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2026 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

Christopher Angstmann

Modelling with fractional differential equations

Fractional derivatives are a type of nonlocal operator that generalise the concept of a derivative away from integer order. Whilst they are rather esoteric in their initial definition, they have had an increasing place in modelling a wide variety of physical phenomena. By exploring a connection between random walks and fractional derivatives it is possible to derive a wide range of models. This project will develop fractional order PDE models with applications to biomathematics.

��Semi-Markov compartment models

Compartment models are a widely used class of models that are useful when considering the flow of objects or people or energy between different labelled states, referred to as compartments. Recently we have constructed a general framework for fractional order compartment models, where the governing equations involve fractional order derivatives, via the consideration of a semi-Markov stochastic process. This project will explore the generalisation of the framework to incorporate a wider range of nonlocal operators.

Adelle Coster

��Nutrient transport in crowded cellular environments

Gram-negative bacteria have two cell membranes – the outer membrane largely contains passive transporter channels and the inner�� �� membrane active transporters. The periplasmic space separating these membranes is small in volume and is a very crowded environment. Investigations of the flow of nutrients through these membranes in and out of the cell can be investigated using a variety of dynamical systems and computational techniques.

Starch Grain Analysis

Starch grains grown in the interstitial spaces in plants often have characteristic shapes dependent on the species. It is possible to�� characterise and classify the origin of some populations of grains based on their shapes. Here extensions to common characterisation�� methods are planned to define further shape measures which might be used to separate the different species – beyond the normal�� �� characteristics of size, area, and perimeter. For example, how triangular must a shape be, before it should be classified as a triangle?�� �� How can we define these shapes? Can we put some bounds on the uncertainties of our classifications based on the different definitions for the shapes?�� This investigation is supported by data from a large database of microscope images of starch grains of known origin.

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.

Daniel Han

The space-dependent variable-order time-fractional diffusion equation

The space-dependent variable-order time-fractional diffusion equation describes, in the diffusion limit, a discrete space random walk where the time spent on each site is power law distributed with a spatially varying power law exponent. The distribution of such random walks exhibits properties that defy traditional statistical mechanics principles such as asymptotic aggregation and non-ergodicity. The project will aim to understand how such an equation might arise from random walk models, the proper limits one must take to arrive at the governing equation, physical interpretation of the governing equation, statistical properties of the solution to the governing equation and first passage/first hitting time problems associated. This project will involve analytical calculations, computational simulations and has the potential to involve real-world particle trajectory datasets for experimental validation.

Bill McLean

Adaptive time stepping for diffusion and sub-diffusion equations

A spatial discretisation of a parabolic PDE leads to a large, sparse and stiff system of ODEs, which is typically integrated via an implicit time stepping scheme.�� The initial goal of this project is to design automatic step-size control algorithms for such schemes, guaranteeing a desired accuracy tolerance.�� A further aim is to extend these algorithms to evolution equations modelling sub-diffusion.�� Such equations are non-local in time, that is, they are integro-differential equations, typically formulated in terms of fractional-order time derivatives.

Scott Sisson and SPLINK NZ Industry Partner

• Prediction of relative hormonal environment for all common menstrual cycle 'types' (natural and pharmaceutically modified)��

Michael Watson

Modelling cell and lipid dynamics in atherosclerotic plaques�� ��

Atherosclerotic plaques are fatty, cellular lesions that form in artery walls and can lead to heart attack or stroke. Plaques are initiated by blood-borne lipid particles (“bad cholesterol”) that become trapped in the artery wall and trigger an immune reaction. Subsequent plaque progression involves a complex interplay between these lipids and the cells that are recruited to the lesion site. Mathematical modelling of plaque cell and lipid dynamics can provide novel insights into the mechanisms of plaque progression that are unobservable in experimental or clinical settings.����

Specific examples of potential projects in this area include:��

1) compartment modelling of plaque formation with volumetric growth;����

2) lipid-structured modelling of phenotype change in plaque smooth muscle cells;��

3) multiphase modelling of plaque formation.��

These projects all involve the development, analysis, and numerical solution of differential equation models. Projects involving the development of stochastic, individual cell-based models may also be available.��

Fluid Dynamics, Oceanography and Meteorology

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.

Gary Froyland

Lagrangian Coherent Structures in Ocean, Atmosphere, and Climate Models

The ocean, atmosphere, and climate system 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 inthese geophysical flows. Moreover, higher-dimensional systems arising from climate models could also be tackled. 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.

Amandine Schaeffer

����������������General physical oceanography, please contact if interested.

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 sub-optimally 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

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.

Upanshu Sharma

Constraints in stochastic (or partial) differential equations

Modern chemistry deals with extremely high-dimensional system of stochastic (or corresponding partial) differential equations. However, in practice, one is only interested in the behaviour of few of these differential equations. This project deals with the important question: 'How does one rigorously constrain a higher dimensional set of differential equations?’ This project involves stochastic differential equations and functional analysis.

Clustering in ordinary differential equations

Chemical reactions are often modelled using a higher-dimensional system of linear equations, but in practice one is often interested in the behaviour of certain particular reactants. This project asks the question ‘Given a set of linear equations for variables (x,y,z), how can one derive a closed approximation for a lower-dimensional set of variables (x+y,z)?’. This problem is closely related to clustering of�� Markov chains. This project involves ordinary differential equations, linear algebra and rudimentary probability.�� ����

Path-finding algorithms.

The aim of this project is to develop and analyse algorithms which ‘find the shortest path between two points in a region full of hard obstacles. While this problem can be solved by several classical algorithms (such as Dijkstra’s algorithm), these deterministic�� approaches suffer in higher-dimensional setting, which suggests the need for stochastic algorithms, for instance using inspiration from�� biology (ant-trail formation). ��This project can have strong numerical/implementation and/or analysis flavour.����

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.

There exist deep and far-reaching connections between integrable systems and classical differential geometry. The geometric study of�� ��integrable systems has proven to be very profitable to both soliton theory and differential geometry. Moreover, 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.

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

This project explores the use of machine learning in either (i) prediction of dynamical systems or (ii) in the construction of efficient linear operator representations of the dynamics. In the latter case, the project will focus on those linear operators that are generated by the�� dynamical system and which allow a spectral analysis of the dynamics.

��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 (���������¼� Mech. and Manufact. Engineering), you will develop and apply new mathematical techniques for�� low 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.

Chunxi Jiao

Bubbling in the presence of noise

Singularities can arise in physical systems such as ferromagnetics and liquid crystals. The corresponding blow-up phenomenon of these models follows from that of harmonic map heat flow. In dimension two, the blow-up is known as bubbling, with backward bubbling closely related to non-uniqueness of weak solutions. At microscopic scales, systems are susceptible to thermal noise which is often modelled by space-time stochastic processes and can contribute to the total energy. This project aims to explore how random noise affects the bubbling, using techniques from PDE and stochastic analysis.�� �� �� �� �� ����

Optimization

Jeya Jeyakumar

Robust Optimization Methods for Managing risk and quantifying uncertainty in complex decision-making

Project abstract 1 Robust optimization is a relatively new distribution-free method for solving real-world decision-making problems in the face of data uncertainty. Developing mathematical theories and methods for quantifying uncertainty and obtaining solutions with risk prevention guarantees remains a major mathematical challenge. This project aims to study and develop optimization methods to address this challenge.

Stochastic Optimization for data-driven decision-making under uncertain conditions

Project abstract 1: Stochastic optimization, which assumes uncertainty has a probability distribution, is one of the well-established methods for modeling practical decision-making problems in the face of uncertain conditions. Despite widespread use, it has significant limitations for applications to large-scale decision-making problems. This project aims to study a hybrid solution approach by combining the strengths of both distribution-free and stochastic methods with applications.

REQUIREMENTS: Due to the technical nature of these projects, interested students should have completed both MATH3161 and�� MATH3171 at the HD levels, and should have done mathematical computing courses.

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.

Gary Froyland

Machine-learning optimal function bases for linear operators

This project will explore the use of machine learning to find optimal basis functions to represent discrete approximations of linear operators.

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