Research

My early research was in classical groups and finite geometry, linking topological features of discrete geometries with algebraic properties of their automorphism groups, using techniques such as the amalgam method.

Current projects

• Axial algebras: a new class of non-associative algebras linked to VOAs.
• Code algebras: non-associative algebras formed from linear codes.
• Vahlen groups and their connections to higher dimensional automorphic forms.
• Sidki's conjecture for presentations of orthogonal groups.

I have written a magma package to construct axial algebras with a given Miyamoto group.

I have written a program in GAP to calculate the fundamental group of a geometry.

Please email me if you make use of it, have any comments, or find any bugs in either of these.

Axial algebras

Axial algebras are a new class of commutative non-associative algebras, related to groups, which were introduced recently by Hall, Rehren and Shpectorov. This concept grew out of an axiomatisation of some important properties of vertex operator algebras (VOAs), which are mathematical counterparts of structures developed by physicists working on quantum mechanics. In particular, the most famous example of axial algebra, the $196,884$-dimensional Griess algebra for the Monster sporadic simple group, arises in the Monster VOA, which is a key ingredient in Borcherd's Fields-medal-winning research on the Monstrous Moonshine conjecture. Other examples of axial algebras include Jordan algebras (classified by another Fields medalist, Zelmanov), which are related to the classical groups and groups of type $G_2$ and $F_4$, and Matsuo algebras arising from the $3$-transposition groups.

An axial algebra is a commutative non-associative algebra $A$ generated by idempotents, called axes, which are primitive and semisimple. That is, the multiplicative action of a given idempotent $a$ on $A$ splits the algebra into a direct sum of eigenspaces $A = \bigoplus A_\lambda$. We require that elements in these eigenspaces multiply according to certain fusion rules.

If the fusion rules are $\mathbb{Z}_2$-graded, then associated to each axis $a$ there is an involution $\tau_a$. The group generated by these involutions is called the Miyamoto group. For the Griess algebra, there is a bijection $a \mapsto \tau_a$ between the axes and the $2A$ involutions in the Monster. Since these generate the Monster, we see that the Miyamoto group of the Griess algebra is the Monster.

The most simple fusion rules, called Jordan-type, give us Jordan and Matsuo algebras. The next simplest, hyper-Jordan-type, add the Griess algebra and its subalgebras, capturing the majority of sporadic simple groups. Hence axial algebras offer the promise of a unified theory for finite simple groups - something that group theorists have craved for generations.

I am working on several projects in this area. One is joint with Sergey Shpectorov to construct axial algebras with a given Miyamoto group. You can find our paper on this here and we have implemented our algorithm in a magma package. Another is studying the structure of axial algebras with Sanhan Khasraw and Sergey Shpectorov. We are also looking at enumerating $3$-generated axial algebra of Monster-type. Sergey and I are also interesting in completing the classification of axial algebras of Jordan type.

Code algebras

Inspired by the axiomatic approaches to VOAs described above, Rehren, Castillo-Ramirez and I have recently introduced the new concept of a code algebra, which is a new class of commutative non-associative algebra constructed from a binary linear code. Our construction is an axiomatisation of the construction of code VOAs. These were first studied independently by Miyamoto, and by Dong, Griess and Höhn, and they form an important class of VOAs whose representation theory is governed by two binary linear codes. Lam and Yamauchi showed that every framed VOA $V$ (such as the one for the Monster) has a uniquely defined code sub VOA and $V$ is a simple current extension of its code sub VOA. Every code VOA has a code algebra embedded in it, but code algebras form a richer class of algebras and they merit further investigation in their own right.

Code algebras have some obvious idempotents and for these we can easily associate a fusion table. However, these are not enough to generate the whole algebra. We introduce a so-called $s$-map construction which produces idempotents from a subcode. In general it is difficult to describe the eigenvectors for such an idempotent, let alone the fusion table. However, for minimal subcodes $\langle \alpha \rangle$ generated by a single codeword $\alpha$, this is possible. Moreover, we have calculated the fusion table. Under some mild conditions on the code, code algebras are axial algebra with respect to these idempotents.

We have classified when the fusion table is $\mathbb{Z}_2$-graded and hence when it leads to automorphisms of the algebra. In particular, this leads to an infinite family of $(\mathbb{Z}_2 \times \mathbb{Z}_2)$-graded algebras. This is especially exciting as it provides the first example of an axial algebra with a grading other than a $\mathbb{Z}_2$-grading.

Vahlen groups

Vahlen groups are sets of $2 \times 2$ matrices whose entries lie in a Clifford algebra. Previously, they were only defined over $\mathbb{R}$, $\mathbb{C}$, of $\mathbb{F}_p$ with a specific quadratic form in an ad-hoc way. In a recent paper published in Mathematische Zeitschrift, I introduce them in a uniform way in their most general setting, over arbitrary rings and with an arbitrary quadratic form.

Even in this most general setup, I was able to show analogous result to the classical case which is roughly that for the special Vahlen group and the special paravector Vahlen groups we have

where $M = H \perp L$ is a splitting and $L$ is a hyperbolic line with respect to the quadratic form $q$.

In the 1980s, Vahlen groups were found to play a central role in understanding higher dimensional harmonic automorphic forms. In the usual $2$-dimensional case, roughly speaking, modular and Maaß forms $f$ are functions on the upper half plane which satisfy some analytic properties and a functional equation

where $k \in \mathbb{N}$ and $\begin{pmatrix}a&b\\c&d \end{pmatrix}$ is an element of a congruence subgroup of the group of Möbius transformations. Natural examples of these are theta functions, which encode representation numbers of quadratic forms, and Eisenstein functions, which encode Riemann-zeta values and power divisor sums. To consider higher dimensions, one allows $f$ to be Clifford algebra valued, rather than $\mathbb{R}$- or $\mathbb{C}$-valued. These then act on the upper half space and Vahlen groups are used in place of the group of Möbius transformations.

Initially, Vahlen groups were only studied for Clifford algebras with a specific quadratic form over $\mathbb{R}$, $\mathbb{C}$, or fields of odd characteristic. In dimension two, modular forms over a number field and a local field are also of interest and it is natural to consider the existence of Clifford valued modular forms in such situations. Here, Vahlen groups defined over a more general ring may play an essential role. For example, do such modular forms span a finite-dimensional space and, if so, what is its dimension? Moreover, do generalised theta or Eisenstein series exist? This is joint work with Dan Fretwell and Tom Oliver.

Sidki's conjecture

In 1982, Sidki generalised a classical presentation for the alternating group to the following:

where $n, m \in \mathbb{N}$ and he conjectured it is finite for all $m$ and $n$. This is still open. Sergey Shpectorov, Said Sidki (Brasília) and I have generalised this to a question of identifying some infinite groups as subgroups of orthogonal groups over rings. This suggests an intriguing link with Kac-Moody groups acting on some substructures of twin buildings. There are also connections with Clifford algebras and other related algebras.