# Honours and PhD Student Projects

## Graduate study and scholarship information

I am always interested in supervising excellent students. Please see information for Future Students regarding Scholarships and Admission for Australian and International students.

## Summary of topics

**Topics in dynamical systems, ergodic theory, and Riemannian geometry.**Ergodic theory is the study of the dynamics of ensembles of points, in contrast to topological dynamics, which focusses on the dynamics of single points. A number of theoretical Honours projects are available in dynamical systems, ergodic theory, and/or differential geometry, aiming at developing new mathematics to analyse the complex behaviour of nonlinear dynamical systems. Depending on your background, these projects may involve mathematics from Ergodic Theory, Functional Analysis, Measure Theory, Riemannian Geometry, Stochastic Processes, and Nonlinear and Random Dynamical Systems.**Operator-theoretic and differential geometric approaches to machine learning**This project will develop new mathematical and computational approaches to analyse high-dimensional data. Operator-theoretic methods will be explored, including the use of transfer operators, dynamic Laplace operators, and Laplace-Beltrami operators, which extract dominant dynamic and geometric modes from the data. In the theoretical direction, this project will tackle the mathematisation of aspects of machine learning. In a combined theoretical and numerical direction, this project will investigate the construction of these operators from high-dimensional data using dynamic and geometric kernel methods. A possible application is to analysing global scalar fields obtained from satellite imagery such as sea-surface temperature to extract climate oscillations such as the El Nino Southern Oscillation and the Madden-Julian Oscillation. This project will use ideas from dynamical systems, functional analysis, and Riemannian geometry.**Differential and spectral geometry with applications to fluid mixing.**Techniques from differential geometry and spectral geometry (via Laplace-type operators) have recently been shown to be particularly effective for analysing complex dynamics in a variety of theoretical and physical systems. This project will focus on developing and extending powerful techniques to extract important geometric and probabilistic dynamical structures from fluid-like models. If desired, application areas include the ocean (an incompressible fluid) and the atmosphere (a compressible fluid). This project will involve dynamical systems, differential geometry, and PDEs.**Stability of linear operator cocycles.**Classical perturbation theory yields continuity of the spectrum and eigenprojections of compact and quasi-compact linear operators. The situation is dramatically different when one creates a cocycle of different operators, driven by some ergodic process. This dramatic difference even occurs in finite-dimensions (cocycles of matrices). This project will discover theory for which one can expect continuity of the corresponding spectral objects, namely Lyapunov exponents and Oseledets spaces. The project will use mathematics from probability and statistics, functional analysis, and connects to dynamical systems and ergodic theory.**Lagrangian Coherent Structures in Ocean and Atmosphere Models.**The ocean and atmosphere display complex nonlinear behaviour, whose underlying evolution rules change over time due to external and internal influences. Mixing processes of in the atmosphere and the ocean are also complex, but carry important geometric transport information. Using the latest models or observational data, and methods from dynamical systems, and elliptic PDEs, this project will identify and track over time those geometric structures that mix least. Known examples of such structures are eddies and gyres in the ocean, and vortices in the atmosphere, however, there are likely many undiscovered coherent pathways in these geophysical flows. There is also the possibility for the project to further develop mathematical theory and/or algorithms to treat one or more specific challenges arising in these application areas. This could a joint project with Mark Holzer or Shane Keating.**Transfer operator computations in high dimensions.**Many real-world dynamical systems operate in phase spaces that are very high dimensional and/or unknown. For example, the dynamics of ocean-atmosphere circulation at various spatial and temporal scales (e.g. from local weather to global climate) is invariable extremely high dimensional. On the other hand, there is increasing availability of spatial datasets from e.g. satellite imagery, which provide high resolution spatial images as ``movies'' in time. One can hope to construct dynamics of a projected system from the dynamics of these images, which are themselves operating in a high-dimensional space (dimension >= number of pixels in the image). This project will investigate recent ideas in constructing transfer operator for high-dimensional systems, and use ideas from dynamical systems, stochastic processes, functional analysis, and Riemannian geometry.**Lagrangian coherent structures in haemodynamics.**Haemodynamics (the dynamics of blood flow) is believed to be a crucial factor in aneurysm formation, evolution, and eventual rupture. Turbulent motion near the artery wall can weaken already damaged arteries, as can oscillations between turbulent and laminar flow. Simulations of 3D blood flow is either derived by (i) computational fluid dynamics (CFD) from patient-specific mathematical models obtained from angiographic images or (ii) laser scanning of real flow through a patient-specific physical plastic/gel cast. In this project, joint with Prof. Tracie Barber (UNSW Mech. and Manufact. Engineering), you will apply the latest mathematical techniques for flow analysis, based on dynamical systems and elliptic PDEs to separate and track regions of turbulent and regular blood flow. Prof. Barber will provide the realistic flow data from her laboratory, from both CFD simulations and physical casts. There is also the opportunity to further develop mathematical theory to solve problems specific to haemodynamics.**Optimising fluid mixing.**Combining techniques from dynamical systems and optimisation, this project aims to develop new mathematical algorithms and practical strategies for enhancing or controlling mixing in fluids, with applications in environmental (e.g. biology or pollution) and industrial settings. The project will use mathematics from dynamical systems, functional analysis, and probability.