Postgraduate Handbook 2021 - School of Mathematics and Statistics College of Engineering - University of Canterbury
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School of Mathematics and Statistics
College of Engineering
Postgraduate Handbook 2021
21SCHOOL OF MATHEMATICS AND STATISTICS
POSTGRADUATE PROGRAMME 2021
Welcome from our School. Postgraduate other subjects are suitable for inclusion
study enables you to study selected math- within your program of study. In addi-
ematical and statistical topics in depth. tion, there are a various joint programs
There is a change of emphasis from the pre- between our School and other Depart-
ceding undergraduate years: courses tend ments/Schools detailed below.
to be more focused on a specific problem
or class of problems, rather than attempt- At the Masters and PhD level you will un-
ing to give a broad coverage of a branch of dertake research, often focussed on deep
mathematics and statistics. study of a specialised topic. You will learn
skills in undertaking systematic investiga-
There is the possibility of taking one or tions, contextualising your work within
more projects in which you investigate the current state of understanding, so that
some problem with the assistance of a your research outcomes can extend be-
member of staff. Depending on the nature yond the forefront of human knowledge.
of the problem, this may involve literature
searches, the use of various computing
packages (for example MATLAB, Maple or
R), resources on the internet, proving new HONOURS OR PG DIPLOMA?
theorems or data analysis. You will pro-
duce a written dissertation and may give
an oral presentation. The School offers both Honours and Post-
graduate Diploma programs of study,
Any proposed programme of study re- which can be undertaken under Science or
quires the approval of the 400 level coor- Arts. The most appropriate program is best
dinator. It is highly unlikely that any pro- decided on a case-by-case basis which you
posed programme that has a high work- should discuss with the 400 level Coordi-
load in one semester will be approved, so nator, Prof. Michael Plank. The following
you should try to construct a programme guide provides some general advice about
that balances your workload evenly over your options. You are welcome to get in
both semesters. touch as soon as possible, but you must
do so before the formal enrolment period.
You can include some courses from other If you are undertaking honours, you must
subjects (e.g. COSC480 is recommended have arranged a supervisor of your project
for developing programming skills). This in advance of enrolment. The 400 level co-
is a good way to ensure you have a broad ordinator can help you with this.
program of study. Check with the School’s
400 level coordinator that the courses from In addition, we also offer a PostgraduateCertificate in Arts for students interested Prof. Jenni Adams (Physics);
in a 60 points programme of study, or a
Certificate of Proficiency for undertaking a • Computational and Applied Mathe-
course or courses of interest. matical Sciences, see Prof. Michael
Plank;
Who should think about Honours? If • Mathematics and Philosophy, see
you view Mathematics/Statistics as more Prof. Clemency Montelle;
than a means to an end, then doing Hon-
ours will be a year well spent. In addition • Finance and Mathematics, see Assoc.
to taught courses, the honours program Prof. Rua Murray;
has a full year 30 point project which will
not only deepen your understanding of • Finance and Statistics, see Assoc.
a specialised topic but will also develop Prof. Marco Reale; or
many of the soft skills desired by employ- • Financial Engineering, see Assoc.
ers or for further Postgraduate study, like Prof. Marco Reale.
self-motivation, independent learning, re-
search, written and oral communication.
These Honours programs require com-
The Honours subject majors are listed be- pletion of papers totalling 90 points at
low. Formal details are in the UC Calen- 400 level or above (typically six 15 point
dar. To enter Honours in Mathematics, you courses) in addition to the 30 point
will need at least 60 points from MATH301- MATH/STAT/CAMS449 project. In the case
394, plus at least 30 points from 300-level of data science, the project is 45 points.
MATH, STAT or other approved courses. For
Honours in Statistics, you need at least 60
points of 300-level STAT301-394, plus at
least 30 points from 300-level STAT, MATH
BA (Hons) Major Subjects
or other approved courses. Normally you
will have maintained at least a B+ average In the Arts faculty, Honours from our
in these papers. School may be completed in Mathemat-
ics or Statistics. BA (Hons) consists of a
project (MATH/STAT449) and six papers
BSc (Hons) Major Subjects (120 points in total).
Who should think about a Postgraduate
In the Science faculty, Honours from our Diploma? The Postgraduate Diploma can
School may be completed in: consist entirely of taught courses, as there
is no requirement that any project is under-
taken. The entry requirements are as for
• Mathematics and/or Statistics, see Honours, except that you are not required
Prof. Michael Plank; to have a B+ average. It is very strongly
• Data Science; see Prof. Jennifer recommended that your average grade in
Brown; your majoring subject at stage 3 is at least
a C+. The PGDipSc can also be used as Part
• Mathematical Physics, see Assoc. I of a two part research MSc.
2PGDIPSCI AND PGDIPARTS MASTERS IN FINANCIAL
ENGINEERING
In both the Science and Arts faculty the
Postgraduate Diploma can be taken in
mathematics or statistics. In addition a Financial engineering is a cross-
Postgraduate Diploma in data science may disciplinary field combining financial the-
be taken in the Science faculty. These ory, mathematics, statistics and compu-
diplomas require completion of papers to- tational tools to design and develop new
talling 120 points at 400-level or above financial or actuarial products, portfolios
(typically eight 15 point courses). and markets. It also has an important
role to play in the financial industry’s reg-
ulatory framework. Financial engineers
manage financial risk, identify market op-
MASTERS IN APPLIED DATA portunities, design and value financial or
actuarial (insurance) products, and opti-
SCIENCE mize investment strategies.
The year long 180 point program consists:
Data science is a new profession emerg-
ing along with the exponential growth
in size, and availability of ‘big data’. A • 135 points from taught courses.
data scientist provides insight into future There is a core set of required
trends from looking at past and current courses in finance, mathematics &
data. Data science is an essential skill in a statistics and computer science. Fur-
world where everything from education to ther, there are a suite of suggested
commerce, communication to transport, courses from these topic areas, that
involves large scale data collection and dig- make up the majoring subject of
italisation. New Zealand and other coun- Financial Engineering. Depending
tries are currently experiencing a skills on your prior education, it is envis-
shortage in this area, and the need for data aged that around half of the taught
savvy professionals with applied experi- courses will be MATH400 or STAT400
ence is growing. papers and the other half will be
FINC600 papers; and
This 180 point conversion master’s is de-
signed to accommodate students from a • the 45 point paper FENG601 Applica-
range of undergraduate backgrounds (not tion of Financial Engineering which
just those with Mathematics, Statistics and provides the opportunity to apply
Computer Science majors), who want to the techniques learned through the
enhance or build their data science capa- programme to real-world financial
bilities and combine these with the skills engineering problems.
and knowledge they bring from their pre-
vious studies. So long as you are data-
There are minimum entry requirements
hungry and industry-aware; this degree
into the program, which if not met you will
can add to your employability and career
prospects.
3be required to take FIEC601 in January- PHD RESEARCH
February prior to commencement of the
program proper. You will be required to
complete COSC480 Introduction to Pro- The PhD programme is the highest degree
gramming, if you do not have equivalent offered in UC. How do you know if you are
programming skills (e.g. from COSC121, ready to purse a PhD in any of the following
MATH170, EMTH171 or STAT221). Full de- subjects we offer?
tails are provided in the UC Calendar.
• mathematics;
• statistics;
RESEARCH MASTERS
• computational and applied mathe-
matical sciences (CAMS);
A research Masters in Science (MSc) or Arts
(MA) consists of two parts: • mathematical physics; and
• mathematics and philosophy.
• Part I - a 120 points of papers (typi-
cally eight 15 point courses); and
The simplest answer is: if you are passion-
• Part II - a 120 points research thesis. ate about a subject and you want to get a
deeper understanding of a field of study or
want to use sophisticated tools from math-
Students can enter directly to Part II, if they ematical sciences to solve real world prob-
have completed a Postgradute Diploma or lems, then you are ready!
Honours degree in the same majoring sub-
ject. For full details see the UC Calendar. If you want to upscale your knowledge in
the subject you love then a PhD in math-
Our School offers the research MSc and ematics or statistics is the programme for
MA in mathematics or statistics. An MSc you. On the other hand, if you have an in-
is also offered in computational and math- terdisciplinary project in mind then a PhD
ematical sciences and data science. En- in CAMS could be a good option for you.
rolment in a Master’s programme requires
approval from the Postgraduate Coordina- Further details are available from the
tor, Dr Daniel Gerhard. At least one staff PG Office website, including scholar-
member must have agreed to supervise ship information, here: http://www.
your Part II research study before approval canterbury.ac.nz/postgraduate/
of your programme of study. phd-and-doctoral-study/ Excellent
performance in a BSc (Hons) or BA (Hons)
may provide sufficient training to under-
take a PhD, thus obviating the need for a
Masters degree. However, a PGDipSc or
PGDipArts would not normally be suffi-
cient.
4400-600 LEVEL COURSES PROJECTS IN MATHEMATICS
The courses for 2021 are outlined on the Cryptographic Schemes and Diophan-
CIS system https://www.canterbury. tine equations
ac.nz/courseinfo/GetCourseDetails. Felipe Voloch
aspx. In order to see all the offerings for, A Diophantine equation is a polynomial
say, Mathematics, search for MATH4. The equation in many variables where the co-
School reserves the right to cancel any efficients and the solutions are restricted
course that has a low enrolment. This will in some way, e.g., to be integers or rational
be determined at the beginning of each numbers.
semester.
Solving a Diophantine equation is typically
It is also possible (and often desirable) hard but it is easy to build an equation so
to include courses from other subjects, as to have a predetermined solution. This
see the Regulations in the Calender for is an example of a “one-way function” with
details with each degree. Note that any potential applications to cryptography. It
STAT courses may be included in a Math- is essential for some of these applications
ematics degree and vice versa. For multi- that we know that the equation we build
disciplinary programmes like Financial En- does not have additional solutions.
gineering and Data Science (which have
courses across subjects) consult Schedule There are a number of choices to go with
A of the BSc Hons regulation in the Calen- that for an honours project depending
dar for a list of potential courses. on the student’s background and interest.
One can look at the theoretical questions
related to counting solutions to Diophan-
tine equations or solving them. One can
400-LEVEL PROJECTS look at computational methods to solve
these equations. Finally, one can look
at the cryptography side of it, building or
A broad range of possible projects are breaking or implementing these cryptosys-
outlined below. However, this list is tems.
not exhaustive and other possibilities for
projects are certainly possible. Project su-
pervision is by mutual agreement of the su- Designs and their code
pervisor and student. You should arrange Geertrui Van de Voorde
your project by the end of the first week of In this project, we will look at connections
term in 2021. It is suggested that you seek between design theory and coding the-
out possible supervisors before enrolment ory. A famous example is provided by the
week. (sporadic) Golay codes, which correspond
to the Witt designs (and sporadic Mathieu
You will hand in a written report on 6 Octo- groups). More generally, the support of the
ber 2021 which will contribute 80% of the codewords in a perfect codes always form
grade; the remaining 20% will be an oral a t-design. This project can take multiple
presentation in Term 4.
5directions, according to your interests; in- alised polygon is either a di-gon (n = 2),
cluding recent research on rank-metric a projective plane (n = 3), a generalised
equivalents of these classical links. quadrangle (n = 4), a generalised hexagon
(n = 6) or a generalised octagon (n = 8).
This project studies generalised quadran-
Axiomatic planes gles and/or polygons and can take multi-
Geertrui Van de Voorde ple directions, according to your interests.
In the real (Euclidean) plane, we know
that there is exactly one line through two
different points and that there is exactly Computations in group cohomology
one line through a point that is parallel Brendan Creutz
to a given line. Now these two properties In this project you will study properties of
can be taken as axioms and a new class of the group GL2 (Z) of invertible 2 x 2 ma-
planes, called axiomatic affine planes, can trices with integer coefficients and related
be constructed. In particular, it is perfectly groups obtained by reducing the entries
possible to construct such planes that modulo n. The main tool for this will be
have only a finite number of points and group cohomology, which sounds compli-
lines. Probably the most important conjec- cated but can be computed fairly easily by
ture in this area is the question what the hand or using computer algebra software.
possibilities are for the number of points In the project you will learn how to carry
in an axiomatic plane. This project can out such computations and then collect
take multiple directions, according to your data about these groups with the hope of
interests. uncovering interesting patterns which you
could then try to prove theoretically. The
motivation for studying these particular
Generalised quadranges/polygons groups comes from elliptic curves, and
Geertrui Van de Voorde possible follow up projects at postgrad-
The incidence graph of a generalised quad- uate level could explore this connection
rangle is characterized by being a con- further. This project would be suitable for
nected, bipartite graph with diameter four a student with basic knowledge of groups
and girth eight. We can think of them (as seen in MATH240) and finite fields or
as a set of points and lines without trian- modular arithmetic (as seen for example
gles (hence the name). Generalised quad- in MATH220, MATH321 or MATH324).
rangles have their own rich theory, which
dates back to the work of Tits on groups of
Lie type. Data driven analysis of dynamical sys-
tems
In this project, we will study finite gener- Rua Murray
alised quadrangles: we look for construc- Dynamical systems arise as solutions of
tions, classifications and characterisations. differential equations, or in any situation
Generalised quadrangles fall into the more where the state of a system updates it-
general class of generalised polygons (di- eratively with the passage of time steps
ameter n, girth 2n). A classic theorem of (e.g., a descent algorithm for training a
Higman and Feit shows that a finite gener- deep learning network). The local and
6global behaviour of dynamical systems covering, packing, and partitioning will be
is often determined by invariants of var- looked at, as well as network interdiction
ious kinds: fixed points, periodic orbits, problems. Variants of genetic algorithms
invariant manifolds, invariant probability will be looked at, and tested on a selection
distributions. When the system is complex of these applications. MATH303 or similar
(due to very strong nonlinearities and/or required.
high dimension), these objects are hard
to find and analyse. In the last decade, a
new family of tools has developed, loosely Maths Craft
under the umbrella name of “Dynamic Jeanette McLeod (Maths & Stats), Phil
mode decomposition”. These methods Wilson (Maths & Stats), David Pomeroy
use samples from the dynamical system (Teacher Education)
to build approximate transfer operators, Do you love mathematics and craft? Are
from which eigenvectors can be extracted. you interested in the history, sociology, or
The theory behind these methods remains psychology of mathematics? We are of-
undeveloped, there is a plethora of pos- fering multiple interdisciplinary honours
sible computational strategies, and any projects for keen students who answer
dynamical system can be analysed in this yes to those questions. Maths Craft New
way. The emphasis in this project can be Zealand is a non-profit public engagement
tailored to student interest. initiative which makes mathematics ac-
cessible through craft. It was founded and
is run by Drs Jeanette McLeod and Phil
Exploring Links between Topology and Wilson from the School of Mathematics &
Combinatorics Statistics. Since Maths Craft’s inception in
Mike Steel 2016, we have run numerous festivals and
Topological methods turn out to have un- workshops across New Zealand, and have
expected applications in discrete math- reached over 11,000 people, making us the
ematics. One example is the use of the largest maths outreach programme in the
“Ham sandwich theorem” to show that country. Together with Dr David Pomeroy
two thieves can always divide up a neck- from the School of Teacher Education, we
lace with k kinds of jewels using no more are studying the mathematical, historical,
than k cuts. Another example is the link sociological, psychological, and pedagog-
between the Möbius function of a partially ical aspects of our craft-based approach
ordered set and the Euler characteristic to mathematics. If you like to think out-
of an associated topological space. This side the box and can combine a love of
project will suit a student who has taken mathematics and craft with a love of (or
MATH320 and is taking MATH428. enthusiasm to learn about) social research,
then please discuss project options with
us.
Integer Programming
Chris Price
This project looks at various applications
of integer programming, and stochastic so-
lution techniques. Applications such as set
7Population dynamics: Bugs, beetles, late evolutionary history of a set of extant
plants, animals and diseases species is typically represented by a phy-
Alex James logenetic network, a particular type of
Populations, be they of bugs, plants or hu- rooted acyclic directed graph. Although
mans infected with viruses show a remark- reticulation has long been recognised in
able range of behaviours. Find out more evolutionary biology, mathematical in-
about them in a project that uses dynami- vestigations into resolving questions con-
cal system models to try and understand cerning the combinatorial and algorithmic
the dynamics of an example population. properties of phylogenetic networks are
Examples systems include data driven dy- relatively new. Questions include, for ex-
namic models of New Zealand birds, the ample, how hard is it to decide if a given
effect of initial viral load on COVID-19 mor- gene tree is embeddable in a given net-
tality and assessing vaccination strategies work? If we select a network with a certain
for emerging diseases. number of leaves uniformly at random,
how many reticulations do we expect it
to have when the number of leaves is suf-
Lax Pairs ficiently large? When is a network deter-
Mark Hickman mined by the path-length distances be-
tween its leaves? The aim of this project is
Given a non-linear differential equation, a to explore such questions. It involves dis-
Lax pair is a pair of linear differential oper- crete mathematics but there are no formal
ators L, M whose commutator vanishes prerequistes.
only on solutions of the differential equa-
tion. A Lax pair allows one to potentially
solve the differential equation by reducing Tutte’s 5-Flow Conjecture
the problem to an eigenvalue problem (if Charles Semple
the operator L is second order, this is a
Sturm-Liouville problem) and a time evo- Network flow problems are an important
lution of the eigenfunction; the so-called class of problems in combinatorial opti-
inverse scattering method. If L is first or- misation and represent a large variety of
der then the Lax pair gives a conservation real-world occurrences. A particular type
law of the differential equation. In this of network problem gives rise to Tutte’s 5-
project, we will be looking at a method to flow conjecture (1954), amongst the most
compute the Lax pair of prescribed order outstanding conjectures in modern-day
for a differential equation (if it exists). This graph theory. The purpose of this project
will involve Maple and would suit a student is to investigate the progress that has been
who has completed MATH302. made on this conjecture and its connec-
tions to other areas of combinatorics. To
whet your appetite, the Four Colour Theo-
Combinatorics of reticulate evolution rem says that every planar graph without
Charles Semple isthmuses has a 4-flow. While some prior
knowledge of graph theory would be help-
Evolution is not always a treelike process ful, it is not a prerequisite for the project.
because of non-treelike (reticulate) events
such as lateral gene transfer and hybridisa-
tion. In computational biology, the reticu-
8Topics in Group Theory a plane and also continuous maps from
Gunter Steinke the unit interval onto the unit square. This
Groups naturally occur as collections of led to the question of whether or not m-
symmetries of algebraic structures or ge- space Rm and n-space Rn can be topolog-
ometries or algebraic structures or other ically the same for different m and n. To
objects. Knowing the structure of the answer this question various topological
group of symmetries of an object often invariants have been devised.
leads to useful information about the un-
derlying object. Groups come in very dif- Obviously, any useful invariant should as-
ferent sizes and forms and are also fasci- sign n-space dimension n. While it is often
nating in their own right. easy to verify that n-space has dimension
at most n, it is harder to establish equality.
In one project we may look at abstract
groups and how they can be seen as The project investigates some possible
groups of symmetries. By making addi- definitions of the dimension of a (met-
tional assumptions on transitivity we try ric) space, their properties, when these
to determine which groups can arise. For dimensions agree and what examples of
example, the symmetric and alternating topological spaces there are for which they
groups are the only finite groups that are are different.
highly 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 arise
PROJECTS IN STATISTICS
as groups of symmetries of (perfect and
unbounded) crystals. In two dimensions Historical heights of army and navy re-
they are referred to as wallpaper groups. cruits
One looks at the underlying principles that Elena Moltchanova
allow to classify crystallographic groups
The population distribution of human
and carry out a classification, for example,
heights reflects prevailing environmental
of wallpaper groups.
and sociological conditions. The histor-
ical records available, however, always
present a biased picture. For example,
Dimension Theory
only people “tall enough” were enlisted
Gunter Steinke
in the army, and the definition of “tall
While one has a precise notion of dimen- enough” varied with demand for soldiers.
sion for vector spaces, there often is an For the navy, the limitations also applied
intuitive understanding of the dimension to people who were too tall to comfort-
of a space (not necessarily a vector space) ably live on a ship. Using the information
as the number of coordinates or parame- available from the historical army and navy
ters used to describe the space. However, records to recreate the population height
this notion proved to be imprecise as dis- distribution throughout the 17th-20th cen-
coveries in the early 20th century showed. tury thus presents some interesting chal-
There exist bijections between a line and lenges. Among many attempts taken to
9model such data, Bayesian methods are impact the fairness, openness, reliability,
of particular interest due to their flexibil- trust, and social benefit of them. We will in-
ity and ability to easily include temporal vestigate mathematical models and com-
autocorrelation in the model. You will putational techniques to assess how state-
need to have taken a course in Bayesian of-the-art recommender systems perform
inference and have solid knowledge of and, eventually, propose alternative sys-
calculus (deriving conditional, marginal, tems.
etc. distributions) and good programming
skills in either R, PYTHON or C/C++. (Other Familiarity with a scientific programming
projects using Bayesian statistics are avail- language (R, PYTHON, JULIA, . . . ) is rec-
able, such as using reversible jump Markov ommended.
chain Monte Carlo to model the Old Bai-
ley’s data or monitoring epi- demics and
manufacturing processes.) Social network and online communi-
ties analysis
Giulio Valentino Dalla Riva
This is a open ended project. If you are
interested in using and developing math-
PROJECTS IN DATA SCIENCE ematical, statistical, data scientific tools
and notions to analyse the behaviour of
Ethics of data science / Data science for online communities in social networks, we
ethics can discuss it. We are probably going to
Giulio Valentino Dalla Riva use a variety of approaches: data wran-
gling, scraping, anonymisation, networks
Using a variety of mathematical, statisti-
modelling, natural language processing,
cal, computational, ..., approaches, we are
image analysis, ...The projects can have a
going to analyse from an hybrid ethical-
varying degree of theoretical - applied con-
technical point of view some fundamental
tent.
data scientific algorithm. As an example,
think about the “Recommended” videos
Familiarity with a scientific programming
on YouTube, the “watch next” movies on
language (R, PYTHON, JULIA, . . . ) is recom-
Netflix, the “Discovery” songs on Spotify,
mended. Knowledge of complex networks
the “related coverage” news in the New
is welcome, but not strictly necessary (we
York Times: what do they all have in com-
can work around it). Original research
mon? They all suggest you, based on your
projects in the area are encouraged.
history and characteristics, which bit of in-
formation to consider next. They define a
user-dependent priority on the available
Data Science Investigation of Historical
information. They filter information for
Mathematical Tables
you, and they shape the way you see the
Giulio Valentino Dalla Riva & Clemency
world (or, at least, part of it).
Montelle
Recommender systems are ubiquitous ma- The history of computational algorithms,
chine learning algorithms for prioritising numerical methods, and data analysis has
information. Different technical decisions long been under-studied in the history
10of mathematics. Indeed, historians have
largely been put off by the sheer volume
of evidence, the majority of which is in the
form of numerical tables. From trigono-
metric functions, to instants of syzygies in
the calendar, to subtle corrections of plan-
etary positions, these tables of numerical
data, sometimes containing thousands of
data points, are the direct product of an
historical author-scientist carrying out an
algorithm with the explicit and implicit set
of mathematical assumptions that charac-
terises their scientific culture of practise.
Classical investigation of these tables has
relied on the expertise of the historians
in solving complex tasks such as identi-
fying the relationships between tables
(“was this table computed starting from
this other table?”). In this project the stu-
dent will try to develop neural network
classifiers, generative models and other
data science techniques to investigate the
tables. Familiarity with a Scientific Pro-
gramming language (R, Python, Julia, ...)
is requested.
11University of Canterbury Te Whare Wananga o Waitaha Private Bag 4800 Christchurch 8140 New Zealand Telephone: +64 366 7001 Freephone in NZ: 0800 VARSITY (0800 827 748) Facsimile: +64 3 364 2999 Email: info@canterbury.ac.nz www.canterbury.ac.nz
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