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.

Interval-censored Transformer Hawkes: Detecting Information Operations using the Reaction of Social Systems

Social media is being increasingly weaponized by state-backed actors to elicit reactions, push narratives and sway public opinion. These are known as Information Operations (IO). The covert nature of IO makes their detection difficult. This is further amplified by missing data due to the user and content removal and privacy requirements. In this talk, we advance the hypothesis that the very reactions that Information Operations seek to elicit within the target social systems can be used to detect them. We will present an Interval-censored Transformer Hawkes (IC-TH) architecture and a novel data encoding scheme to account for both observed and missing data. We derive a novel log-likelihood function that we deploy together with a contrastive learning procedure. We showcase the performance of IC-TH on three real-world Twitter datasets and two learning tasks: future popularity prediction and item category prediction. The latter is particularly significant. Using the retweeting timing and patterns solely, we can predict the category of YouTube videos, guess whether news publishers are reputable or controversial and, most importantly, identify state-backed IO agent accounts. Additional qualitative investigations uncover that the automatically discovered clusters of Russian-backed agents appear to coordinate their behavior, activating simultaneously to push specific narratives.

Mathematical oranges and deep space communication

Imagine you were trying to pack a crate full of oranges. How should you arrange the oranges in order to maximize the number of oranges you can pack into your crate?
This is one example of what mathematicians call a sphere packing problem: how to arrange spheres (or polyhedra) of various dimensions to maximize density. In general these problems are notoriously hard, but have deep implications for materials science, signals communication and physics.
In this talk I’ll explain some easy sphere packing problems, explain how the sphere packing problem in dimension 24 was key to the deep space communication that allowed NASA to transmit pictures data from the Voyager space craft, and why Maryna Viazovska just won the 2022 Fields Medal for her solution of the sphere packing problem in dimension 8.

Mathematics of protecting biodiversity in the GBR, Antarctica, and beyond

Ecosystems are incredibly complex, and we often have limited or patchy data about them. At the same time, we gain a multitude of benefits from the environment – needs that are increasing as human populations grow. Sustainable use of the environment can be expensive, and limited funds are available for preserving biodiversity and function. So we have complex, uncertain systems, with a high impetus for protection but resource limitations. Mathematical modelling and decision science are therefore important tools for understanding and protecting environmental systems.

In ecological decision science, we draw from mathematical modelling (social and environmental network models, systems of DEs, Bayesian network models among many more) and operations research (MDPs, spatial prioritisation, sequential decision-making) to tackle these problems. In this talk, I will present work from my research group the Applied Mathematical Ecology Group at QUT focused on applications in the Great Barrier Reef, Antarctica, and threatened species conservation in Australia.

22 May 2023
2:10 pm 

Please note this talk is on a Monday in Lower Napier LG28

Describing groups using first-order logic

Some properties of groups, such as being abelian, can be expressed by a sentence of first-order logic. Many others, such as being torsion, solvable, simple, or finitely generated, cannot be expressed even by a collection of first-order sentences.
Nonetheless, within a reference class C of groups, even a single sentence of first-order logic can have a surprisingly strong expressive power. A group is called finitely axiomatizable (FA) with respect to C if there is a single first-order sentence describing it up to isomorphism among the groups in C. We introduced this notion for the class C of finitely generated group in a 2003 paper (Separating classes of groups by first order sentences, Intern. J. of Algebra and Computation 13). We discuss the ensuing research, as well as more recent work with Segal and Tent when C is the class of separable profinite groups (Finite axiomatizability for profinite groups. Proc. of the LMS 2021, arxiv.org/abs/1907.02262). For instance, the Heisenberg group UT3(Z) over the integers is FA within the f.g. groups, and the Heisenberg group over the p-adic integers is FA within the profinite groups (arxiv.org/abs/1804.05331, p.24-).

Pattern Formation via Blackboards and Web Browsers

Motivated by a range of problems in embryology and ecology, I will present recent extensions to Turing's classical reaction-diffusion paradigm for pattern formation. This will start by reviewing reaction-diffusion systems and their analysis via classical linear instability theory, followed by a range of generalizations to more realistic scenarios of reaction-transport models in complex domains. Such extensions are motivated by the evolving and heterogeneous landscapes of pattern formation in nature. Throughout this discussion, numerical simulations will play key roles in validating and extending the near-equilibrium theory. To drive home this last point, I will present VisualPDE, a new web-based simulator for lightning-fast interactive explorations of these systems. Such accessible numerical tools are invaluable for rapidly prototyping models of complex biological phenomena. Importantly, accessible simulations underscore the need for sound theory which goes beyond phenomenological modelling in biology.

1 Sep 2023
2:10pm

Please note this talk is at Eng Nth 218

Dr Dionne Ibarra

Monash University

Invariants of knots and $3$-manifolds

We will give an expository talk on the colored Jones polynomial, Witten-Reshetikhin-Turaev 3-manifold invariants, Turaev-Viro 3-manifold invariants and its extension to knot complements. We will end with a discussion on a volume conjecture proposed by Q. Chen and T. Yang as well as work in progress related to the conjecture with E. Mcquire and J. Purcell.

Mechanics, Geometry and Topology of Health and Disease

Experimental biologists traditionally study healthy biological functions and the progression of diseases predominantly through their abnormal molecular or cellular features. For example, they investigate genetic abnormalities in cancer, hormonal imbalances in diabetes, or an aberrant immune system in vascular diseases. However, many diseases also have a mechanical component which is critical to their deadliness. Notably, cancer kills typically through metastasis, where the cancer cells acquire the capability to remodel their adhesions, generate forces and migrate to distant organs. Solid tumours are also characterised by physical changes in the extracellular matrix – the material surrounding the cells. While such physical changes are long known, and e.g. enable doctors to feel a tumour by palpation, only relatively recent research revealed that cells can sense altered physical properties and transduce them into chemical information. An example is the YAP/TAZ signalling pathway that can activate in response to altered matrix mechanics and that can drive tumour phenotypes such as the rate of cell proliferation, or metastatic behaviour. Another example is blood vessel cells that sense blood flow, material properties of the surrounding environment and forces from neighbouring cells.

In this talk, I will argue that physical signatures are a critical part of many biological systems and therefore, need to be incorporated into mathematical models. Crucially, physical disease signatures bi-directionally interact with molecular and cellular signatures, presenting a major challenge to developing such models. I will present several examples of recent and ongoing work aimed at uncovering the relations between mechanical and molecular/cellular signatures in health and disease. I will discuss how the heart’s molecular state changes during ageing, which, consequently, affects heart muscle function. Next, I will discuss how blood vessel cells interact mechano-chemically with each other to regulate the passage of cells and nutrients between blood and tissue, and how we model these interactions through a contact mechanics model.  Finally, I will discuss how cellular and subcellular geometry, such as cell shape, affects intracellular reaction networks – the networks that control cell behaviour.

Dr Cody Nitschke
University of Sydney/Flinders University

Investigating the Evolution of the Human Life History Using Mathematical Models

Humans are significantly different from our closest primate relatives. This includes characteristics such as a longer lifespan, the duration of development, menopause, larger brains, and the social structure of mating. The historically favored theory to explain these differences, called the hunting hypothesis, claims that big game hunting and aggressive scavenging led to the evolution of our genus. However, challenges to those claims are substantial, making this an evolutionary puzzle. In this talk, I will discuss several mathematical models that have been developed and give an overview of recent results that support an alternative hypothesis.