar results for
(i) partial geometries with certain order conditions;
(ii) thick generalised quadrangles of order $(s,s^2)$;
(iii) the dual polar spaces
$\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$;
(iv) the Penttila-Williford scheme.
In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.

##
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.*)

*Bulletin of the London Mathematical Society*

**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 $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$
acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$.
Furthermore, we show that such groups are non-spreading for all prime powers $q$.

##
4.
The non-existence of block-transitive subspace designs,
*Combinatorial Theory*, 2022.
With Daniel R. Hawtin.
(*See arXiv or Journal or abstract.*)

*Combinatorial Theory*

**Abstract:** Let $q$ be a prime power and $V\cong\mathbb{F}_q^d$. A \emph{$t$-$(d,k,\lambda)_q$ design}, or simply a *subspace design*, is a pair $\mathcal{D}=(V,\mathcal{B})$, where $\mathcal{B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $\lambda$ elements of $\mathcal{B}$. Subspace designs are the \emph{$q$-analogues} of balanced incomplete block designs. Such a design is called *block-transitive* if its automorphism group $\mathrm{Aut}(\mathcal{D})$ acts transitively on $\mathcal{B}$. It is shown here that if $t\geqslant 2$ and $\mathcal{D}$ is a block-transitive $t$-$(d,k,\lambda)_q$ design then $\mathcal{D}$ is trivial, that is, $\mathcal{B}$ is the set of all $k$-dimensional subspaces of $V$.

##
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.*)

*Journal of Combinatorial Theory, Series A*

**Abstract:** We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \geqslant 2$ and all odd prime powers $q$, that admit $\Omega_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \geqslant 4$.

##
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.*)

*Designs, Codes and Cryptography*

**Abstract:** We generalise the work of Segre (1965), Cameron – Goethals – Seidel (1978), and Vanhove (2011) by showing that nontrivial $m$-ovoids of the dual polar spaces $\mathsf{DQ}(2d,q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$ $(d>3)$ are hemisystems. We also provide a more general result that holds for regular near polygons.

##
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.*)

*Journal of the Australian Mathematical Society*

**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.