Queer and Trans Mathematicians

A sequence covering array is a set of permutations of the alphabet $\{0, \dots, v-1\}$ such that every possible sequence of $t$ elements of this alphabet is a subsequence of at least one permutation. The absolute minimum number of permutations in such an array is $t!$ and Levenshtein conjectured that this bound can only be met when $v \in \{t,t+1\}$. I researched Levenshtein's conjecture during my PhD candidature, however this research was interrupted when I began transitioning in 2022. While Levenshtein's conjecture is a gripping problem at the intersection of permutation groups, finite projective geometry, coding theory and partial orders, sometimes other priorities are forced to take centre stage.

In this talk, I will discuss my own perspective of being a trans woman in the Australian tertiary sector, talk about the challenges of doing research amidst a gender awakening, and perhaps even go over some maths along the way. The mathematical portions of this talk contain joint work with Daniel Horsley and Ian Wanless.

TBA

The \emph{frequency} of an unlabelled graph $H$ in a graph $G$ is the number of induced subgraphs of $G$ that are isomorphic to $H$. The graph $H$ is a \emph{Most Frequent Connected Induced Subgraph (MFCIS)} of $G$ if the frequency of $H$ in $G$ is maximum among all unlabelled graphs occurring as induced subgraphs of $G$. All graphs up to $3$ vertices are most frequent connected induced subgraphs of some graphs. However, we proved $C_n,n>3$ cannot be the most frequent connected induced subgraphs of any graph.

In general, maximum likelihood estimation involves parameter estimation (such as p) based on the final outcome. For instance, in sports games like tennis, there might be a situation where one player (Player A) initially leads another player (Player B) and then gets caught up by Player B. In such scenarios, making a reasonable estimation of the skill levels of both Player A and Player B is fair to both parties. In cases of small samples or limited instances of such events, this paper proposes a possible method to seek reasonable estimates for the aforementioned process, with the obtained results showing alignment with actual data.

Thick-thin decompositions provide a controlled way to study hyperbolic 3 manifolds. Each component of the thin part consists of Margulis tubes and cusps and there are finitely many homeomorphism types of thick parts of all hyperbolic 3 manifolds up to some volume V>0.

In this talk we introduce a combinatorial thick thin decomposition of closed orientable 3 manifolds with non-trivial Z_2 homology. We do this by building on previous work of Jaco, Rubinstein, Spreer, and Tillmann which provides lower bounds on the complexity of such 3 manifolds.

This is joint work with Stephan Tillmann

Let A a subset of (Zp)^2 be a set of size 2p + 1 for prime p ≥ 5. We prove that A +^ A = {a1 + a2 | a1, a2 in A, a1 ≠ a2} has cardinality at least 4p. This result is the first advancement in over two decades on a variant of the Erdos-Heilbronn problem studied by Eliahou and Kervaire.

A Steiner triple system (STS) is a partition of a point set into triples so that each pair occurs in just one triple. If the triples can furthermore be partitioned into resolution classes so that each point is in precisely one triple in each resolution class, this is a Kirkman Triple system (KTS). These properties are useful in experimental design when we want to test pairwise interactions between treatments in an efficient way.

Frames are constructions which, when combined with specific STSs and KTSs, can be used to make KTSs of larger orders. We review the standard frame methods from the literature, and apply these to colourings KTSs. Our main result is that for any v congruent to 3 (mod 6), there exists a KTS on v points and a colouring of the points by 3 colours such that no block is monochromatic.

(Joint work with: Andrea Burgess, Peter Danziger and David Pike, ""Weak colourings of Kirkman Triple System"", Designs Codes and Cryptography, 2025)"

Constructing a single latin square is just like filling a Sudoku grid; time consuming yet simple. Constructing an infinite family of latin squares requires a bit more paper. It turns out that finding families of latin squares with disjoint subsquares can be made easier by first finding squares with a combinatorist's enemy: rational numbers.

The cyclic sieving phenomenon has been a mainstay of the enumerative and algebraic combinatorics landscape since its inception twenty years ago. We give a full account of the phenomenon in the case of Lyndon-like sieving, and observe that the sizes of the objects satisfy Gauss congruence. This incorporates many well-known results of the theory and provides methods for creating novel examples.

This talk will be based on my recent preprint https://arxiv.org/abs/2410.05678

Centraliser algebras of monomial representations of finite groups can be constructed and analysed using techniques similar to those in permutation group theory. Building on results from D. G. Higman and others, we explicitly construct a basis for the centraliser algebra of a monomial representation. The character table of this algebra is then derived and used in conjunction with a Gr\"obner basis algorithm to construct combinatorial structures, such as complex Hadamard matrices. This is joint work with Padraig \'{O} Cath\'{a}in, Heiko Dietrich and Ronan Egan.

The full-domain partition monoid $\mathcal{P}_n^{\mathrm{fd}}$ has been discovered independently in two recent studies on connections between diagram monoids and category theory. It is a right restriction Ehresmann monoid, and contains both the full transformation monoid and the join semilattice of equivalence relations. In this paper we give presentations (by generators and relations) for the full-domain partition monoid , its singular ideal, and its planar submonoid. The latter is not an Ehresmann submonoid, but it is a so-called 'grrac monoid' in the terminology of Branco, Gomes and Gould. In particular, its structure is determined in part by a right regular band in one-one correspondence with planar equivalences.