Talks — Anthony Pisani

Abstracts and slides for the more notable talks I've given are archived below. In some circumstances, files may be stored on other websites; if you can't access these for any reason, please don't hesitate to request a copy.

2025

The Saxl hypergraph of a permutation group (Forthcoming)

(at the 63rd meeting of the Australian Mathematical Society, Latrobe University, Australia)

A base for a permutation group 𝐺 ≤ Sym(Ω) is a subset 𝐵 of Ω with trivial pointwise stabiliser in 𝐺; the minimum cardinality of a base is termed the base size 𝑏(𝐺) of 𝐺. In the notable case 𝑏(𝐺) = 2, Burness and Giudici introduce the Saxl graph of 𝐺, the graph on Ω with 2-point bases as edges. Freedman et al. later generalise this concept to groups for which 𝑏(𝐺) > 2, taking as edges the 2-element subsets of 𝑏(𝐺)-element bases. In this talk, we consider properties of an alternative generalisation, the Saxl hypergraph (where edges are the bases of size 𝑏(𝐺) themselves), including the groups with complete Saxl hypergraphs and analogues of a striking conjecture of Burness and Giudici. This is joint work with Melissa Lee.

Computing the Character Table of a 2-local Maximal Subgroup of the Monster

(at the 9th Australian Algebra Conference, Latrobe University, Australia)

The maximal subgroups of the Monster, the largest of the 26 sporadic simple groups, have recently been the subject of renewed interest. In 2023, Dietrich, Lee, and Popiel used Seysen’s revolutionary mmgroup package for fast computations in the Monster to address three open cases for maximal subgroups, completing their classification. Burness and Hulpke computed several character tables later that year, while Popiel and Pisani calculated four outstanding sets of class fusions in 2024. In this talk, we discuss the use of mmgroup and Dietrich and Hulpke's hybrid group representation to compute the last remaining character table and class fusions of a maximal subgroup of the Monster.

2024

The Maximal Subgroups of the Monster Group

(at the 8th Australian Algebra Conference, ANU, Australia)

The Monster is the largest of the 26 sporadic finite simple groups, and was until recently the only one without a fast computational model, leaving several basic questions unanswered. Seysen's mmgroup package is finally breaking this deadlock; notably, Dietrich, Lee, and Popiel addressed the open cases for maximal subgroups of the Monster in 2023. In this talk we describe related work implementing other maximal subgroups in mmgroup, including a correction to the list thereof.