## NZMS Seminar Series

The NZMS is running a regular seminar series with talks held via Zoom.

## 2021

**Wednesday 29 September 2021, 3pm**

**Sione Ma'u, University of Auckland: The extremal function of a real convex body**

https://massey.zoom.us/j/84190393386?pwd=RkVlMmIxbHFOcTUxcGdUVHpKeDhjdz09

The extremal function (or pluricomplex Green function of logarithmic growth) associated to a regular compact set K is important in polynomial approximation. In one complex variable it is a classical Green function for the complement with pole at infinity: zero on K, harmonic on the complement of K in the complex plane, and grows logarithmically as one goes to infinity. In several variables the extremal function is zero on K, maximal on the complement of K and grows logarithmically as one goes to infinity.

In this talk I will describe basic properties of the extremal function in one and several variables: its importance in approximation, and what "maximal" means (which is not the same as harmonic, except in the complex plane).

Then I will explain how to compute the extremal function of a convex body in $R^n$ (considered as a subset of $C^n$). In one variable the extremal function of a real interval e.g. -1,1 is well-known, given in terms of the inverse Joukowski function.

**Wednesday 13 October 2021, 3pm**

**Shaun Cooper, Massey University: The Rogers-Ramanujan continued fraction**

Video: https://youtu.be/FO5jpSCZJfc

Just over 100 years ago, an unknown clerk from India wrote some letters to Cambridge mathematician G. H. Hardy. About certain results in one of the letters, Hardy later wrote

*“(they) defeated me completely; I had never seen anything in the least like them before”*

and concluded

*"A single look at them is enough to show that they could only be written down by a mathematician of the highest class.”*

I will describe these particular results, put them in a modern context, and outline some recent developments.

**Wednesday 27 October 2021, 3pm**

**David Bryant, University of Otago: Minkowski diversities**

https://otago.zoom.us/j/98054569910?pwd=ODJKS2RDaStDcUpZR2docElMK0ZXQT09

The *circumradius* of a set of points in R^n is the minimum amount you need to scale a unit ball in order to contain them. Replace the ball with a convex body K and you have the *generalised circumradius*. We characterise which functions on bounded subsets of R^n correspond to a generalised circumradius for some choice of K. We then discuss the harder question of determining when a function on a finite powerset can be realised as the generalised circumradius defined on some finite set. The results mix convex analysis, the theory of mathematical diversities and ideas from metric geometry.

**Wednesday 10 November, 3pm**

**Dion O'Neale, University of Auckland: Modelling COVID-19 in Aotearoa NZ on a bipartite contact network of 5 million individuals **

Video: https://youtu.be/LK-B_zMKH_o

Many of the models used for rapid policy advice during the COVID-19 pandemic have relied on simplifying assumptions about the homogeneity of individuals. However, we know that risk factors for exposure, transmission, and poor outcomes are not evenly distributed across society. We have built a stochastic model of infection dynamics that runs on an empirically derived bipartite contact network of the ~5 million people in Aotearoa New Zealand. The contact network includes spatial information, and individual demographic information, along with distinct ‘transmission contexts’ including dwellings, workplaces, and schools, built from linked data in the Statistics NZ Integrated Data Infrastructure. This network is the underlying structure on which we run a stochastic contagion process to model the spread of COVID-19, which includes explicit representation of the testing and contact tracing processes. We have used this model to estimate the probable outcomes of COVID outbreaks in Aotearoa and to evaluate the effect of non-pharmaceutical interventions including 'Alert Level' changes. In particular, we find that this heterogeneity (network structure) means that the effect of different interventions does not combine linearly.

** Wednesday 1 December, 9.00 am – 4.30 pm**

**Manawatū–Wellington Applied Mathematics Conference**

Organised by Robert McLachlan. Video: https://youtu.be/w8aXAayfbsI

** Thursday 2 December, 2 pm**

**Juliette Unwin, Imperial College London: Modelling COVID-19: A British perspective**

Video: https://youtu.be/NElieXNZT0U

Late 2019 / early 2020, reports of a novel pathogen spreading around China began to be discussed in the Department of Infectious Disease Epidemiology at Imperial College that I worked in. Little did we know at that point, the scale to which this global pandemic response would reach. During this seminar I’m going to give an overview of the modelling that went on at Imperial over the past few years from the perspective of a member of the COVID-19 response team. I’ll share most insights about the particular projects I was involved in - developing a new method for modelling Rt (the time varying reproduction number), estimating proportions of cases not detected in an outbreak, and now global orphanhood estimates from COVID.

## 2022

** Wednesday 2 February, 3 pm**

**Florian Beyer, University of Otago: The nonlinear stability of “big bang” formation for solutions of the Einstein equations in mathematical cosmology**

Recording: https://www.youtube.com/watch?v=ssBljFpGG2I

The Einstein equations are a complicated nonlinear system of geometric wave-type partial differential equations defined on Lorentzian “spacetime" manifolds. In cosmology, these equations describe the dynamics of gravity and therefore the history of the universe as a whole. While the observationally very successful “Lambda-CDM” cosmological model restricts to very simple settings, the full nonlinear regime of these equations is still full of mysteries. Several fundamental issues (“Mixmaster behavior”, “cosmic censorship” etc) especially related to the “early universe” in the vicinity of the “initial big bang singularity” (a domain of the spacetime manifold near a spacelike singularity of the Riemann curvature tensor) remain open. Tackling these issues boils down to the singular analysis of these nonlinear partial differential equations, while exploiting their geometric nature, with the hope to obtain sufficiently detailed rigorous estimates of the global behaviour of generic solutions.

In this talk I present recent global nonlinear stability results providing sufficiently sharp estimates which (to a point) confirm that the simple description of the early universe given by the Lambda-CDM model is a good approximation. However our results also indicate the existence of new dynamical phenomena not covered by that model. This is work done in collaboration with Todd Oliynyk (Monash University).

** Wednesday 16 February, 3 pm**

**Melissa Tacy, University of Auckland: Adapting analysis/synthesis pairs to pseudodifferential operators**

Video: https://youtu.be/I_crLu6rb84

Many problems in harmonic analysis are resolved by producing an analysis/synthesis of function spaces, such as the Fourier or wavelet decompositions. In this talk I will discuss how to use Fourier integral operators to adapt analysis/synthesis pairs (developed for the constant coefficient PDE case) to the pseudodifferential setting. I will demonstrate how adapting a wavelet decomposition can be used to prove L^{p} bounds for joint eigenfunctions.