Discipline of Mathematical Sciences Colloquium 2023

Expanding black hole cosmologies

In general relativity, the Kerr de Sitter family of solutions to Einstein’s equations with positive cosmological constant are a model of a black hole in the expanding universe. In this talk, I will give an overview of the stability problem for the expanding region of the spacetime, which can be formulated as a characteristic initial value problem to the future of the cosmological horizons of the black hole. Unlike in the stability of Kerr spacetimes, the solution in the cosmological region does not globally converge to an explicit family of solutions, but displays genuine asymptotic degrees of freedom. I will describe my work on the decay of the conformal Weyl curvature in this setting, and the global construction of optical functions in de Sitter, which are relevant for my approach to this problem. This talk is accessible to a general audience.

The mathematics of minimal surfaces, red blood cells and bush fires

The mathematics of minimal surfaces, red blood cells and bush fires

Abstract: Geometric analysis is the intersection between geometry and partial differential equations. Curvature flows are one of the main topics of my research. I will present several theoretical results with direct applications to minimal surfaces problems and modelling a fire line or a biological membrane.

Using Field Data to Inform Agent-Based and Continuous Models of Locust Hopper Bands

An outstanding problem in mathematical biology is using laboratory and field observations to tune a model’s functional form and parameter values. These problems lie at the intersection of dynamical systems and data science. In this talk I will discuss an ongoing project developing models of the Australian plague locust for which excellent field and experimental data is available.

Under favourable environmental conditions flightless locust juveniles may aggregate into coherent, aligned swarms referred to as hopper bands. These bands are often observed as a propagating wave having a dense front with rapidly decreasing density in the wake. A tantalizing and common observation is that these fronts slow and steepen in the presence of green vegetation. This suggests the collective motion of the band is mediated by resource consumption. Exploiting the alignment of locusts in hopper bands, I will first describe a one dimensional model of density variation perpendicular to the front. We develop two models in tandem; an agent-based model that tracks the position of individuals and a partial differential equation model that describes locust and resource density. We first estimate biological realistic ranges for the ten input parameters of our models. Then by examining 4.4 million parameter combinations, we identify a set of parameters that reproduce field observations.

I will then discuss two ongoing efforts to improve this model. The first uses ideas from dynamical systems and continuum mechanics to extend this model into two dimensions by modelling the known tendency of locusts to align using ideas from the Kuramoto model of oscillator synchronization. The second, firmly based in data science, uses motion tracking of tens of thousands of locusts to shed light on how locust movement is informed by interactions with other individuals.

( Co-authors: Michael Culshaw-Maurer, University of California, Davis; Rebecca Everett, Haverford College; Maryann Hohn, Pomona College; Christopher Strickland, University of Tennessee, Knoxville; Jasper Weinburd, Hamline University)

The Topology and Geometry of Ancient Ricci Flows

The Ricci flow is a weakly parabolic partial differential equation which has been used to solve many important problems in geometry and topology, including the famous Poincare conjecture. Central to Perelman's original proof of the Poincare conjecture was an understanding of the so-called "ancient Ricci flows"; these are solutions of Ricci flow which can be extended indefinitely backwards in time. In this talk, I will discuss how Ricci flow was applied to the Poincare conjecture, and will also discuss existence and uniqueness results for ancient Ricci flows, both classical and modern.