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College of EngineeringSchool of Mathematics and Statistics

Postgraduate Handbook 2021

SCHOOL OF MATHEMATICS AND STATISTICS

POSTGRADUATE PROGRAMME 2021

Welcome from our School. Postgraduatestudy enables you to study selected math-ematical and statistical topics in depth.There is a change of emphasis from the pre-ceding undergraduate years: courses tendto be more focused on a specific problemor class of problems, rather than attempt-ing to give a broad coverage of a branch ofmathematics and statistics.

There is the possibility of taking one ormore projects in which you investigatesome problem with the assistance of amember of sta�. Depending on the natureof the problem, this may involve literaturesearches, the use of various computingpackages (for example MATLAB, Maple orR), resources on the internet, proving newtheorems or data analysis. You will pro-duce a written dissertation and may givean oral presentation.

Any proposed programme of study re-quires the approval of the 400 level coor-dinator. It is highly unlikely that any pro-posed programme that has a high work-load in one semester will be approved, soyou should try to construct a programmethat balances your workload evenly overboth semesters.

You can include some courses from othersubjects (e.g. COSC480 is recommendedfor developing programming skills). Thisis a good way to ensure you have a broadprogram of study. Check with the School’s400 level coordinator that the courses from

other subjects are suitable for inclusionwithin your program of study. In addi-tion, there are a various joint programsbetween our School and other Depart-ments/Schools detailed below.

At the Masters and PhD level you will un-dertake research, o�en focussed on deepstudy of a specialised topic. You will learnskills in undertaking systematic investiga-tions, contextualising your work withinthe current state of understanding, so thatyour research outcomes can extend be-yond the forefront of human knowledge.

HONOURS OR PG DIPLOMA?

The School o�ers both Honours and Post-graduate Diploma programs of study,which can be undertaken under Science orArts. The most appropriate program is bestdecided on a case-by-case basis which youshould discuss with the 400 level Coordi-nator, Prof. Michael Plank. The followingguide provides some general advice aboutyour options. You are welcome to get intouch as soon as possible, but you mustdo so before the formal enrolment period.If you are undertaking honours, you musthave arranged a supervisor of your projectin advance of enrolment. The 400 level co-ordinator can help you with this.

In addition, we also o�er a Postgraduate

Certificate in Arts for students interestedin a 60 points programme of study, or aCertificate of Proficiency for undertaking acourse or courses of interest.

Who should think about Honours? Ifyou view Mathematics/Statistics as morethan a means to an end, then doing Hon-ours will be a year well spent. In additionto taught courses, the honours programhas a full year 30 point project which willnot only deepen your understanding ofa specialised topic but will also developmany of the so� skills desired by employ-ers or for further Postgraduate study, likeself-motivation, independent learning, re-search, written and oral communication.

The Honours subject majors are listed be-low. Formal details are in the UC Calen-dar. To enter Honours in Mathematics, youwill need at least 60 points from MATH301-394, plus at least 30 points from 300-levelMATH, STAT or other approved courses. ForHonours in Statistics, you need at least 60points of 300-level STAT301-394, plus atleast 30 points from 300-level STAT, MATHor other approved courses. Normally youwill have maintained at least a B+ averagein these papers.

BSc (Hons) Major Subjects

In the Science faculty, Honours from ourSchool may be completed in:

• Mathematics and/or Statistics, seeProf. Michael Plank;

• Data Science; see Prof. JenniferBrown;

• Mathematical Physics, see Assoc.

Prof. Jenni Adams (Physics);

• Computational and Applied Mathe-matical Sciences, see Prof. MichaelPlank;

• Mathematics and Philosophy, seeProf. Clemency Montelle;

• Finance and Mathematics, see Assoc.Prof. Rua Murray;

• Finance and Statistics, see Assoc.Prof. Marco Reale; or

• Financial Engineering, see Assoc.Prof. Marco Reale.

These Honours programs require com-pletion of papers totalling 90 points at400 level or above (typically six 15 pointcourses) in addition to the 30 pointMATH/STAT/CAMS449 project. In the caseof data science, the project is 45 points.

BA (Hons) Major Subjects

In the Arts faculty, Honours from ourSchool may be completed in Mathemat-ics or Statistics. BA (Hons) consists of aproject (MATH/STAT449) and six papers(120 points in total).

Who should think about a PostgraduateDiploma? The Postgraduate Diploma canconsist entirely of taught courses, as thereis no requirement that any project is under-taken. The entry requirements are as forHonours, except that you are not requiredto have a B+ average. It is very stronglyrecommended that your average grade inyour majoring subject at stage 3 is at leasta C+. The PGDipSc can also be used as PartI of a two part research MSc.

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PGDIPSCI AND PGDIPARTS

In both the Science and Arts faculty thePostgraduate Diploma can be taken inmathematics or statistics. In addition aPostgraduate Diploma in data science maybe taken in the Science faculty. Thesediplomas require completion of papers to-talling 120 points at 400-level or above(typically eight 15 point courses).

MASTERS IN APPLIED DATASCIENCE

Data science is a new profession emerg-ing along with the exponential growthin size, and availability of ‘big data’. Adata scientist provides insight into futuretrends from looking at past and currentdata. Data science is an essential skill in aworld where everything from education tocommerce, communication to transport,involves large scale data collection and dig-italisation. New Zealand and other coun-tries are currently experiencing a skillsshortage in this area, and the need for datasavvy professionals with applied experi-ence is growing.

This 180 point conversion master’s is de-signed to accommodate students from arange of undergraduate backgrounds (notjust those with Mathematics, Statistics andComputer Science majors), who want toenhance or build their data science capa-bilities and combine these with the skillsand knowledge they bring from their pre-vious studies. So long as you are data-hungry and industry-aware; this degreecan add to your employability and careerprospects.

MASTERS IN FINANCIALENGINEERING

Financial engineering is a cross-disciplinary field combining financial the-ory, mathematics, statistics and compu-tational tools to design and develop newfinancial or actuarial products, portfoliosand markets. It also has an importantrole to play in the financial industry’s reg-ulatory framework. Financial engineersmanage financial risk, identify market op-portunities, design and value financial oractuarial (insurance) products, and opti-mize investment strategies.

The year long 180 point program consists:

• 135 points from taught courses.There is a core set of requiredcourses in finance, mathematics &statistics and computer science. Fur-ther, there are a suite of suggestedcourses from these topic areas, thatmake up the majoring subject ofFinancial Engineering. Dependingon your prior education, it is envis-aged that around half of the taughtcourses will be MATH400 or STAT400papers and the other half will beFINC600 papers; and

• the 45 point paper FENG601 Applica-tion of Financial Engineering whichprovides the opportunity to applythe techniques learned through theprogramme to real-world financialengineering problems.

There are minimum entry requirementsinto the program, which if not met you will

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be required to take FIEC601 in January-February prior to commencement of theprogram proper. You will be required tocomplete COSC480 Introduction to Pro-gramming, if you do not have equivalentprogramming skills (e.g. from COSC121,MATH170, EMTH171 or STAT221). Full de-tails are provided in the UC Calendar.

RESEARCHMASTERS

A research Masters in Science (MSc) or Arts(MA) consists of two parts:

• Part I - a 120 points of papers (typi-cally eight 15 point courses); and

• Part II - a 120 points research thesis.

Students can enter directly to Part II, if theyhave completed a Postgradute Diploma orHonours degree in the same majoring sub-ject. For full details see the UC Calendar.

Our School o�ers the research MSc andMA in mathematics or statistics. An MScis also o�ered in computational and math-ematical sciences and data science. En-rolment in a Master’s programme requiresapproval from the Postgraduate Coordina-tor, Dr Daniel Gerhard. At least one sta�member must have agreed to superviseyour Part II research study before approvalof your programme of study.

PHD RESEARCH

The PhD programme is the highest degreeo�ered in UC. How do you know if you areready to purse a PhD in any of the followingsubjects we o�er?

• mathematics;

• statistics;

• computational and applied mathe-matical sciences (CAMS);

• mathematical physics; and

• mathematics and philosophy.

The simplest answer is: if you are passion-ate about a subject and you want to get adeeper understanding of a field of study orwant to use sophisticated tools from math-ematical sciences to solve real world prob-lems, then you are ready!

If you want to upscale your knowledge inthe subject you love then a PhD in math-ematics or statistics is the programme foryou. On the other hand, if you have an in-terdisciplinary project in mind then a PhDin CAMS could be a good option for you.

Further details are available from thePG O�ice website, including scholar-ship information, here: http://www.canterbury.ac.nz/postgraduate/phd-and-doctoral-study/ Excellentperformance in a BSc (Hons) or BA (Hons)may provide su�icient training to under-take a PhD, thus obviating the need for aMasters degree. However, a PGDipSc orPGDipArts would not normally be su�i-cient.

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400-600LEVELCOURSES

The courses for 2021 are outlined on theCIS system https://www.canterbury.ac.nz/courseinfo/GetCourseDetails.aspx. In order to see all the o�erings for,say, Mathematics, search for MATH4. TheSchool reserves the right to cancel anycourse that has a low enrolment. This willbe determined at the beginning of eachsemester.

It is also possible (and o�en desirable)to include courses from other subjects,see the Regulations in the Calender fordetails with each degree. Note that anySTAT courses may be included in a Math-ematics degree and vice versa. For multi-disciplinary programmes like Financial En-gineering and Data Science (which havecourses across subjects) consult ScheduleA of the BSc Hons regulation in the Calen-dar for a list of potential courses.

400-LEVEL PROJECTS

A broad range of possible projects areoutlined below. However, this list isnot exhaustive and other possibilities forprojects are certainly possible. Project su-pervision is by mutual agreement of the su-pervisor and student. You should arrangeyour project by the end of the first week ofterm in 2021. It is suggested that you seekout possible supervisors before enrolmentweek.

You will hand in a written report on 6 Octo-ber 2021 which will contribute 80% of thegrade; the remaining 20% will be an oralpresentation in Term 4.

PROJECTS IN MATHEMATICS

Cryptographic Schemes and Diophan-tine equationsFelipe VolochA Diophantine equation is a polynomialequation in many variables where the co-e�icients and the solutions are restrictedin some way, e.g., to be integers or rationalnumbers.

Solving a Diophantine equation is typicallyhard but it is easy to build an equation soas to have a predetermined solution. Thisis an example of a “one-way function” withpotential applications to cryptography. Itis essential for some of these applicationsthat we know that the equation we builddoes not have additional solutions.

There are a number of choices to go withthat for an honours project dependingon the student’s background and interest.One can look at the theoretical questionsrelated to counting solutions to Diophan-tine equations or solving them. One canlook at computational methods to solvethese equations. Finally, one can lookat the cryptography side of it, building orbreaking or implementing these cryptosys-tems.

Designs and their codeGeertrui Van de VoordeIn this project, we will look at connectionsbetween design theory and coding the-ory. A famous example is provided by the(sporadic) Golay codes, which correspondto the Witt designs (and sporadic Mathieugroups). More generally, the support of thecodewords in a perfect codes always forma t-design. This project can take multiple

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directions, according to your interests; in-cluding recent research on rank-metricequivalents of these classical links.

Axiomatic planesGeertrui Van de VoordeIn the real (Euclidean) plane, we knowthat there is exactly one line through twodi�erent points and that there is exactlyone line through a point that is parallelto a given line. Now these two propertiescan be taken as axioms and a new class ofplanes, called axiomatic a�ine planes, canbe constructed. In particular, it is perfectlypossible to construct such planes thathave only a finite number of points andlines. Probably the most important conjec-ture in this area is the question what thepossibilities are for the number of pointsin an axiomatic plane. This project cantake multiple directions, according to yourinterests.

Generalised quadranges/polygonsGeertrui Van de VoordeThe incidence graph of a generalised quad-rangle is characterized by being a con-nected, bipartite graph with diameter fourand girth eight. We can think of themas a set of points and lines without trian-gles (hence the name). Generalised quad-rangles have their own rich theory, whichdates back to the work of Tits on groups ofLie type.

In this project, we will study finite gener-alised quadrangles: we look for construc-tions, classifications and characterisations.Generalised quadrangles fall into the moregeneral class of generalised polygons (di-ameter n, girth 2n). A classic theorem ofHigman and Feit shows that a finite gener-

alised polygon is either a di-gon (n = 2),a projective plane (n = 3), a generalisedquadrangle (n = 4), a generalised hexagon(n = 6) or a generalised octagon (n = 8).This project studies generalised quadran-gles and/or polygons and can take multi-ple directions, according to your interests.

Computations in group cohomologyBrendan CreutzIn this project you will study properties ofthe group GL2(Z) of invertible 2 x 2 ma-trices with integer coe�icients and relatedgroups obtained by reducing the entriesmodulo n. The main tool for this will begroup cohomology, which sounds compli-cated but can be computed fairly easily byhand or using computer algebra so�ware.In the project you will learn how to carryout such computations and then collectdata about these groups with the hope ofuncovering interesting patterns which youcould then try to prove theoretically. Themotivation for studying these particulargroups comes from elliptic curves, andpossible follow up projects at postgrad-uate level could explore this connectionfurther. This project would be suitable fora student with basic knowledge of groups(as seen in MATH240) and finite fields ormodular arithmetic (as seen for examplein MATH220, MATH321 or MATH324).

Data driven analysis of dynamical sys-temsRua MurrayDynamical systems arise as solutions ofdi�erential equations, or in any situationwhere the state of a system updates it-eratively with the passage of time steps(e.g., a descent algorithm for training adeep learning network). The local and

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global behaviour of dynamical systemsis o�en determined by invariants of var-ious kinds: fixed points, periodic orbits,invariant manifolds, invariant probabilitydistributions. When the system is complex(due to very strong nonlinearities and/orhigh dimension), these objects are hardto find and analyse. In the last decade, anew family of tools has developed, looselyunder the umbrella name of “Dynamicmode decomposition”. These methodsuse samples from the dynamical systemto build approximate transfer operators,from which eigenvectors can be extracted.The theory behind these methods remainsundeveloped, there is a plethora of pos-sible computational strategies, and anydynamical system can be analysed in thisway. The emphasis in this project can betailored to student interest.

Exploring Links between Topology andCombinatoricsMike SteelTopological methods turn out to have un-expected applications in discrete math-ematics. One example is the use of the“Ham sandwich theorem” to show thattwo thieves can always divide up a neck-lace with k kinds of jewels using no morethan k cuts. Another example is the linkbetween the Möbius function of a partiallyordered set and the Euler characteristicof an associated topological space. Thisproject will suit a student who has takenMATH320 and is taking MATH428.

Integer ProgrammingChris PriceThis project looks at various applicationsof integer programming, and stochastic so-lution techniques. Applications such as set

covering, packing, and partitioning will belooked at, as well as network interdictionproblems. Variants of genetic algorithmswill be looked at, and tested on a selectionof these applications. MATH303 or similarrequired.

Maths Cra�Jeanette McLeod (Maths & Stats), PhilWilson (Maths & Stats), David Pomeroy(Teacher Education)Do you love mathematics and cra�? Areyou interested in the history, sociology, orpsychology of mathematics? We are of-fering multiple interdisciplinary honoursprojects for keen students who answeryes to those questions. Maths Cra� NewZealand is a non-profit public engagementinitiative which makes mathematics ac-cessible through cra�. It was founded andis run by Drs Jeanette McLeod and PhilWilson from the School of Mathematics &Statistics. Since Maths Cra�’s inception in2016, we have run numerous festivals andworkshops across New Zealand, and havereached over 11,000 people, making us thelargest maths outreach programme in thecountry. Together with Dr David Pomeroyfrom the School of Teacher Education, weare studying the mathematical, historical,sociological, psychological, and pedagog-ical aspects of our cra�-based approachto mathematics. If you like to think out-side the box and can combine a love ofmathematics and cra� with a love of (orenthusiasm to learn about) social research,then please discuss project options withus.

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Population dynamics: Bugs, beetles,plants, animals and diseasesAlex JamesPopulations, be they of bugs, plants or hu-mans infected with viruses show a remark-able range of behaviours. Find out moreabout them in a project that uses dynami-cal system models to try and understandthe dynamics of an example population.Examples systems include data driven dy-namic models of New Zealand birds, thee�ect of initial viral load on COVID-19 mor-tality and assessing vaccination strategiesfor emerging diseases.

Lax PairsMark HickmanGiven a non-linear di�erential equation, aLax pair is a pair of linear di�erential oper-ators L, M whose commutator vanishesonly on solutions of the di�erential equa-tion. A Lax pair allows one to potentiallysolve the di�erential equation by reducingthe problem to an eigenvalue problem (ifthe operator L is second order, this is aSturm-Liouville problem) and a time evo-lution of the eigenfunction; the so-calledinverse scattering method. If L is first or-der then the Lax pair gives a conservationlaw of the di�erential equation. In thisproject, we will be looking at a method tocompute the Lax pair of prescribed orderfor a di�erential equation (if it exists). Thiswill involve Maple and would suit a studentwho has completed MATH302.

Combinatorics of reticulate evolutionCharles SempleEvolution is not always a treelike processbecause of non-treelike (reticulate) eventssuch as lateral gene transfer and hybridisa-tion. In computational biology, the reticu-

late evolutionary history of a set of extantspecies is typically represented by a phy-logenetic network, a particular type ofrooted acyclic directed graph. Althoughreticulation has long been recognised inevolutionary biology, mathematical in-vestigations into resolving questions con-cerning the combinatorial and algorithmicproperties of phylogenetic networks arerelatively new. Questions include, for ex-ample, how hard is it to decide if a givengene tree is embeddable in a given net-work? If we select a network with a certainnumber of leaves uniformly at random,how many reticulations do we expect itto have when the number of leaves is suf-ficiently large? When is a network deter-mined by the path-length distances be-tween its leaves? The aim of this project isto explore such questions. It involves dis-crete mathematics but there are no formalprerequistes.

Tutte’s 5-Flow ConjectureCharles SempleNetwork flow problems are an importantclass of problems in combinatorial opti-misation and represent a large variety ofreal-world occurrences. A particular typeof network problem gives rise to Tutte’s 5-flow conjecture (1954), amongst the mostoutstanding conjectures in modern-daygraph theory. The purpose of this projectis to investigate the progress that has beenmade on this conjecture and its connec-tions to other areas of combinatorics. Towhet your appetite, the Four Colour Theo-rem says that every planar graph withoutisthmuses has a 4-flow. While some priorknowledge of graph theory would be help-ful, it is not a prerequisite for the project.

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Topics in Group TheoryGunter SteinkeGroups naturally occur as collections ofsymmetries of algebraic structures or ge-ometries or algebraic structures or otherobjects. Knowing the structure of thegroup of symmetries of an object o�enleads to useful information about the un-derlying object. Groups come in very dif-ferent sizes and forms and are also fasci-nating in their own right.

In one project we may look at abstractgroups and how they can be seen asgroups of symmetries. By making addi-tional assumptions on transitivity we tryto determine which groups can arise. Forexample, the symmetric and alternatinggroups are the only finite groups that arehighly transitive, but there are many inter-esting groups that are 2-, 3- or 4-transitive.

In another project one may investigate so-called crystallographic groups. They ariseas groups of symmetries of (perfect andunbounded) crystals. In two dimensionsthey are referred to as wallpaper groups.One looks at the underlying principles thatallow to classify crystallographic groupsand carry out a classification, for example,of wallpaper groups.

Dimension TheoryGunter SteinkeWhile one has a precise notion of dimen-sion for vector spaces, there o�en is anintuitive understanding of the dimensionof a space (not necessarily a vector space)as the number of coordinates or parame-ters used to describe the space. However,this notion proved to be imprecise as dis-coveries in the early 20th century showed.There exist bijections between a line and

a plane and also continuous maps fromthe unit interval onto the unit square. Thisled to the question of whether or not m-space Rm and n-space Rn can be topolog-ically the same for di�erent m and n. Toanswer this question various topologicalinvariants have been devised.

Obviously, any useful invariant should as-sign n-space dimension n. While it is o�eneasy to verify that n-space has dimensionat most n, it is harder to establish equality.

The project investigates some possibledefinitions of the dimension of a (met-ric) space, their properties, when thesedimensions agree and what examples oftopological spaces there are for which theyare di�erent.

PROJECTS IN STATISTICS

Historical heights of army and navy re-cruitsElena MoltchanovaThe population distribution of humanheights reflects prevailing environmentaland sociological conditions. The histor-ical records available, however, alwayspresent a biased picture. For example,only people “tall enough” were enlistedin the army, and the definition of “tallenough” varied with demand for soldiers.For the navy, the limitations also appliedto people who were too tall to comfort-ably live on a ship. Using the informationavailable from the historical army and navyrecords to recreate the population heightdistribution throughout the 17th-20th cen-tury thus presents some interesting chal-lenges. Among many attempts taken to

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model such data, Bayesian methods areof particular interest due to their flexibil-ity and ability to easily include temporalautocorrelation in the model. You willneed to have taken a course in Bayesianinference and have solid knowledge ofcalculus (deriving conditional, marginal,etc. distributions) and good programmingskills in either R, PYTHON or C/C++. (Otherprojects using Bayesian statistics are avail-able, such as using reversible jump Markovchain Monte Carlo to model the Old Bai-ley’s data or monitoring epi- demics andmanufacturing processes.)

PROJECTS IN DATA SCIENCE

Ethics of data science / Data science forethicsGiulio Valentino Dalla RivaUsing a variety of mathematical, statisti-cal, computational, ..., approaches, we aregoing to analyse from an hybrid ethical-technical point of view some fundamentaldata scientific algorithm. As an example,think about the “Recommended” videoson YouTube, the “watch next” movies onNetflix, the “Discovery” songs on Spotify,the “related coverage” news in the NewYork Times: what do they all have in com-mon? They all suggest you, based on yourhistory and characteristics, which bit of in-formation to consider next. They define auser-dependent priority on the availableinformation. They filter information foryou, and they shape the way you see theworld (or, at least, part of it).

Recommender systems are ubiquitous ma-chine learning algorithms for prioritisinginformation. Di�erent technical decisions

impact the fairness, openness, reliability,trust, and social benefit of them. We will in-vestigate mathematical models and com-putational techniques to assess how state-of-the-art recommender systems performand, eventually, propose alternative sys-tems.

Familiarity with a scientific programminglanguage (R, PYTHON, JULIA, . . . ) is rec-ommended.

Social network and online communi-ties analysisGiulio Valentino Dalla RivaThis is a open ended project. If you areinterested in using and developing math-ematical, statistical, data scientific toolsand notions to analyse the behaviour ofonline communities in social networks, wecan discuss it. We are probably going touse a variety of approaches: data wran-gling, scraping, anonymisation, networksmodelling, natural language processing,image analysis, ...The projects can have avarying degree of theoretical - applied con-tent.

Familiarity with a scientific programminglanguage (R, PYTHON, JULIA, . . . ) is recom-mended. Knowledge of complex networksis welcome, but not strictly necessary (wecan work around it). Original researchprojects in the area are encouraged.

Data Science Investigation of HistoricalMathematical TablesGiulio Valentino Dalla Riva & ClemencyMontelleThe history of computational algorithms,numerical methods, and data analysis haslong been under-studied in the history

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of mathematics. Indeed, historians havelargely been put o� by the sheer volumeof evidence, the majority of which is in theform of numerical tables. From trigono-metric functions, to instants of syzygies inthe calendar, to subtle corrections of plan-etary positions, these tables of numericaldata, sometimes containing thousands ofdata points, are the direct product of anhistorical author-scientist carrying out analgorithm with the explicit and implicit setof mathematical assumptions that charac-terises their scientific culture of practise.

Classical investigation of these tables hasrelied on the expertise of the historiansin solving complex tasks such as identi-fying the relationships between tables(“was this table computed starting fromthis other table?”). In this project the stu-dent will try to develop neural networkclassifiers, generative models and otherdata science techniques to investigate thetables. Familiarity with a Scientific Pro-gramming language (R, Python, Julia, ...)is requested.

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University of CanterburyTe Whare Wananga o WaitahaPrivate Bag 4800Christchurch 8140New Zealand

Telephone: +64 366 7001Freephone in NZ: 0800 VARSITY (0800 827 748)Facsimile: +64 3 364 2999Email: info@canterbury.ac.nzwww.canterbury.ac.nz