# Honours and PhD Student Projects

Please consult the UNSW Graduate Research School page for scholarship information for Australian and International students.

## Writing well

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## Summary of topics

1. 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 A/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. A/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.

2. Topics in dynamical systems and ergodic theory. 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 projects are available in dynamical systems and ergodic theory, 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, Nonlinear Time Series Analysis, Nonlinear and Random Dynamical Systems, Markov chains, Graph Theory, and Coding and Information Theory.

3. Transfer operator analysis with applications to fluid mixing. A transfer operator is a linear operator that completely describes the evolution of probability densities of a nonlinear dynamical system. Transfer operators are therefore fundamental objects like discrete time maps and continuous time flows, but operate on ensembles rather than single points. Spectral techniques using transfer operators have recently been shown to be particularly effective for analysing complex dynamics in a variety of theoretical and physical systems, and are an active research area internationally. This project will focus on developing powerful transfer operator 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).

4. 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.

5. Nonlinear and mixed integer linear optimization with application to radiotherapy. The clinical aim of this project is to reduce imaging dose, or alternatively improve image quality, in radiotherapy treatments for lung cancer when imaging the thorax or upper abdomen using a technique known as four dimensional cone beam computed tomography. For the same image quality, we aim to reduce imaging dose by at least 50%. The mathematical component of this project involves scheduling of the 4D cone beams, taking into account a variety of geometric constraints, so as to achieve a good combination of image quality and imaging dose, and will require mathematical research in nonlinear, integer linear, and possibly integer nonlinear, optimization. (Part of a Cancer Australia Priority-driven Collaborative Cancer Research Scheme, Investigators: R. O'Brien (Medicine - Radiation Physics, USydney), G. Froyland (Mathematics, UNSW), and J.-J. Sonke (Netherlands Cancer Institute): "Reducing Thoracic Imaging Dose and Improving Image Quality in Radiotherapy Treatments"

6. Extreme value statistics for chaotic systems. Accurately estimating the probability of rare events is particularly challenging in models with long memory, such as systems with a high level of determinism and a low level of randomness. This project will develop mathematical theory and accurate, rigorous numerical schemes to handle such systems. The project will also apply these new methods to estimate the likelihood of rare events from real data. The project will use mathematics from probability and statistics, functional analysis, and connects to dynamical systems and ergodic theory.

7. 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.

8. Topics in integer programming and combinatorial optimisation. Integer programming is a mathematical framework for solving large decision problems. Usually there is some underlying discrete structure for the problem such as a network or graph. You will learn new mathematical techniques in discrete mathematics, algebra, and geometry. If desired, application areas may include radiotherapy, scheduling airlines, rail, or mining processes. Skedgo are offering top-up scholarships for research topics related to optimisation problems in public transport.

9. Stochastic integer programming. Almost all real world models have significant uncertainty in their measured data. A naive approach is to replace probability distributions of data with their mean value and create a single deterministic model. However, optimising this deterministic model typically results in decisions that are far from optimal. In order to make better decisions, the underlying probability distributions must be properly incorporated into the optimisation process. This is the aim of stochastic programming. The aim of this project is to develop rigorous optimization methods that include uncertainties in the forecast data and evaluate all possible options in light of the latest information. Familiarity with probability theory is essential. If desired, application areas may include radiotherapy, scheduling airlines, rail, traffic, or mining processes.