Te reo Māori (the Māori language) is the language of the Indigenous people of Aotearoa New Zealand, however, the number of speakers of this language has undergone a decline since colonisation. As part of the movement to revitalise the Māori language, various interventions have been suggested and put into practice over the past few decades, such as introducing Māori language week, and the establishment of Kohanga Reo pre-schools (immersion-based “language nests”). In recent years, the New Zealand Government has released a series of goals for the language, including the audacious goal of achieving one million Māori language speakers by 2040. However, the likelihood of achieving the Government’s goals remains uncertain. In this presentation, we introduce the context (historical and contemporary) for the need to model the language, before presenting a compartmental ODE model to analyse the trajectory of the Māori language, in terms of the number of speakers at different levels of proficiency. By incorporating existing data on language proficiency and usage, the model will evaluate the future trajectory of the prevalence of the language. We will show how this model can be used to inform strategies for achieving the goal of one million speakers by 2040, and whether achieving this goal gives a reliable indicator that the language is on the trajectory to recovery.

People play ‘ordinary solitaire’ (Klondike) millions of times a day. Yet, we mathematicians can’t figure out ‘what is the chance of winning?’ or ‘how can I play well?’

Indeed, even the chess and go playing programs ‘Alpha zero’ can’t solve these problems (thank goodness that there are some things the computer can’t do). I’ll introduce an easier solitaire where we can figure things out. The math involved is some of the deepest of the past 50 years. It has applications to the way fire burns and the rings left by coffee cups. AND of course, it’s beautiful in its own right. I’ll try to explain all to a general audience ‘in English’.

The dimer model is a classical model of statistical mechanics, involving the study of random perfect matchings (dimer covers) of a graph. When the graph is planar, determinantal and other linear algebra techniques can be used to compute many quantities of probabilistic interest.  

We extend this model to planar graphs with a GL(n) connection, showing how the associated determinant enumerates traces of “multiwebs”, which are representation-theoretic objects.  

All terms will be defined, and some applications will be discussed.

A wide array of physical systems spanning atomic to atmospheric scales, including many we encounter in daily life as physical, chemical or biological systems, display a variety of the same regular repeating patterns, e.g., stripes, squares, hexagons, etc. How are such diverse systems able to organise into the same ordered structures? Does the display of a particular pattern at the large scale inform us of structural/physical restrictions at a smaller scale? Questions like these motivate the investigation of patterns as an entity independent of the specific system in with they arise. 

The study of pattern formation concerns itself with developing a range of tools to model and analyse minimal mechanisms and then to characterise the obtained patterns, irrespective of any particular application. It borrows methods and tools heavily across multiple areas of mathematics, such as partial differential equations, dynamical systems and numerical computing. 

My talk will start with the idea of how to model a prototypical pattern-forming system and then to analyse the possible patterns that can arise within it in two and three dimensions. I will then explore some of the limitations of this model and what ingredients can be added to allow  more complicated patterns to arise, e.g., superlattice patterns and quasipatterns. The rest of my talk will focus on my use of two new connections within mathematics that help explore and classify patterns. The first is the use of computational algebraic geometry to obtain all equilibria with a chosen symmetry. The second is the use of topological data analysis to characterise obtained patterns quantitatively.   

Finding proofs and finding examples are critical parts of the mathematical process. Over the last few years there has been enormous effort in academia and industry to try to use AI to discover proofs. The focus of my talk will be different: can we use AI to find interesting examples?

I will focus on three different areas where AI has proven useful: extremal combinatorics, combinatorial geometry and knot theory. I will explain the role played by the neural network in each search task. A common theme is that although the neural network improves search in every example (and in several cases allows one to discover the best known examples), the effect is often subtle and interesting. By looking at several examples in parallel, I hope to give a sense of where neural networks might help the mathematician interested in finding new phenomena using computers and AI.

This is joint work with lots of people: the extremal combinatorics work is joint with Charton, Ellenberg and Wagner. The knot theory work is joint with Charton, Narayanan and Yacobi, and the combinatorial geometry work is joint with Davies, Gupta, Racanière, Swirszcz, Weber and Wagner.