# Postgraduate study and research

Postgrad life in the School of Mathematics and Statistics at UC is challenging and rewarding. The academic staff are experts in their field of research and will be able to guide you through your postgraduate studies.

## Close-knit research community

The postgraduate community is supportive, close-knit, and a vital part of the School. Our weekly postgrad talks are an opportunity to practice our presentation skills and learn about new subjects. We have a culture of excellence in presentations, winning numerous awards for talks and posters at various conferences.

### Conference opportunities

Every year, postgrad students can attend the New Zealand Mathematics and Statistics Postgrad Conference with other postgrad students from all over New Zealand - an annual highlight. Our students are generally well funded and encouraged to travel to conferences in their field of research, making UC a fantastic place to start your academic career with a postgrad degree.

## Getting started

### Postgraduate Diploma in Science (PGDipSc)

This is a one-year full-time course. The course may be taken by any BSc graduate with 90 points in 300-level courses approved by the Head of School, and must include 60 points from MATH 310-399 or from STAT 310-399. A pass in eight one-semester courses chosen at the 400-level is normally required. Your choice requires the approval of the Head of School. A completed Postgraduate Diploma in Science (PGDipSc) can be substituted for the course work year of a two-year Master of Science (MSc).

### Bachelor of Arts or Bachelor of Science with Honours - BA(Hons) / BSc(Hons)

The Honours degree is a one-year (if studied full-time) programme of study consisting of an Honours project and eight 400-level courses. The assessment of the Class of Honours is based on overall performance in the programme.

#### Eligibility

Students need to be eligible to graduate with a BA or BSc (360 points) and have the appropriate prerequisites, which generally means at least 60 points at 300-level from your chosen subject, and a further 30 points at 300-level from either MATH or STAT courses. Students are also expected to have a GPA of at least 6.0 (B+ average) in courses relevant to their chosen subject. Final approval for entry is given by the Head of School.

#### Course requirements

For details of the 400-level courses on offer, see the School’s Postgraduate handbook (also available from reception). Every year there will be at least one course offered in analysis, algebra, discrete mathematics, functional analysis, differential equations and computational mathematics. Every Statistics 300-level course is offered as a 400-level course and courses in generalised linear models and bioinformatics are offered each year. A broad range of honours projects are listed in the Postgraduate Handbook. This list is not exhaustive, and there is plenty of scope for other possible projects. It is expected that a student will have arranged their project by the end of the first week of term. Assessment is based on a written report (80%), which is to be submitted in September, and an oral or a poster presentation in early October (20%).

### Specialised honours programmes

In addition to the single honours degrees, there are a number of joint honours programmes that you can study to combine Mathematics with another subject. To keep your options open to enter these courses you must ensure you study a broad base of courses at lower levels, especially the core mathematics courses, so that you have the appropriate prerequisites. For all joint honours programmes it is very important that you check the calendar regulations to ensure you are taking all the required courses.

Find out more about possible joint programmes and specialisations.

### Degree regulations

- Find out more about undertaking masters level research at UC.
- Find out more about undertaking PhD research at UC.

#### Masters theses - Master of Science (MSc) and Master of Arts (MA)

Masters theses are expected to display a good general knowledge of the field of study. Theses are judged by two examiners who submit independent reports. A masters thesis usually takes one year to complete.

Students who have a bachelors degree in Mathematics or Statistics can enrol for a master’s degree. The minimum period of study is two years full-time after a BSc/BA degree, or one year full-time after the respective Honours degree.

The first year of the two–year masters degree consists of at least eight one–semester 400-level courses. Your choice requires the approval of the Head of School. The second year (and also the one year masterate) consists of a thesis only. We almost always ask that you enrol for a BSc(Hons)/BA(Hons) or PGDipSc in your first year.

#### PhD theses

A PhD is a higher degree awarded for a thesis presenting original research that is a significant contribution to scientific knowledge. It must meet recognised international standards. Students with a good honours or masters degree can enrol for the degree of Doctor of Philosophy.

You will work under the supervision of an academic staff member with whom you have a shared research interest. This normally involves at least three years full–time study.

You must then present a thesis embodying the results of this research for examination. It is judged by two examiners, one of whom is usually from New Zealand. The examiners submit independent reports on the thesis. You then sit an oral examination conducted by one of the examiners. There is no coursework requirement.

#### Enrolment criteria

To qualify to enrol for a PhD you must have a bachelors or masters degree from a New Zealand university with first or second class honours (or the equivalent). Students from outside New Zealand must have met an equivalent standard.

To qualify to enrol for a research Masters degree you must have completed a bachelors degree with honours or a Postgraduate Diploma or Masters part I (this is equivalent to a PGDipSci). Students from outside New Zealand must have met an equivalent standard.

If English is not your first language you must also meet the University's English language requirements.

First study the currently available research projects, listed above. If any particular project interests you contact the supervisor for more information.

If you have a project or research area in mind and think a particular academic may be a suitable supervisor contact them directly. The School’s key research areas are outlined on our research page. There are often other projects available that are not listed here.

If you wish to make a more general inquiry please email us, outlining your interests. Attach your CV and academic transcript. Your email will be forwarded to appropriate members of staff, who will contact you if they have a suitable project. Try to be as specific as possible about the areas you are interested in, as this will help us match you with a suitable supervisor.

Once we have identified a supervisor for you they will be able to guide you through the application procedure in more detail.

- Find out more about admission and enrolment processes at UC.

As a research student you can expect:

- a senior supervisor who directs your research and training and spends at least 1 hour a week with you
- a supervision team with at least one other staff member to monitor progress and offer advice
- regular progress meetings with your supervision team
- regular progress reviews for which you prepare a report
- a programme of training in research and transferable skills tailored to your needs
- opportunities to attend seminars by leading academics from around the world
- opportunities to present your work both nationally and internationally
- general courses available to all students which include project planning, writing and presentation
- specific mathematical and statistical courses through our 400-level courses
- a thriving peer-support network.

There are a range of scholarships available to postgraduate students in Mathematics and Statistics who need help to fund their studies.

- Explore the full UC Scholarships database.

### Funding your studies

The most common form of funding for PhD candidates are UC doctoral scholarships. These awards pay tuition fees and give a generous living allowance for three years (PhD). UC also offers masters scholarships. There are two application rounds each year with deadlines of October 15th and May 15th.

In some cases, alternative funding may be available for specific projects and the project supervisor will advise you about this.

Other sources of funding include UC College of Engineering scholarships (for women and Maori students). The university also has agreements with some countries (including China and Malaysia) concerning funding. To find out more about these contact the university scholarships office.

#### International students

International students pay fees at the domestic rate for their PhD studies at UC and similar UC doctoral scholarships are available for international students - with the same application process. Applications to the university scheme are primarily judged by the GPA (or equivalent) of the student.

### Masters and PhD research projects

There are a number of active research groups working on problems in Pure Mathematics, Applied Mathematics and Statistics.

### Topic 1: New combinatorial and algorithmic tools for network reconstruction

It is now 150 years since the publication of Darwin’s Origin of Species. In the intervening years, phylogenetic trees have been successfully used to represent and analyse ancestral relationships. Mathematics underlies modern tree reconstruction methods. However, mathematics faces a new challenge because, at various evolutionary scales, evolution is not described by a tree, but by a network.

This project will address some unanswered questions concerning phylogenetic network reconstruction. Questions include quantifying lateral gene transfer in early evolution and axiomatising network reconstruction methods. The development of tools for answering these questions will involve combinatorics, algorithmics, and graph theory.

#### Student background

Honours degree (or equivalent) in mathematics. Intended for a Masters or PhD student.

#### Research areas

Algebra, Combinatorics, and Logic, Mathematical Biology

#### Contact

### Topic 2: Combinatorial aspects of small world networks

The idea that any two people on the planet are connected via a short chain acquaintances (in most cases only 5) has become known as the 'six degrees of separation' theory, or the 'small world effect'. It is an idea that has engaged two fields that usually have little to do with each other, namely combinatorics (particularly random graph theory) and sociology. Moreover, the idea has also been applied to other interaction networks - from the links between webpages on the internet, to metabolic networks. This project will survey this field from a combinatorial perspective, and consider some questions such as 'how far apart can two people be?'

#### Student background

The project will suit a student with a solid background in discrete mathematics. Probability theory and/or programming experience may also be helpful. Intended for a Masters student.

#### Research area

Algebra, Combinatorics, and Logic

#### Contact

### Topic 3: Finite Laguerre planes

Finite Laguerre planes are combinatorial structures that generalize the geometry of polynomials of degree at most 2 over finite fields. They can be regarded as extensions of affine planes plus a family of parabolic ovals and are related to certain generalized quadrangles. There are many open problems. One of them is the existence of planes that are not ovoidal, that is, that cannot be obtained from an ovoidal cone in 3-dimensional projective space by intersecting with planes of 3-space that do not pass through the vertex of the cone.

In this project one addresses some of the open problems. For example, one can make assumptions on the automorphism group of a Laguerre plane, which in particular leads to the so-called elation Laguerre planes, and try to use group theory to construct new models. Another approach is to use the classification of certain finite affine planes and do a computer search for parabolic ovals in these affine planes. If one finds sufficiently many parabolic ovals one can then try to extend them to a Laguerre plane.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Algebra, Combinatorics, and Logic, Analysis and Geometry

#### Contact

Associate Professor Günter Steinke

### Topic 4: Möbius near planes of small orders

A finite Möbius or inversive plane of order n is a 3-(n2+1,n,1) design and removing one point leads to a Möbius near plane. Conversely one can start with an axiomatic definition of a Möbius near plane and ask whether or not it comes from a Möbius plane by removing one point. The answer to this question is affirmative if n is at least 5. The project is to study and classify the remaining cases. This may require a computer search.

#### Student background

Intended for a Masters student

#### Research area

Algebra, Combinatorics, and Logic

#### Contact

Associate Professor Günter Steinke

### Topic 5: The geometry of interpolating systems

An interpolating system of rank n is a system of curves such that given n distinct points no two of which have the same first coordinates there is a unique curve in the system passing through these n points. Various other conditions are imposed if one has differentiable curves. In applied mathematics interpolating systems are investigated as to how well they approximate functions or what happens as n tends to infinity, etc. However, every interpolating system has a geometric aspect to it: interpolating systems of rank 2 are essentially projective planes and interpolating systems of rank 3 are closely related to Laguerre planes, both of which are well-studied geometric objects.

In this project one looks at the geometry of interpolating systems of rank at least 4 over the real numbers, where topological considerations also are involved. There are a number of possible topics such as the construction of new interpolating systems, the characterization of certain interpolating systems by configurational conditions, a determination of the automorphism groups of interpolating systems and classification of the most homogeneous ones, or a characterization of certain interpolating systems by groups of transitive automorphisms.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Algebra, Combinatorics, and Logic, Analysis and Geometry

#### Contact

### Topic 1: Differentiability in topological circle planes

Topological geometries (such as the Euclidean affine plane) are defined by their geometric properties and topological conditions that require the geometric operations to be continuous. (For example, in the Euclidean affine plane joining two distinct points by a line depends continuously on the two points.) The standard geometries not only have a topological structure but also a differentiable one with respect to which the geometric operations are even differentiable. In this project one looks at topological circle planes (Möbius, Laguerre or Minkowski planes), imposes differentiability conditions and asks what models can result.

#### Student background

Intended for a Masters or PhD student

#### Research area

Analysis and Geometry

#### Contact

Associate Professor Günter Steinke

### Topic 2: Finite Laguerre planes

Finite Laguerre planes are combinatorial structures that generalise the geometry of polynomials of degree at most 2 over finite fields. They can be regarded as extensions of affine planes plus a family of parabolic ovals and are related to certain generalised quadrangles. There are many open problems. One of them is the existence of planes that are not ovoidal, that is, that cannot be obtained from an ovoidal cone in 3-dimensional projective space by intersecting with planes of 3-space that do not pass through the vertex of the cone.

In this project one addresses some of the open problems. For example, one can make assumptions on the automorphism group of a Laguerre plane, which in particular leads to the so-called elation Laguerre planes, and try to use group theory to construct new models. Another approach is to use the classification of certain finite affine planes and do a computer search for parabolic ovals in these affine planes. If one finds sufficiently many parabolic ovals one can then try to extend them to a Laguerre plane.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Algebra, Combinatorics, and Logic, Analysis and Geometry

#### Contact

Associate Professor Günter Steinke

### Topic 3: Semiclassical Minkowski planes

Minkowski planes are geometries that generalise the geometry of linear fractional maps over fields. In the case of the real numbers one also has topological properties.

A general principle in topological geometry is to take some classical object, cut it in two (or more) halves and reassemble the pieces so that the defining geometric and topological properties are still preserved. In this project this cut-and-paste method is applied to the standard Minkowski plane over the real numbers and all possible models obtained in this way are to be determined.

#### Student background

Intended for a Masters or PhD student

#### Research area

Analysis and Geometry

#### Contact

Associate Professor Günter Steinke

### Topic 4: The geometry of interpolating systems

An interpolating system of rank n is a system of curves such that given n distinct points no two of which have the same first co-ordinates there is a unique curve in the system passing through these n points. Various other conditions are imposed if one has differentiable curves. In applied mathematics interpolating systems are investigated as to how well they approximate functions or what happens as n tends to infinity, etc. However, every interpolating system has a geometric aspect to it: interpolating systems of rank 2 are essentially projective planes and interpolating systems of rank 3 are closely related to Laguerre planes, both of which are well-studied geometric objects.

In this project one looks at the geometry of interpolating systems of rank at least 4 over the real numbers, where topological considerations also are involved. There are a number of possible topics such as the construction of new interpolating systems, the characterisation of certain interpolating systems by configurational conditions, a determination of the automorphism groups of interpolating systems and classification of the most homogeneous ones, or a characterisation of certain interpolating systems by groups of transitive automorphisms.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Algebra, Combinatorics, and Logic, Analysis and Geometry

#### Contact

### Topic 1: Efficient survey designs

Development of efficient survey designs for environmental monitoring. This project involves looking at optimal survey designs for monitoring changes in animal and plant population abundances. The project could involve some field work.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Applied Statistics, Mathematical Biology

#### Contact

### Topic 2: Spatial patterns in ecological models

Incorporating information on spatial pattern into ecological models. Statistics on spatial patterns will be used to model optimal network designs for environmental monitoring. Aspects of adaptive sampling will be used to develop optimal designs.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Applied Statistics, Mathematical Biology

#### Contact

### Dynamical systems and ergodic theory

Dynamical systems is the study of any mathematical system where spatial structure evolves with time. Modern approaches may be geometric, topological, probabilistic or computational. I can supervise a range of projects in theoretical and/or computational dynamical systems and ergodic theory. I am also able to co-supervise projects in applied dynamics.

#### Student background

Intended for a Masters or PhD student

#### Research area

Dynamical Systems and Differential Equations

#### Contact

### Project 1: Fluid-particle interactions for granular flows

A Masters project aimed at extending a simple model of falling bodies interacting with the surrounding medium, bounding walls, and each other. The project involves fluid dynamics, asymptotic analysis, and some computation.

#### Student background

Intended for a Masters student

#### Research area

Financial and Industrial Mathematics

#### Contact

### Project 2: Pseudo-wake structure in colliding turbulent boundary layers

A Masters project aimed at examining the global behaviour of turbulent boundary layers colliding in a curved pipe. The project involves fluid dynamics, asymptotic analysis, and some computation.

#### Student background

Intended for a Masters student

#### Research area

Financial and Industrial Mathematics

#### Contact

### Project 1: Crisis and Change

Crisis and Change-the historical and mathematical analysis of a significant episode in the history of mathematics.

#### Student background

Intended for a Masters or PhD student

#### Research area

History and Philosophy of Mathematics, and Mathematics Education

#### Contact

Associate Professor Clemency Montelle

### Project 2: Ethnomathematics

#### Student background

Intended for a Masters or PhD student

#### Research area

History and Philosophy of Mathematics, and Mathematics Education

#### Contact

Associate Professor Clemency Montelle

### Project 3: Mathematical themes from ancient mathematics

Various mathematical themes from ancient mathematics, including the Ancient Near East, Ancient Greece, India, or the Islamic Near East.

#### Student background

Intended for a Masters or PhD student

#### Research area

History and Philosophy of Mathematics, and Mathematics Education

#### Contact

Associate Professor Clemency Montelle

### Project 4: Textual and mathematical analysis of a historical work

Textual and mathematical analysis of a historical work, either in translation or original language (Cuneiform, Greek, Latin, Sanskrit, Arabic, French).

#### Student background

Intended for a Masters or PhD student

#### Research area

History and Philosophy of Mathematics, and Mathematics Education

#### Contact

### Project 1: Efficient survey designs

Development of efficient survey designs for environmental monitoring. This project involves looking at optimal survey designs for monitoring changes in animal and plant population abundances. The project could involve some field work.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Applied Statistics, Mathematical Biology

#### Contact

### Project 2: Spatial patterns in ecological models

Incorporating information on spatial pattern into ecological models. Statistics on spatial patterns will be used to model optimal network designs for environmental monitoring. Aspects of adaptive sampling will be used to develop optimal designs.

#### Student background

Intended for a Masters or PhD student

#### Research areas

Applied Statistics, Mathematical Biology

#### Contact

### Project 3: Combining size and species in marine ecosystems

There are two main paradigms for the study of ecosystems: food webs and size spectra. The traditional food web approach uses an individual's species as the main indicator of its feeding requirements. More recently, marine ecosystems have been characterised by the relationship between body mass and abundance: small organisms are much more abundant than large ones, and this relationship is usually quite predictable. In this framework, an individual's size, rather than its species, determines its feeding preferences. Increasingly, it is being recognised than both size and species are important factors, and it is not sufficient to consider one in isolation. This project will systematically investigate the effects of adding a size structure to traditional species-based models, or alternatively of introducing different species into a size spectrum model.

#### Student background

Intended for a Masters or PhD student

#### Research area

Mathematical Biology

#### Contact

Associate Professor Michael Plank

### Project 4: Epidemic spread on a network

Information on the spread of an epidemic through a population is often collected in the form of contact tracing data – essentially a tree showing who passed the disease onto whom. A natural way to model the spread of an epidemic that complements this form of data is a network model: the population is represented as a network with each node representing an individual, and links between nodes representing social contacts. Different networks have different mathematical properties, such as the mean and variance of the number of contacts per individuals, and the level of clustering of connections. These properties can be crucial in determining the spread or extinction of the epidemic. This project will investigate the relationship between network properties and epidemic progress, comparing the results with standard SIR-type models.

#### Student background

Intended for a Masters or PhD student

#### Research area

Mathematical Biology

#### Contact

Associate Professor Michael Plank

### Project 5: Weed risk assessment and control

New weeds have to go through a number of stages before they become an invasive pest. These include introduction to a particular country or region, naturalisation (or "going wild") and becoming a pest with some negative economic or other impact. The number of species present in each of these categories typically decreases by a factor of approximately 10 from one stage to the next. This poses a dilemma for weed management agencies. Should they focus their limited resources on controlling a few established pests (which may be widespread and impossible to completely elimiate), or should they divert some attention to the many species in the earlier stages, in an attempt to nip potential weeds in the bud before they become a problem? This question can be approached in the context of a bioeconomic optimisation problem, with the goal of minimising the "cost" of the system, consisting of the monetary cost of control, plus the negative impacts of weeds in the environment. Formulating the problem demands knowledge of factors involved in weed invasion and control. But once this is done, finding the optimum solution is pretty straightforward mathematically and has the potential to provide a general strategic framework for weed managers.

#### Student background

Intended for a Masters student

#### Research area

Mathematical Biology

#### Contact

Associate Professor Michael Plank

### Project 6: New combinatorial and algorithmic tools for network reconstruction

It is now 150 years since the publication of Darwin’s Origin of Species. In the intervening years, phylogenetic trees have been successfully used to represent and analyse ancestral relationships. Mathematics underlies modern tree reconstruction methods. However, mathematics faces a new challenge because, at various evolutionary scales, evolution is not described by a tree, but by a network.

This project will address some unanswered questions concerning phylogenetic network reconstruction. Questions include quantifying lateral gene transfer in early evolution and axiomatising network reconstruction methods. The development of tools for answering these questions will involve combinatorics, algorithmics, and graph theory.

#### Student background

Students should have an honours degree (or equivalent) in mathematics. Intended for a Masters or PhD student.

#### Research areas

Algebra, Combinatorics, and Logic, Mathematical Biology

#### Contact

### Project 7: Discrete random models in evolutionary biology

The last 3 decades have seen spectacular advances in our understanding of evolutionary biology, due largely to the wealth of molecular data (genes and genomes) being generated. Stochastic models are a fundamental tool to analyse this data, and the development of better models, and better methods of analysis requires a careful interplay of mathematics, algorithm development, statistics, and communication with biologists.

This project will aim to develop models and methods required to analyse new types of genomic data that are becoming available, and to explore approaches that build a 'network of life' rather than 'tree of life'.

#### Student background

The precise project will depend on the skills and interests of the student, but any of the following background would be useful:

- probability theory and statistics
- discrete mathematics
- algorithms and computer science
- programming
- some background in modern molecular evolutionary biology

Intended for a PhD student

#### Research area

Mathematical Biology

#### Contact

### Project 8: Stochastic properties of speciation and extinction models

How do species evolve (and go extinct) through time? The 'shape' of evolutionary trees reconstructed from genetic data, and calibrated with fossil evidence, provides some clues. However to test competing hypotheses requires understanding the mathematical and statistical properties of different speciation and extinction models. This project will survey what is know in this area, then identify, and possibly answer one of several remaining questions. This project will suit a student with a solid background in probability theory, and linear ordinary differential equation modeling. Programming experience may also be helpful.

#### Student background

Intended for a Masters student

#### Research areas

Mathematical Biology, Theoretical and Computational Statistics

#### Contact

### Stochastic properties of speciation and extinction models

How do species evolve (and go extinct) through time? The 'shape' of evolutionary trees reconstructed from genetic data, and calibrated with fossil evidence, provides some clues. However to test competing hypotheses requires understanding the mathematical and statistical properties of different speciation and extinction models. This project will survey what is know in this area, then identify, and possibly answer one of several remaining questions. This project will suit a student with a solid background in probability theory, and linear ordinary differential equation modeling. Programming experience may also be helpful.

#### Student background

Intended for a Masters student

#### Research areas

Mathematical Biology, Theoretical and Computational Statistics

#### Contact

In general, I am happy to discuss Masters or PhD projects in the modelling of: lipid bilayers; cytoskeletons; cortical spreading depression; cerebral blood flow autoregulation.

#### Student background

Intended for a Masters or PhD student