If you are an Advanced Mathematics or Advanced Science student, then Honours is built into your program. For all other students, if you are keen on Mathematics and have achieved good results in years 1 to 3, you should consider embarking on an Honours year.

Below you can find some specific information about Applied Mathematics Honours.

For other information about doing Honours in Applied Mathematics, see the Honours Page.

### Honours Coordinator - Applied

If you have any questions about the Honours year, don't hesitate to contact the Honours Coordinator listed below. In particular, if you are just starting third year and vaguely thinking ahead to Honours, then it is important to choose a sufficiently wide variety of third year courses. Please see the Honours Coordinator to discuss your choice of courses.

E: a.schaeffer@unsw.edu.au
T: 9385 1679
Office: Room 4102, Red Centre (Centre Wing), UNSW

### Suggested Honours Topics

The following are suggestions for possible supervisors and Honours projects in Applied Mathematics. Other projects are possible, and you should contact any potential supervisors to discuss your options. You can find a full list of our Applied Mathematics staff via our Staff Directory. Please feel welcome to contact any staff member whose research is of interest.

You can get in touch with the potential supervisors below via their details on our Staff Directory

### 2023 Honours Projects in Applied Mathematics

This info below contains descriptions of thesis projects offered for Honours year students in Applied Mathematics. Please note that the list is not exhaustive, and feel free to contact supervisors for other projects in their field.

Honours candidates are strongly encouraged to contact their preferred supervisor as early as possible to discuss potential projects and to make sure they have any requisite background knowledge. More information about the Honours year is available by emailing the Applied Mathematics Honours Coordinator, or via our Honours Year webpage.

#### Mathematical Modelling

• The insulin signalling pathway in adipocytes: a mathematical investigation

The insulin signalling pathway in adipocytes (fat cells) is the main controller of the uptake of glucose in the cells. Understanding of this system is vital in the investigation of diabetes, which is a deficiency in this control system. Projects in this area include the development a mathematical description of the movement of the vesicles (small membrane spheres) containing glucose transporter proteins, and also the diffusion of the proteins if the vesicle fuses with the cell surface, and also the analysis of the biochemical signalling pathway from the insulin receptor. These will involve analysis of differential equations and possibly some computational simulations of the system. Experimental data for comparison with the models will be available.

• Do glucose transporters queue to get to the cell surface?

Cells transport glucose into their interior via protein channels. In adipocytes (fat cells) glucose uptake is regulated by insulin, as a dynamic balance between exocytosis (outward transport) and endocytosis (inward transport) of the proteins. The proteins are, however, packeted into small vesicles (spheres of membrane) when transported. One characterisation of the observed transport behaviour utilises queueing theory. This idea stems from the presence of the microtubules that cross the cytoplasm of the cells.

Microtubules are implicated in the sorting of different endocytic vesicular contents and have well characterised molecular motors which control the movement of vesicles down their lengths. Vesicles carrying the proteins have been observed to be transported along microtubules. Could these act as a scaffold for the vesicles, which would then form queues waiting for exocytosis? This project will model the recycling system as a closed Markovian queueing network and explore what characteristics of the network may be responsible for the transitory behaviour when insulin is applied, initiating protein transport. Experimental data for comparison will be available and the investigation will have both theoretical and computational aspects.

• Modelling of drug epidemics

In recent years, use of methamphetamine has grown rapidly. It’s a dangerous drug for the user and those with whom they interact. At present we have no working model of the methamphetamine epidemic that could be used to explore various policy options to deal with the epidemic or simply project its likely course. This project will explore different models developed for drug use and develop a credible model of the methamphetamine epidemic in Australia. This work would be jointly supervised by Prof Don Weatherburn at the National Drug and Alcohol Research Centre.

• Recidivism in the Criminal Justice System

Researchers are often called upon to evaluate programs designed to reduce the rate of reoffending. Typically, however, there are not observations of every offence a person commits, but rather only for those that result in an arrest or a conviction. How big a change would you need in actual re-offending rates to see a measurable difference in re-arrest or re-conviction rates? What interventions could be made, and what would be their effects?

The investigation will look at different mathematical approaches to model this problem and intervention strategies. This work would be jointly supervised by Prof Don Weatherburn at the National Drug and Alcohol Research Centre.

• Modelling Myelinated Nerve Function

Pacemaking in cardiac and neuronal cells is primarily controlled by the interaction between different voltage gated ion channels. An existing mathematical model of the human motor axon utilises coupled differential equations to describe the electrical activity within the nerve. This model will be explored to interpret the measured responses to extended hyperpolarisation and the contribution of different ion currents to the transmission of signals. It will also investigate the responses of sensory neurons to strong longlasting hyperpolarisation and contrast with motor neurons. Extensions to the model will then be made to incorporate additional inhomegeneities within the neuron structure. This project will include analysis and development of coupled differential equations, numerical simulations and the optimisation of the model using data from the experimental literature.

Steffen Docken (UNSW Kirby Institute)

• Mathematical and Statistical analysis of data relating to HIV infection
• Stochastic models, differential equations, and hypothesis testing

Gary Froyland

• 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 develop and apply new mathematical techniques for flow analysis, based on dynamical systems and spectral methods 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.

Michael Watson

• Modelling Phagocytic Capacity in Atherosclerotic Plaque Macrophages

Atherosclerosis, the primary cause of heart attacks and strokes, is characterised by the growth of plaques in the walls of arteries. Plaques are initiated by “bad cholesterol”, which accumulates in the artery wall and triggers an immune response. Macrophages (specialised immune cells) are recruited to consume (phagocytose) and remove this cholesterol.

However, when cholesterol-loaded macrophages die in the artery wall, the problem remains unresolved and a plaque full of fat and cellular debris can develop. Plaque formation can be studied mathematically by using structured population models to track the dynamic accumulation and recycling of fat within plaque macrophage populations. Recent models of this type assume that there is no upper limit to how much fat a macrophage can internalise. While this assumption is mathematically convenient, it is clearly physically unrealistic. This project will develop the theory required for new and improved models that include an upper limit on macrophage phagocytic capacity. The project will analyse and computationally solve differential equation models to reveal the impact of macrophage phagocytic capacity on real life atherosclerotic plaques.

• Structured Models of Atherosclerotic Plaque Formation with Volumetric Growth

Atherosclerotic plaques are growths in artery walls that lead to heart attacks and strokes. Plaques are initiated by fat (“bad cholesterol”) that enters the artery wall from the bloodstream. Immune cells called macrophages are recruited to the artery wall to consume and remove the fat. However, when macrophages die in the wall, they deposit fatty debris that stimulates further macrophage recruitment. This leads to a chronic cycle of cell and debris accumulation that causes the plaque to grow. Recent progress in understanding plaque formation has been made with the use of structured population models that study the dynamics of macrophages, their internalised fat and the debris they deposit when dying. These models predict, for example, that increasing plaque macrophage reproduction can limit debris accumulation. But what if this accumulation of cells ultimately makes the plaque larger? Is the benefit worth the cost? This project will address this question and more by developing a novel mathematical framework that couples structured population models to models of volumetric plaque growth.

#### Computational Mathematics

Josef Dick

• Well distributed points in high dimension with applications to numerical Integration
• Generation of non-uniform quasi random numbers
• Approximation properties of neural networks

Gary Froyland

• Operator-theoretic and differential-geometric kernel methods for 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.

Frances Kuo

• The theoretical development and/or practical application of Quasi-Monte Carlo Methods

Quasi-Monte Carlo methods, or QMC methods in short, are numerical methods for high dimensional integration and related problems. The prefix “quasi” indicates that these methods rely on cleverly designed pointsets or sequences in high dimensions, as opposed to the regular Monte Carlo method which is based on sequences of pseudorandom numbers. We now know how to construct good QMC methods efficiently in thousands of dimensions, with fast error convergence independently of dimension! A potential honours project would be on the theoretical development and/or practical application of QMC methods.

Bill McLean

• Numerical techniques for reaction-diffusion systems

Reaction-diffusion phenomena are modelled by coupled systems of semilinear parabolic PDEs, and have attracted much attention due to the striking patterns exhibited by their solutions. The project will compare different approaches for time stepping, such as exponential integrators and implicit Runge-Kutta schemes, and should suit students with an interest in hands-on scientific computing.

• Approximating the fractional powers of an elliptical differential operator
• Adaptive error control using discontinuous Galerkin time stepping

Thanh Tran

• The role of the Landau-Lifschitz equation in micromagnetism

One of the hallmarks of modern society is the increasing demand for the large data storage which can be rapidly and efficiently accessed. The most important devices for information storage are magnetic memories which are used in, for example, mobile phones, credit cards, televisions, and computer hard drives.

Submicron-sized ferromagnetic elements are the main building blocks of data storage devices. They preserve magnetic orientation for a long time, allowing bits of information to be encoded as directions of the magnetisation vector. The stored information can be modified by an external magnetic field. A well-known model for magnetisation is the Landau-Lifshitz-Gilbert equation. The equation possesses complex mathematical properties such as nonconvex side-constraints, strongly nonlinear terms and the appearance of singularities. These properties demand sophisticated numerical approximations.

In this project you will learn different numerical methods to solve the equation. Depending on your needs and interests, there are open problems to cut your teeth on.

• Stochastic differential equations

In this project, you will learn how different basic numerical methods are developed for stochastic differential equations. You will then learn how to apply them to solve practical problems and compare the performance of the methods.

• Problems in random domains

In many industries (e.g., in aerospace engineering) random discrepancies between the ideal geometries conceived in the design phase and their actual realisation may lead to considerable variations in expected outcomes.

The effect of randomness in domains is even more dramatic in manufacturing of nanodevices (e.g. data storage devices governed by the Landau-Lifshitz-Gilbert equation). Indeed, under certain resolution, surfaces of these devices become rough, and a minor discrepancy results in relatively large adverse effects. In this project you will learn how shape calculus can be used to deal with problems on random domains.

• Boundary element methods

Boundary element methods have long been used in engineering to solve boundary value problems. These problems are formulated from many physical phenomena, ranging from mechanical engineering (e.g. in car design) to petroleum engineering (e.g. for simulation of fractured reservoirs).

In this project, you will first learn basic concepts of boundary element methods, how to implement and analyse efficiency and accuracy of the methods. Then, depending on your needs and interests, you will use the methods to solve practical problems in engineering or geodesy. Problems in geodesy will involve programming with data collected by a NASA satellite, which may contain up to almost 30 million points.

Jan Zika

• Truncating climate errors: Developing new methods to improve how numerical climate models describe fluid flow and improve their projections of global warming

Climate Models are the key tool to predict how the climate will change. Accurate projections rely on having models which realistically mix carbon and heat into the deep ocean. Numerical models often spuriously mix due to imprecise numerical methods. Quantifying this spurious mixing is key to developing better climate projects. In this project we will develop and apply novel techniques to diagnose and reduce such numerical errors in climate models.

#### Fluid Dynamics, Oceanography and Meteorology

Gary Froyland

• 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 spectral theory, 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 be a joint project with Shane Keating.

Shane Keating

• Simulating fractal curves in turbulent fluids

Material immersed in a turbulent fluid, such as oil, chlorophyll or dye, will be stirred and deformed into strikingly beautiful fractal-like filaments and curves. The geometry of these curves has a strong influence on how quickly the material is mixed and diffused throughout fluid. However, numerical models of turbulence often lack the computational power to fully resolve the dynamics of these curves and must represent or “parameterize” the average effect of the flow on the material using simplified models, In this project you will explore a novel approach to parameterizing mixing in a turbulent fluid using a stochastic method to simulate fractal curves directly. The geometry of these curves will the be used to estimate the diffusion of material by the turbulent fluid. Strong coding ability with Matlab or Python is required.

• Tilting of mesoscale ocean eddies in the Tasman Sea

Mesoscale ocean eddies are the "weather systems" of the ocean --- rotating columns of fluid, hundreds of kilometers across and a kilometer or more deep that live for weeks to months and play an important role in transporting heat, salt, and other tracers throughout the ocean. Although ocean eddies are readily visible in satellite imagery of sea surface temperature or sea surface height, little is known about the vertical structure of these features due to the cost and difficulty of making measurements at sea. In particular, it is not known the degree to which ocean eddies "tilt" or lean over at an angle from the vertical.

In this project, you will investigate the prevalence of tilting in mesoscale ocean eddies in the Tasman Sea using a high-resolution ocean model. Experience with Matlab or Python is a requirement for this project.

Chris Tisdell

• Exploring the theory of Navier-Stokes equations and their applications to fluid flow

Navier-Stokes equations are of immense theoretical and physical interest. These partial differential equations have been used to better understand the weather, ocean currents, water flow in a pipe and air flow around a wing. However, the theory of the equations has not yet been fully formed. For example, it has not yet been proven whether solutions always exist in three dimensions and, if they do exist, whether they are smooth - i.e. they are infinitely differentiable all points in the domain. The Clay Mathematics Institute has identified this as one of the seven most important open problems in mathematics and has offered a US\$1 million prize for a solution or a counter example.

In this project we will examine existence and smoothness of solutions to problems derived from the Navier-Stokes equations that arise in laminar fluid flow in porous tubes and channels. Channel flows - liquid flows confined within a closed conduit with no free surfaces - are everywhere. In plants and animals, they serve as the basic ingredient of vascular systems, distributing energy to where it is needed and allowing distal parts of the organism to communicate. In engineering, one of the major functions of channels is to transport liquids or gases from sites of production to the consumer or industry. Such a project will involve the nonlinear analysis of boundary value problems and some numerical approximations.

Jan Zika

• Novel machine learning quantifying the ocean's role in a changing climate

The ocean is turbulent and hosts dynamics on a range of spatial and temporal scales leading to rich variability and apparently ‘noisy’ signals in data. However, when mapped into certain phase spaces, underlying structures emerge. These suggest the ocean can be parameterised and the noisy data filtered to clarify underlying long-term trends and their drivers. In this project we will use novel machine learning techniques to charactirse the ocean’s underlying structure and improve methods for analysing (big) ocean data.

• How does heat and carbon get into the ocean? An investigation of the physical mechanisms that control the ocean's uptake of heat and its effect on climate

Heat and carbon uptake by the ocean are central to climate change. However, the physical mechanisms by which the ocean transports such properties from the near surface to the deep ocean are not well understood. In this project we will project ocean data into novel phase diagrams to quantify the underlying energetic and thermodynamic drivers of ocean vertical transport.

#### Biological Modelling

Quoc Thong Le Gia

• Approximate cloaking simulation
• Analysis of changing data and applications
• Long Short-Term Memory network and its applications

In machine learning, artificial neural networks are popular and effective algorithms that have many applications ranging from image recognition to simulations of financial markets. There are a few types of popular neural networks based on their structures, namely fully connected neural network (FCN), convolutional neural network (CNN) and long short-term memory network (LSTM).

In this project, we will investigate the structure of long short-term memory network and its use in the problem of portfolio optimization.

#### Mathematics Education

Chris Tisdell

• Improving the ways we teach and learn mathematics

Research into learning and teaching mathematics at universities is a relatively new and suboptimally theorized field. It has largely remained sheltered from critical debate due to dominant views of mathematics and its teaching as a universal, absolute and unchanging state within tertiary institutions. As such, inherited long-standing ways of teaching and learning therein have gained a lustre of naturalized value, forming what appears to be a state of global pedagogical agreement.

Responding to this over-stabilization, this project explores the following research questions:

1. What are the limitations and hidden consequences of traditionally dominant pedagogy within university mathematics education?

2. How might we constructively reframe and renew these situations by offering alternative pedagogical perspectives?

#### Dynamical Systems

Gary Froyland

Topics in dynamical systems, ergodic theory, or differential 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.

• 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 and differential/spectral geometry.

• 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.

• Machine-learning dynamical systems by regression

It has been recently demonstrated that machine-learning techniques such as reservoir computers are equivalent to nonlinear regression. This is an important result because it brings “black box” machine learning output into a clearly human-understandable framework. This project will explore the advantages and limitations of predicting dynamical systems through optimal regression techniques, and investigate further improvements.

• 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.

• 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.

John Roberts

• Arithmetic dynamics

The topic is broadly taken to be the intersection of algebra, number theory, and dynamical systems. This interdisciplinary area of research is cutting edge and exciting, and has important applications to, e.g., cryptography, random matrix theory, materials science and engineering.

• Discrete integrable systems

The study of integrable (partial) difference equations and integrable maps is presently a very active field of research. In the first instance, this is due to the increasingly numerous areas of physics in which such systems feature. The study of discrete integrable systems also has intrinsic mathematical appeal, broadly speaking to do with finding analogues of concepts or properties (e.g., the Painleve property, Lax pairs, Hamiltonian structure) that exist in integrable systems with continuous time. Increasingly, I am interested in using algebraic geometry and ideal theory to understand discrete integrable dynamical systems.

Wolfgang Schief

• Topics in Soliton Theory

Solitons constitute essentially localised nonlinear waves with remarkable novel interaction properties. The soliton represents one of the most intriguing of phenomena in modern physics and occurs in such diverse areas as nonlinear optics and relativity theory, plasma and solid state physics, as well as hydrodynamics. It has proven to have important technological applications in optical fibre communication systems and Josephson junction superconducting devices.

Nonlinear equations which describe solitonic phenomena (soliton equations’ or integrable system’) are ubiquitous and of great mathematical interest. They are privileged in that they are amenable to a variety of solution generation techniques. Thus, in particular, they generically admit invariance under symmetry transformations known as Bäcklund transformations and have associated nonlinear superposition principles (permutability theorems) whereby analytic expressions descriptive of multi-soliton interaction may be constructed. Integrable systems appear in a variety of guises such as ordinary and partial differential equations, difference and differential-difference equations, cellular automata and convergence acceleration algorithms.

It is by now well established that there exist deep and far-reaching connections between integrable systems and classical differential geometry. For instance, the interaction properties of solitons observed in 1953 by Seeger, Donth and Kochendörfer in the context of Frenkel and Kontorova’s dislocation theory and later rediscovered by Zabusky and Kruskal (1965) in connection with the numerical treatment of the important Fermi-Pasta-Ulam problem are encoded in the geometry of particular classes of surfaces governed by the sine-Gordon equation and Korteweg-de Vries (KdV) equation respectively. The geometric study of integrable systems has proven to be very profitable to both soliton theory and differential geometry.

Integrable systems play an important role in discrete differential geometry. The term ‘discrete differential geometry’ reflects the interaction of differential geometry (of curves, surfaces or, in general, manifolds) and discrete geometry (of, for instance, polytopes and simplicial complexes). This relatively new and active research area is located between pure and applied mathematics and is concerned with a variety of problems in such disciplines as mathematics, physics, computer science and even architectural modelling. Specifically, theoretical and applied areas such as differential, discrete and algebraic geometry, variational calculus, approximation theory, computational geometry, computer graphics, geometric processing and the theory of elasticity should be mentioned.

Soliton theory constitutes a rich source of Honours topics which range from applied to pure. Specific topics will be tailored towards the preferences, skills and knowledge of any individual student.

Chris Tisdell

• Advanced Studies in differential equations

Many problems in nonlinear differential equations can reduced to the study of the set of solutions of an equation of the form f(x) = p in an appropriate space. This project will give the student an introduction to the applications of analysis to nonlinear differential equations. We will answer such questions as:

1. When do these equations have solutions?

2. What are the properties of the solution(s)?

3. How can we approximate the solution(s)?

A student who completes this project will be well-prepared to make the transition to research studies in related fields.

• A Deeper Understanding of Discrete and Continuous Systems Through Analysis on Time Scales

Historically, two of the most important types of mathematical equations that have been used to mathematically describe various dynamic processes are: differential and integral equations; and difference and summation equations, which model phenomena, respectively: in continuous time; or in discrete time. Traditionally, researchers have used either differential and integral equations or difference and summation equations | but not a combination of the two areas to describe dynamic models. However, it is now becoming apparent that certain phenomena do not involve solely continuous aspects or solely discrete aspects. Rather, they feature elements of both the continuous and the discrete.

These types of hybrid processes are seen, for example, in population dynamics where nonoverlapping generations occur. Furthermore, neither difference equations nor differential equations give a good description of most population growth. To effectively treat hybrid dynamical systems, a more modern and flexible mathematical framework is needed to accurately model continuous, discrete processes in a mutually consistent manner.

An emerging area that has the potential to effectively manage the above situations is the field of dynamic equations on time scales. Created by Hilger in 1990, this new and compelling area of mathematics is more general and versatile than the traditional theories of differential and difference equations, and appears to be the way forward in the quest for accurate and flexible mathematical models. In fact, the field of dynamic equations on time scales contains and extends the classical theory of differential, difference, integral and summation equations as special cases. This project will perform an analysis of dynamic equations on time scales. It will uncover important qualitative and quantitative information about solutions; and the modelling possibilities. Students who undertake this project will be very well equipped to make contributions to this area of research.

• Advanced Studies in Nonlinear Difference Equations

Difference equations are of huge importance in modelling discrete phenomena and their solutions can possess a richer structure than those of analogous differential equations. This project will involve an investigation of nonlinear difference equations and the properties of their solutions (existence, multiplicity, boundedness, etc). Students who complete this project will be very well-equipped to contribute to the research field.

#### Optimisation

Gary Froyland

• Efficient optimisation methods for optimal transport

Optimal transport concerns the transformation of one probability distribution into another with least cost. The theory of optimal transportation involves connections between optimisation, Riemannian geometry, and measures/probabilities. A key practical aspect is determining the optimal transport transformation, or at least the cost of this optimal transformation. This project will explore highly efficient techniques to determine optimal costs for transportation on large or high-dimensional domains.

• 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 optimisation, dynamical systems, functional analysis, and probability.

• Machine-learning dynamic predictors via optimal regression

It has been recently demonstrated that machine-learning techniques such as reservoir computers are equivalent to nonlinear regression. This is an important result because it brings “black box” machine learning output into a clearly human-understandable framework. This project will explore the advantages and limitations of predicting dynamical systems through optimal regression techniques, and investigate further improvements.

• 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 public or private transport modes in urban environments, medical imaging, scheduling airlines, rail, or mining processes.

Jeya Jeyakumar

• Robust Classification

Motivated by the fact that there may be inaccuracies in features and labels of training data, this project will examine robust optimization techniques to study the uncertainty in data features and labels in classification problems, develop robust optimization formulations for commonly used classification methods, such as support vector machines, logistic regression and decision trees, and compare with regularized and nominal methods.

Guoyin Li

• Rank Optimisation with matrix variables

Notions such as order, complexity, or dimensionality can often be expressed by means of the rank of an appropriate matrix. Therefore, many practical problems can be modelled as rank optimisation problems with matrix variables. Typical examples include the matrix completion problem (which arises in the Netflix prize), minimum-order linear system realisation problem and image compression. The rank optimisation problem can be regarded as an extension of the celebrated compress sensing problem and is often hard to analyse. One of the major difficulties in solving the rank optimisation problem is the nonsmoothness and non-convexity of the rank function. In this project, we will first examine the fundamental mathematical aspects of rank functions using tools from nonsmooth optimisation. We will then develop computationally efficient techniques for solving rank optimisation problems. Finally, we will apply these results to solve important problems such as the matrix completion problem and the minimum-order linear system realisation problem.

• Optimisation approaches for tensor computation: modern techniques for multirelational data analysis

Modern data analysis is the science of correctly collecting data, assessing it for trustworthiness, extracting qualitative information from it, and presenting it in a comprehensible informative way. Traditional techniques in data analysis deal with singlerelational data despite the reality that multi-relational data, whose objects have interactions among themselves based on different relations, often appear in our daily life.

Mathematically, single-relational data can be modelled by a matrix. Likewise, multirelational data can be modelled as a tensor, which is a multidimensional extension of the concept of a matrix. In this project, we aim to develop optimisation approaches to identify and rank the most significant factors from complex and huge-scale multi-relational data, and to express them in a way to highlight similarities and differences. Finally, we will apply these results to problems arise in modern decision-making and image science.

Mareike Dressler

• Topics in polynomial optimisation

A polynomial optimisation problem is a special class of nonconvex nonlinear global optimisation in which both the objective and constraints are polynomials. That is, it aims at finding the global minimiser(s) of a multivariate polynomial on a certain set. Polynomial optimisation has a wide range of applications like dynamical systems, robotics, computer vision, signal processing, and economics. Mathematically, it is well-known that solving polynomial optimisation is very hard in general. One of the most powerful approaches for handling such problems with rigorous and global guarantees is via algebraic techniques.

A potential honours project would be on the theoretical development and/or practical application of these algebraic techniques.

Jan Zika

• Novel optimisation techniques to quantify the ocean’s role in climate

The ocean is warming, carbon concentrations are rising and the concentration of salt in sea water is changing. These changes are the net result of both changes in climate forcing leading to exchanges at the sea surface and changes in how the ocean mixes and redistributes properties. Diagnosing the ‘how’ and ‘why’ of these changes is essential to understanding climate. The ways in which the ocean could have arrived at its present state can be viewed as a constrained set of solutions for which optimisation methods can be used to indicate the most likely subset. This project will develop and implement novel optimisation approaches to help quantify the drivers of ocean change.