Postgraduate Prospectus 2009
Graduate Study in Mathematics
CONTENTS
Teaching Staff and Graduate Studies Coordinator
Introduction
2
3
Purpose of this Document
Student Employment
VUW Graduate Awards
International Students
Enrolling For Graduate Study
Honours Degrees
The Prerequisite For Honours In Mathematics
Workload
Spreading The Course Over More Than One Year
List of Courses and Recommended Background
List of 400-Level Courses
Individual Study, Special Topics, and Projects
Substitution From Other Subjects
Honours in Logic & Computation
Graduate Diploma in Science (GDipSc)
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3
4
4
4
4
5
6
12
13
13
13
Research Topics
Discrete Mathematics, Algebra, Logic, Theoretical Computer Science
Analysis, Topology and Geometry
Applied and Numerical Mathematics
Research Degrees
MSc Part 2, or MA
MSc in Stochastic Processes in Finance and Insurance (Parts 1 and 2)
PhD
General Information
Classes of degree
Examinations and assessment
Postgraduate research supervision
Funding
Postgraduate scholarships and prizes
Official School communications
School-provided facilities
Postgraduate Students Association
Te Rōpu Āwhina Pūtaiao
Student Services Group
Vic OE (Overseas Exchange for Victoria students)
Faculty of Science
Faculty of Humanities and Social Sciences
14
14
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15
16
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1
Postgraduate Prospectus 2009
MATHEMATICS
School:
Location:
Telephone:
Fax:
Email:
Website:
School of Mathematics, Statistics and Operations Research
Te Kura Matai Tatauranga, Rangahau Punaha
School Office: Cotton Building, Floor 3, Room 358
School Office hours: 8:30am to 5:00pm
Staff Members: Cotton Building, Floors 3 & 4
(04) 463-5341 from NZ, +64-4-463-5341 from overseas
(04) 463-5045 from NZ, +64-4-463-5045 from overseas
office@mcs.vuw.ac.nz
www.mcs.vuw.ac.nz/
STAFF
ROOM
CONTACT
Head of School
To be confirmed
Deputy Head of School
To be confirmed
Graduate Teaching Staff
Dr Chris Atkin
Dr Colin Bailey
Dr George Barmpalias
Dr Peter Donelan
Prof Rod Downey
Prof Rob Goldblatt
(on leave for 2009)
Dr Noam Greenberg
Dr Byoung Du Kim
Dr Dillon Mayhew
A/Prof Mark McGuinness
(on leave for 2009)
Dr Hung Le Pham
Prof Matt Visser
Prof Geoff Whittle
Global Analysis
Mathematical Logic, General Algebra
Computability Theory
Singularity Theory, Invariant Theory, Robotics
Computability, Complexity, Combinatorics,
Algebra
Mathematical Logic, General Algebra
353
354
324
441
324
463 6739
463 5658
463 6744
463 5659
463 5067
438
463 5660
Computability Theory, Set Theory
Number Theory, Arithmetic Geometry
Matroid Theory, Graph theory, Combinatorics
and Complexity
Industrial Applied Maths, Modelling
436
434
435
463 6778
463 5665
463 5155
323
463 5059
Functional Analysis
Black Holes, General Relativity, Cosmology
Combinatorics, Matroids, Graph Theory
tba
321
320
463 5115
463 5650
Email: all staff can be reached at the address
firstname.lastname@vuw.ac.nz
where firstname and lastname are as in the list above.
MATH Graduate Studies Coordinator for 2009
Peter Donelan
Cotton room 441, phone 463-5659
email: peter.donelan@vuw.ac.nz
TIMETABLE
The timetable for 400-level MATH courses in 2009 will be set by email discussion prior to
the start of each trimester. Please ensure the MATH Graduate Coordinator has your
email address for this purpose.
2
Victoria University of Wellington
Graduate Study in Mathematics
INTRODUCTION
Purpose of this Document
This Prospectus gives information about Mathematics courses available through the School
of Mathematics, Statistics and Computer Science at the graduate level (Honours, Masters,
Diploma, PhD), and describes areas of research that offer students opportunities for
advanced study. In particular it gives details of 400-level courses that may be available in
2009. Final decisions about which of these will actually be offered are made towards the end
of 2008, in the light of requests from students.
There is considerable flexibility at this level, and students with special interests may be able
to have a programme tailored to those interests through the use of Directed Individual Study
labels (MATH 440 & 460, see page 12) if staffing is available. Contact the Graduate Studies
Coordinator if you wish to explore this option.
The Graduate Studies Coordinator is responsible for the overall organisation and
administration of graduate studies in Mathematics. All students are asked to keep the
coordinator informed at all times about their current programme of study and any course
changes they make. Anyone enrolling in person in February should make a point of seeing
the coordinator at that time. Students are also encouraged to consult any staff member at
any time about individual courses that interest them.
Student Employment
Each year a number of students are employed as part-time sessional assistant markers for
100 and 200-level courses in Mathematics. Preference is given to graduate students.
Further information may be obtained from Dr Peter Donelan (Cotton Building 441, ext. 5659).
Part-time assistants may also be employed on research and consulting projects.
VUW Graduate Awards
The VUW Graduate Awards are open to students taking Honours or Masters Part 1, and
provide a waiver of tuition fees. VUW also offers scholarships that are open to students
taking Masters Part 2 or a PhD.
Applications for these awards for 2009 close on 1 November 2008. Forms are available at
the Scholarships Office, room 120, Hunter Building. Alternatively, see the webpage:
http://www.victoria.ac.nz/home/studying/scholarships_prizes.html
International Students
Students from overseas are welcome in the School. Victoria International is the University’s
office for international students. It has a website at www.victoria.ac.nz/international that
provides much information on application and immigration formalities, scholarships, NZ living
costs, fees, academic programmes and the like. You can contact Victoria International online through this website, or send an email to victoria-international@vuw.ac.nz
Important note on tuition fees for international students:
International students accepted for the PhD degree will pay the same fee as domestic
students. For other degrees and diplomas. International students from Australia, Germany or
France (countries with which New Zealand has a reciprocal arrangement) pay the same
tuition fee as New Zealand students, but students from other countries pay the full
international student fees. Some thesis students may have their fee paid from a scholarship
or from research grants of their supervisors.
Enrolling for graduate study
Students should enrol online for Honours, MSc Part 1 or Graduate Diploma from 1 October
2008. It is advisable to discuss your intended programme first with the graduate Coordinator.
Enrolment for Masters by thesis or PhD requires a separate application form available from
the University’s website or Faculty offices.
3
Postgraduate Prospectus 2009
Honours Degrees
The Mathematics course for BA(Hons), BSc(Hons), or MSc Part 1, consists of 120 points,
typically made up of eight 15-point courses or the equivalent, in an approved combination,
to be chosen from the list given below, subject to availability.
The Honours degree is intended to be a single course based on a coherent programme of
study and is not merely the aggregation of a specified number of unrelated courses. When
courses are substituted from other subjects, they must be relevant and complementary to the
rest of the course. At most 60 points may be substituted, i.e. at least 60 points must be from
those listed below.
Assessment of the Class of Honours to be awarded
programme as a whole, and is not just a matter of
different times. The assessment to be made is of
command of the subject displayed over a range of
limited time specified for the course''.
is based on overall performance in the
totalling marks or grades awarded at
“the candidate's quality of mind and
material and tasks appropriate to the
Those who do MSc Part 1 normally do MSc Part 2 the following year, and obtain the MSc
degree with a class of Honours. However, the School prefers that students do exactly the
same two years' work in mathematics by obtaining a BSc(Hons) degree in the first year, and
then enrolling in MSc Part 2 to complete an MSc degree. There is no MA Part 1; in
Mathematics, MA has the same status as MSc Part 2.
The Prerequisite For Honours In Mathematics
The prerequisite for BA(Hons) or BSc(Hons) in Mathematics is 48 points in approved courses
from 300-level MATH.
Workload
Each 15-point course will have 30 contact hours, typically run over 10 weeks. It will usually
have a final exam at the end of the course in June or October/November. The standard
workload for such a course is around 10 - 12 hours per week (including lectures) during the
teaching period.
The details of lectures, tutorials, number of assignments, mandatory course requirements
etc. will be provided for each course in a Course Information Sheet given out at the first
lecture.
Spreading The Course Over More Than One Year
Students intending to spread their Honours/Masters Part 1 over more than one year should
obtain prior approval for this from the Graduate Studies Coordinator. The maximum time for
BSc(Hons) is two years, for BA(Hons) four years.
4
Victoria University of Wellington
Graduate Study in Mathematics
List of Courses and Recommended Background
The following list contains courses that are likely to be available in 2009, together with some
that may not be available in 2009, but may be offered in 2010. See the individual course
outlines on the following pages for more details of availability. Each course listed is worth
15 points.
Further labels for Directed Individual Study, Special Topic and Project options are listed
on page 12.
The Recommended Background column lists prior MATH courses that provide desirable
background, but this may not be required in individual cases.
While every effort will be made to ensure that all students are able to take courses meeting
their overall requirements we cannot guarantee that any listed course will be offered. This
depends both on student demand and staff availability. Note that some lecturers are
associated with more than one course, and may have to make a choice.
If the entry in the When Taught column is blank, the choice of trimester is flexible. Otherwise
the course is most likely to be taught in the trimester indicated.
MATH
Code
CRN
Title
Recommended
Background
409
431
10004
7672
432
433
434
435
436
437
439
441
442
452
453
7673
7674
7675
7676
7677
7678
13578
7680
7681
591
593
461
462
464
465
482
483
7684
7685
10021
10022
6893
8795
Mathematical Logic
Combinatorics 1:
311
Enumeration of Patterns and Orders
Combinatorics 2: Matroids
214 or 314
Model Theory
309 or 409
Set Theory
Computability and Complexity
309 or equivalent
Algebra 1: Galois Theory
311
Algebra 2: Ideals, Varieties and Algorithms
311
Category Theory
311
Analysis 1: Measure Theory
312
Analysis 2: Functional Analysis
441
Topology 1: General Topology
Topology 2:
207, 311
Lie Groups, Lie Algebras & Their Representations
Differential Equations
301
Chaotic Dynamics
301
Differential Geometry
301 or equivalent
General Relativity and Cosmology
464
Special Topic: Elliptic Curves
206, 207
Special Topic: Knots, Polynomials & Complexity
When
Taught
1/3
1/3
2/3
2/3
2/3
1/3
1/3
2/3
1/3
1/3
2/3
1/3
2/3
1/3
2/3
1/3
2/3
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Postgraduate Prospectus 2009
List of 400-Level Courses
MATH 409 CRN 10004
MATHEMATICAL LOGIC
15 POINTS
Coordinator:
Lectures/Tutorial:
Textbook:
Dr Colin Bailey
Mon, Tue, Thu, Fri 9-10, COLT122
Burris, Logic for Mathematics and Computer Science
[1/3]
An introduction to the semantics and proof theory of symbolic languages, explaining the role
of logic in describing mathematical structures and formalising reasoning about them. Topics
covered include propositional logic; first-order logic of quantifiers and predicates; the
beginnings of model theory, including completeness and compactness theorems; and an
introduction to the theory of computability, including Turing machines and Gödel's
Incompleteness Theorem for formal arithmetic.
Note: This course is co-taught with MATH 309, and may not be taken by anyone who has
already passed MATH 309.
MATH 431 CRN 7672
COMBINATORICS 1:
ENUMERATION OF PATTERNS AND ORDERS
Coordinator:
Dr Colin Bailey
15 POINTS
[1/3]
Patterns often arise from a group acting on a set and the list of possibilities is easily
recoverable from the cycle indicator polynomial of the group. The course will develop some
of the theory behind such polynomials, some ways to compute them and give a number of
applications to counting graphs of various types. Some knowledge of basic group theory is
essential.
MATH 432 CRN 7673
COMBINATORICS 2: MATROIDS
Coordinator:
Dr Dillon Mayhew
15 POINTS
[2/3]
Matroids were introduced by Whitney in 1933 to axiomatise the combinatorial properties of a
finite set of points in a projective space. Many structures, for example graphs, provide
examples of matroids. In a sense matroid theory plays the same role in combinatorics as that
played by group theory in algebra and topology in analysis. The course is an introduction to
structural matroid theory.
MATH 433 CRN 7674
MODEL THEORY
15 POINTS
[2/3]
Coordinator:
Dr Noam Greenberg
Recommended reading: David Marker: Model Theory: An Introduction
Leonid Libkin: Elements of Finite Model Theory
Model Theory is the study of the interaction between symbolic languages and the
mathematical structures that they describe. The course builds upon ideas first presented in
MATH 309/409 in developing methods of constructing models and applications of the
methods in the context of first-order logic. This includes both the more traditional study of
infinite models, emphasizing set theory and applications to algebra and number systems,
and, subject to interest, the more recent finite model theory with applications to theoretical
computer science.
Some knowledge of algebra and of set theory (MATH 434) is helpful, but not essential.
6
Victoria University of Wellington
Graduate Study in Mathematics
MATH 434
CRN 7675
Coordinator:
SET THEORY
15 POINTS
[2/3]
Dr Colin Bailey
Set theory lies at the foundations of mathematics - all objects of mathematical interest can be
construed as sets. Contemporary set theory explores some of the rich structure of the class
of all sets, and the limitations of the theory. We will consider one small model of set theory as
a base point. And we will consider the method of forcing - used to modify existing models
and so obtain limitative results.
The course uses ideas from MATH 309, but is not a strict continuation of that course. There
is no set text, but many texts on set theory may be found in the library.
MATH 435 CRN 7676
COMPUTABILITY AND COMPLEXITY
15 POINTS
[1/3]
Coordinator:
Dr George Barmpalias
Recommended reading: Carl Smith: A Recursive Introduction to the Theory of Computation,
Garey and Johnson: Computers and Intractability, Hartley Rogers,
Jr: Theory of Recursive Functions and Effective Computability,
R.I. Soare: Recursively Enumerable Sets and Degrees
This is a course about the algorithmic content of mathematics. That is, the part of
mathematics that could be, theoretically at least, performed upon a machine.
It will build on the foundation of MATH 309, although it could be attempted by students with
alternative suitable backgrounds. It is about the underlying mathematics of algorithms and
hence the mathematical ideas behind the discipline of computer science. Structural
complexity and computation are studied at a more advanced level. Some study of the theory
of distributed systems may be included.
MATH 436 CRN 7677
ALGEBRA 1: GALOIS THEORY
Coordinator:
Reference:
Dr BD Kim
Ian Stewart, Galois Theory
15 POINTS
[1/3]
Galois theory is one of the most spectacular mathematical theories. It brings together several
branches of mathematics and creates a powerful machine for the study of some historical
problems, such as solubility of polynomial equations by radicals, and duplication of a cube by
ruler and compass. The most famous application of Galois theory is the proof that the
general quintic equation with rational coefficients cannot be solved by radicals. The main
theorem of Galois theory, the fundamental Galois correspondence, is one of the most
beautiful theorems in all of mathematics.
The course begins by discussing the problem of solutions of polynomial equations, and goes
on to cover field extensions, algebraic and transcendental numbers, Galois groups, the
Galois correspondence, etc. It may include applications to finite fields and to the fundamental
theorem of algebra.
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Postgraduate Prospectus 2009
MATH 437 CRN 7678
ALGEBRA 2: IDEALS, VARIETIES AND
ALGORITHMS
15 POINTS
[2/3]
Coordinator:
Dr Peter Donelan
Recommended reading: Cox, Little and O’Shea, Ideals, Varieties and Algorithms
This is a course in algebraic geometry with special emphasis on algorithms and applications.
Many problems in mathematics and applications require finding solutions of sets of
multivariate polynomials with coefficients in some field k (e.g. finite fields, rational, reals,
complex numbers). This is an algebraic problem concerning rings of polynomials and their
ideals. It also has a geometric aspect in that the set of solutions may be visualised as some
curve, surface or hypersurface in a space (affine or projective). Such sets are called
varieties. There is a rich interaction between the geometric and algebraic objects.
In particular we explore computational methods such as Gröbner bases and resultants. The
associated algorithms are implemented in computer algebra systems such as Maple which
we make use of. We will study the theory of Gröbner bases and you will undertake
investigation of an application such as graph colouring, robot kinematics, integer
programming, coding theory etc.
MATH 439 CRN 13578
CATEGORY THEORY
15 POINTS
[1/3]
NOT OFFERED IN 2009
Coordinator:
Useful references:
Prof Rob Goldblatt
Goldblatt: Topoi: the Categorial Analysis of Logic (free copies of
relevant chapters will be provided in class);
Mac Lane: Categories for the Working Mathematician;
Barr & Wells: Category Theory for Computing Science.
Category theory studies the algebra of functions under the operation of composition, and
develops the viewpoint that most mathematical objects can be defined by the way they
connect to other objects in their external environment through functions, rather than by
referring to their internal set-membership structure.
A category might be a single object, like a group, vector space or topology; or it might be the
whole universe of entities representing an entire branch of mathematics, such as the
category of all vector spaces, representing linear algebra. Categories occur everywhere.
Their study reveals new mathematical concepts, and provides a powerful language that has
become essential for describing many parts of mathematics, as well as playing an important
role in the foundations of logic, computer science, theoretical physics, and other subjects.
8
Victoria University of Wellington
Graduate Study in Mathematics
MATH 441
CRN 7680
Coordinator:
Useful references:
Recommended
background:
ANALYSIS 1: MEASURE THEORY
15 POINTS
Dr Chris Atkin
Halmos: Measure Theory; Munroe: Introduction to Measure and
Integration; Saks: Theory of the Integral (chapters 1,2)
MATH 312 (but nothing in MATH 312 is absolutely necessary; you
must be comfortable with set-theoretic manipulations, and with
series and limits in terms of epsilons and deltas).
Much of modern mathematics, both pure and applied, and ranging from number theory to
quantum mechanics, depends on having a method of integrating functions that applies to
more functions and has better properties than the Riemann integral taught in undergraduate
courses. Such a method was invented by Lebesgue; it depends on the idea of “measure'',
which can be thought of as, in origin, an extension of the concepts of “area'' and “volume'',
but which was subsequently seen to be precisely what is needed to found a rigorous theory
of probability. The course will introduce the definition of measure, construct the most useful
examples of measures, discuss integration with respect to a measure, and relate the theory
to basic ideas in probability theory and functional analysis.
MATH 442
CRN 7681
Coordinator:
Recommended
background:
Useful reference:
ANALYSIS 2: FUNCTIONAL ANALYSIS
15 POINTS
Dr Hung Le Pham
MATH 441, or some equivalent introduction to integration.
Simmons: Introduction to topology and modern analysis
Topics in functional, stochastic, or harmonic analysis, depending on circumstances.
MATH 452
Coordinator:
References:
CRN 591
TOPOLOGY 1: GENERAL TOPOLOGY
15 POINTS
[1/3]
Dr Chris Atkin
James R. Munkres, Topology, 2nd edition, Prentice Hall.
General topology or point-set topology is the study of the general abstract nature of
continuity or “closeness" on spaces. Basic topological notions are ones like continuity,
dimension, compactness and connectedness. General topology deals with differing notions
of continuity and compares them, as well as dealing with their properties. It is the foundation
on which algebraic topology and differential topology stand.
9
Postgraduate Prospectus 2009
MATH 453
CRN 593
TOPOLOGY 2: LIE GROUPS, LIE ALGEBRAS
AND THEIR REPRESENTATIONS
15 POINTS
[2/3]
ONLY ONE OF MATH 453/MATH 462 OFFERED IN 2009
Coordinator:
Useful references:
Dr Peter Donelan
William Fulton and Joe Harris: Representation Theory, a First
Course; Brian Hall: Lie Groups, Lie Algebras, and Representations
Hossein Abbaspour and Martin Moskowitz: Basic Lie Theory
Lie (pronounced "lee") groups are sets of transformations preserving some geometric
structure, such as the set of orthogonal transformations of a Euclidean space which preserve
the Euclidean inner product. They have two compatible structures: they satisfy the axioms of
a group and they also depend continuously on parameters so form a differentiable manifold.
Associated to any Lie group is its Lie algebra of infinitesimal transformations. Lie algebras
can be studied in their own right. One can understand a great deal about a Lie group by
studying its representations - actions of the group on vector spaces.
Lie theory is deeply connected with the development of mathematical physics, through
special relativity and quantum theory. The classification of certain types of Lie groups and
algebras is one of the masterpieces of 20th century mathematics. In this course the concepts
of Lie group and algebra will be introduced together with the key examples, the classical
groups. The ideas underlying their classification will be developed.
MATH 461
CRN 7684
DIFFERENTIAL EQUATIONS
15 POINTS
[1/3]
NOT OFFERED IN 2009
Coordinator:
Textbook:
Dr Mark McGuinness
Bender and Orszag, Advanced Mathematical Methods for
Scientists and Engineers.
Much undergraduate work on differential equations is concerned with exact analytical
solutions. However, differential equations arise in the course of modelling the real world, and
often are not amenable to exact methods. In MATH 461, approximate methods which are
powerful enough to be used on nonlinear problems are studied. Techniques allow
characterization of singular solution behaviour, often necessary before solving numerically.
Some background in complex variables is useful.
MATH 462
CRN 7685
CHAOTIC DYNAMICS
15 POINTS
[2/3]
ONLY ONE OF MATH 453/MATH462 OFFERED IN 2009
Coordinator:
Textbook:
Dr Peter Donelan
Alligood, Sauer and Yorke,
Chaos: an introduction to dynamical systems
A gourmet's sampling from the smorgasbord of delights in chaos and dynamical systems,
from the Cantor set to strange attractors, including the iteration of maps, bifurcation theory,
symbolic dynamics, Smale horseshoes and Poincaré sections.
Dynamical systems model aspects of the real world, either discretely with maps or
continuously with differential equations. We study maps in one and two dimensions and use
their properties to understand systems of differential equations via the idea of Poincaré
sections. As a result we are led from fixed points via periodic points to chaos and fractals.
10
Victoria University of Wellington
Graduate Study in Mathematics
MATH 464 CRN 10021
DIFFERENTIAL GEOMETRY
15 POINTS
Coordinator:
Useful references:
Prof Matt Visser
Bernard Schutz, Geometrical Methods of Mathematical Physics
[1/3]
This course introduces the notation and ideas of modern Differential Geometry that form an
essential background to many fields in Mathematics and Physics. It develops the theory of
manifolds and bundles from a largely intuitive standpoint, and discusses the geometric
notions of metric, connexion, geodesic, curvature and sectional curvature. Extensive notes
are supplied. The course is an essential prerequisite for MATH 465.
Topics include:
 Topological Manifolds and differentiable structure.
 Affine connexion and Curvature: the Riemann tensor.
 Exterior differential forms: generalized Stokes' theorem.
MATH 465 CRN 10022
GENERAL RELATIVITY AND COSMOLOGY
15 POINTS
[2/3]
Coordinator:
Useful references:
Prof Matt Visser
Misner, Thorne, and Wheeler: Gravitation;
James B. Hartle, Gravity: An Introduction to Einstein's General
Relativity;
Sean Carroll: Spacetime and Geometry: An Introduction to General
Relativity.
This course introduces Einstein's general relativity, black holes, gravitational waves, some
idealized models of the universe, and a brief discussion of some extensions to the theory.
Extensive notes are supplied.
Topics include:
 Special relativity: R4 with a Lorentzian metric; the Lorentz group; causal structure
 Lorentzian (pseudo-Riemannian) geometry
 General relativity: the Einstein equivalence principle
 Einstein's equations (vacuum); Schwarzschild solution
 Einstein's equations with matter
 Gravitational waves
 Idealized cosmologies; FLRW universes
11
Postgraduate Prospectus 2009
Individual Study, Special Topics, and Projects
DIRECTED INDIVIDUAL STUDY: MATH 440 OR 460
The Directed Individual Study label can be used to provide a reading course for a particular
student when there is no suitable topic label available. The student follows an individual
programme of study under supervision. Several students may be studying different topics
with different supervisors under the same label.
There is one such 15-point label available for each trimester:
MATH 440: CRN 15207 for 1/3, and 14732 for 3/3.
MATH 460: CRN 15208 for 2/3.
SPECIAL TOPICS: MATH 480-483
The Special Topic label can be used to create 30-point or 15-point courses tailored to
particular interests, or to introduce new topics that may be offered in a particular year. One
Special Topic may involve different subject-matter for different students.
There are 4 labels that can be used, two for 30-point full-year courses, and two for 15-point
one-trimester courses that are each available in both 1/3 and 2/3:
MATH 480
MATH 481
MATH 482
MATH 483
Special Topic
Special Topic
Special Topic
Special Topic
Points
30
30
15
15
CRN
6891
6892
6893 for 1/3; or 9758 for 2/3
6894 for 1/3; or 8795 for 2/3
For 2009 the following Special Topic courses are available:
MATH 482 CRN 6893
SPECIAL TOPIC: ELLIPTIC CURVES
15 POINTS
[1/3]
Coordinator:
Recommended
background:
Recommended reading:
Dr Noam Greenberg
MATH 206, 207 are required. Particular 300-level courses are not
prerequisites, but a level of mathematical maturity is expected.
Notes on the website:
http://homepages.mcs.vuw.ac.nz/~greenberg/math408.pdf
The goal of this course is to explain Abel's isomorphism of the complex torus and the
geometric group structure on an elliptic curve. The course mixes algebraic geometry,
topology and complex analysis; methods are hands-on, computational approaches of preDedekind abstraction, such as the use of resultants and the elimination method to prove the
Nullstellensatz and Bezout's theorem.
MATH 483 CRN 8795
KNOTS, POLYNOMIALS AND COMPLEXITY
Coordinator:
Textbook:
Professor Geoff Whittle
D. J. A. Welsh
Complexity, Knots, Colouring and Counting
15 POINTS
[2/3]
The course is an introduction to knot theory with a focus on computational aspects.
Polynomial invariants of knots such as the Jones polynomial and the Kauffman polynomial
are examined. We consider complexity-theoretic aspects associated with their computation.
We also consider the connection with fundamental polynomial invariants of graphs such as
the Tutte polynomial and the chromatic polynomial. Some applications to graphs and
statistical physics will be discussed.
12
Victoria University of Wellington
Graduate Study in Mathematics
PROJECTS: MATH 488 OR 489
A Project is also a possible part of the Mathematics Honours course. It can be done either as
a 15-point course (MATH 488, CRN 7693) or a 30-point course (MATH 489, CRN 7694),
only one of which can be taken. Students wishing to follow this option should read the
description of staff research interests on page 2, and research areas on page 14, and consult
one of the staff listed there on the possibilities for working on a particular subject with them.
A typical project will result in a written report, forming a readable and self-contained
presentation of a single topic in mathematics that a student has learned about, understood,
and explained in his or her own words. At the end, the project may be presented at a
seminar.
Substitution From Other Subjects
Up to half of a programme for Mathematics with Honours can consist of courses from other
subjects, such as Computer Science, Geophysics, Logic, Philosophy, Physics, Statistics and
Operations Research. COMP, OPRE and STAT graduate courses count as “other subjects''
in a Mathematics programme. Information about them is contained in the following
documents available from the School Office:
 Graduate Study in Computer Science
 Graduate Study in Statistics and Operations Research
Details of courses offered in others areas should be sought from the Schools responsible for
them.
Honours in Logic & Computation
The subject Logic and Computation (LOCO) can be taken for either BSc(Hons) or
BA(Hons), and is open to students who have passed 48 points in approved courses in
Computer Science, Mathematics, or Philosophy. The programme consists of 120 points, 60
of which must be from a specified core list of COMP, MATH, and PHIL courses concerned
with the study and application of aspects of logic, and the remaining 60 from COMP 401-499
and/or MATH 401-499. Details of this programme and its core list are contained in the
separate prospectus
 Graduate Study in Logic and Computation
available from the School Office.
Graduate Diploma in Science (GDipSc)
This is a one-year programme of coursework at 200 level and above for students who
already have a Bachelors degree. It requires at least 48 points of courses at 300 level or
above. The Diploma can optionally be endorsed with a particular subject, chosen from a wide
range, two of which are as follows.
• Mathematics, requiring 30 points of MATH courses at 400 level and a 30-point project
(MATH 889, CRN 672) – see the Graduate Studies Coordinator.
• Modelling with Differential Equations, requiring MATH 461, MATH 462, and the
project MATH 889 – see A/Prof Mark McGuinness.
Other possible subjects for the GDipSc are Logic & Computation, and Mathematics of
Finance and Insurance.
13
Postgraduate Prospectus 2009
RESEARCH TOPICS
Particular fields for which supervision may be available are as follows.
Discrete Mathematics, Algebra and Number Theory
Current staff interests encompass combinatorics, matroid theory, graph theory, universal
algebra and coalgebra, category theory, number theory and arithmetic geometry. Staff
involved include Dr Bailey, Prof Downey, Prof Goldblatt, Dr Kim, Dr Mayhew and Prof
Whittle.
Logic and the Theory of Computation
This covers aspects of mathematical and philosophical logic and theorical computer science,
including model theory, set theory, computability theory, complexity of computation,
algorithmic randomness, algebraic logic and semantics of intentional logics. Staff involved
include Dr Bailey, Prof Downey, Dr Greenberg and Prof Goldblatt. Links are maintained with
Philosophy and Computer Science.
Analysis, Topology and Geometry
There are interests in global analysis (Dr Atkin); singularity theory and algebraic invariant
theory with applications to robotics (Dr Donelan); functional and harmonic analysis (Dr
Pham); and differential geometry (Dr Visser).
Applied and Numerical Mathematics
Dr McGuinness has research interests in mathematical modelling with differential equations,
with applications in biomathematics, industrial processes, geophysical processes, and twophase fluid flow in porous media. Dr Visser works in general relativity and quantum field
theory, as well as in differential equations and modelling.
Mathematics Education
Associate Professor Megan Clark (CO 425, phone 463-6738) supervises research on equity
issues and transitions in mathematics education. Graduate students in this area are required
to have practical experience as well as academic skills.
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Victoria University of Wellington
Graduate Study in Mathematics
RESEARCH DEGREES
MSc Part 2, or MA
Students entering this programme will normally have completed BA(Hons) or BSc(Hons) with
a class of Honours of II(2) or better, or MSc Part 1. The whole course takes one to one and a
half years for full-time students, with extensions pro rata for part-time students.
Two types of programme are available:
1. THESIS. This programme requires the presentation of a thesis (MATH 591, CRN 667). It
may also include one or both of the 15-point special topic courses MATH 548 (CRN 649)
and MATH 549 (CRN 651). Entry requires approval of the Graduate Studies
Coordinator, and depends on an initial agreement on a programme of study, supervisor,
and a provisional thesis topic. The thesis is normally an exposition of a piece of
mathematical work and may contain new results or may represent a study of known
material from a fresh point of view, together with some review of the literature. The
thesis is prepared under the direct supervision of a staff member.
The thesis is worth between 75% and 100% of the total programme, depending on
whether one, both or neither of MATH 548, 549 is included.
2.
ADVANCED COURSE OF STUDY: MATH 592 (CRN 5007). This involves a
combination of reading and lecture courses, together with a research project worth
between 25% and 50% of the total programme. Each student's programme will be
individually designed with the prospective supervisor, and will require the approval of the
Graduate Studies Coordinator from the outset. It may include 400-level courses not
previously taken. There is a wide range of graduate texts that are suitable for reading
courses in different areas, and selection should be made by consultation with the
appropriate members of the teaching staff. The research project will involve the
presentation of a written report that is smaller than a thesis. The Advanced Course of
Study results in the award of a single overall grade.
MSc in Stochastic Processes in Finance and Insurance (Parts 1 and 2)
Students interested in this option should contact Professor Estate Khmaladze, and see the
Graduate Study in Statistics and Operations Research Prospectus for more details.
For more information on Masters degrees, see the Faculty of Science Masters Handbook:
http://www.victoria.ac.nz/science/degrees/postgraduate/index.aspx
PhD
The degree of PhD is awarded for a thesis (MATH 690, CRN 670) which demonstrates the
candidate's ability to carry out independent research which makes a significant contribution
to the knowledge or understanding of a field of study. A candidate for the degree pursues a
course of advanced study and research at the University under the immediate direction of a
supervisor, or supervisors. The study is usually full time, and is for a period of at least two
calendar years (and not more than five years unless special permission is obtained) from the
date of registration. Local students will normally have completed a Masters degree before
entering the PhD programme, but entry direct from an Honours degree is possible.
The University has a PhD Handbook covering these and many other matters: you can
download a copy from
www.victoria.ac.nz/home/publications/phd_handbook.pdf
15
Postgraduate Prospectus 2009
GENERAL INFORMATION
Classes of degree
Honours degrees are awarded with first, upper second, lower second, or third class honours.
Lecturers may assign provisional grades for individual pieces of work during the year. In
addition to posting a final class of honours at the end of the year, letter grades (A+, A, A-, B+,
B, B-, C+, C, D, E) will be posted for particular courses. “A” grades correspond to first class
work; “B+” and higher “B” to upper second class work; lower “B” and “B-“ to lower second
class work; and “C” to third class work.
Candidates should be aware that the award of a class honours is based upon overall
assessment of the calibre of work done across all the courses taken. The final assessment is
arrived at by the School in consultation with external examiners.
Those who take MSc Parts 1 and 2 will receive a class of Honours. Candidates taking MSc
Part 2 only (thesis) may be eligible for Distinction or Merit (see VUW Calendar, Personal
Courses of Study Statute Part 2).
Examinations and assessment
Please note that students enrolled in courses that have a final examination are expected to
be available in the relevant examination period. In 2009 these are: 12 June – 1 July and 23
October – 14 November. Exam timetables are normally published after the mid-term break.
Postgraduate research supervision
Academic Board requires all supervisors to provide 6-monthly written reports on students
enrolled in Masters by thesis and PhD courses. These reports are expected to identify what
has been achieved, outline agreed timetables for future work and identify any problems with
a student’s performance that require to be rectified. Copies of the formal written reports are
provided to the student, the School’s postgraduate co-ordinator and relevant Student
Administration Advisers.
Theses are prepared and written in close consultation with a staff member who acts as
supervisor. Research students are expected to participate in and contribute to research-inprogress seminars organised from time to time by the School.
Funding
The Research Funding Guide is published by the University’s Research Policy Office and is
available on the University website at
www.victoria.ac.nz/home/publications/research_funding_guide.pdf
The Postgraduate Students Association has information on StudyLink funding. Faculty
Research Grants may also be available, contact Keith Willett, Faculty of Science, tel 04-463
5508, keith.willett@vuw.ac.nz, for information.
Postgraduate scholarships and prizes
Students should check out the University’s Prizes and Scholarships database, accessible
via: www.victoria.ac.nz/home/studying/scholarships_prizes.html
Official School communications
Official notices of the School are posted on noticeboards in level 2 and 3 corridors of the
Cotton building. Each course will have a specific web presence which may be used for
advising of announcements, check http://www.mcs.vuw.ac.nz/courses/ for a list of all the
courses offered by the School. You may also be communicated with via your MSCS e-mail
account or via a course specific forum.
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Victoria University of Wellington
Graduate Study in Mathematics
School-provided facilities

Office facilities (e.g., room, furniture, fax, phone, photocopier),

computing facilities (PC, software, internet/e-mail access, MSCS services),

printing facilities,

tea/coffee facilities, kitchen facilities and common room availability
Postgraduate Students Association
Room 202, 20 Kelburn Parade, Mon, Wed, Thurs 10am-1pm
Tel 0-4-463 6973, email pgsa-ea@vuw.ac.nz, www.victoria.ac.nz/pgsa
The Victoria Postgraduate Students’ Association (PGSA) is recognised by VUWSA as the
representative body for all postgraduate students at Victoria University. The PGSA provides
representation and other services for all Victoria’s approximately 3,500 postgraduate
students. Services include advice, advocacy for individuals and groups of postgraduate
students, lobbying on issues important to you, representation on a variety of university
committees, social activities, seminars, training workshops and information.
In addition the PGSA organises Victoria’s teaching awards (the Victorias), and publishes a
postgraduate journal Third Degree. The Association is always eager for postgraduates to get
involved, and there are always things for people to do!
Subscribe to the PGSA email list by emailing pgsa-members-subscribe@vuw.ac.nz
Te Rōpu Āwhina Pūtaiao
Awhina is the comprehensive whanau mentoring programme for Maori and Pacific Nations
Science, Architecture, Design and Engineering students enrolled in 100- and 200-level
courses. Awhina also supports non Maori and Pacific Nations students who wish to be
included. Graduate and postgraduate Maori and Pacific Nations students (and other
interested students) are encouraged to become involved as mentors to these students and
help them get their university studies off to a good start. If you are interested in becoming a
mentor please contact Liz Richardson, Deputy Dean (Equity) for the Faculties of Science,
Architecture & Design and Engineering (463 5748, liz.richardson@vuw.ac.nz). For further
information check the Science Faculty website: www.victoria.ac.nz/science/Awhina
Student Services Group
Student Services provides a range of services to ALL students to help you make the most of
your time at university. Contact the following services for assistance directly or visit the
website www.victoria.ac.nz/st_services/ to find out more. Many of these services are
available at all campuses – the location details of the services on the Kelburn campus are
listed here and the main phone number. Make contact to choose which available location
best suits you.
Accommodation Service
14 Kelburn Parade
Phone: 04-463 5896
Email:
accommodation@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/accommodation
If you need a flat, flatmates or Hall of Residence information, the Accommodation Service is
a great place to start. The website has an online letting service with a range of vacancy
listings to suit all budgets and tastes and staff are happy to advise you on tenancy issues.
Career Development and Employment (Vic Careers)
14 Kelburn Parade
Phone: 04-463 5393
Email:
careers-service@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/careers
Contact Vic Careers if you need some independent career advice to help you find the right
career, to find out what sorts of careers are available with your degree or to get some help in
17
Postgraduate Prospectus 2009
writing a CV. If you are looking for a job watch out for Graduate Recruitment programmes
and check-out Victoria CareerHub (careerhub.victoria.ac.nz) your 24/7 jobs online service
– logon using your ITS account.
Counselling Service
2 Wai-te-ata Road
Phone: 04-463 5310
Email:
counselling-service@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/counselling
Counsellors are available to discuss personal and academic issues that affect your general
sense of wellbeing, your relationships or your learning. Ring to make an appointment for this
free, confidential service.
Crèches and ECEs
Phone: 04-463 5151
Email:
childcare@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/creches
The University crèches can provide your child/children with the best possible education and
care while you study. The Student Crèche has three centres on Kelburn Campus and one at
the Law School, Pipitea Campus.
Disability Support Services (DSS)
Level 1, Robert Stout Building
Phone: 04-463 6070
Email:
disability@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/disability
At Victoria, disability is self-defined and includes people with permanent, temporary or
recurring impairments, injuries or chronic medical conditions. Contact DSS’s Student
Advisers to confidentially discuss your individual needs.
Financial Support and Advice
14 Kelburn Parade
Phone: 04-463 6644 for information, 04-463 7474 for an appointment
Email:
student-hardship@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/finadvice
Finance Advisers can provide you with practical advice on budgeting and coping financially,
help you with Student Loan and Allowance applications and the preparation of financial
statements for Scholarship applications. Through the Hardship Fund they are also able to
provide emergency financial assistance if you are facing hardship.
Health Service
4 Wai-te-ata Road
Phone: 04-463 5308 or 04-463 7474
Email:
student-health@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/health
The Health Service offers you a general practice medical service on campus which is free or
very low cost for most students. It deals with illnesses, accidents and prescriptions, and
offers specialist services such as psychiatry, nutrition, dermatology and physiotherapy.
Kaiwawao Mäori / Mäori Student Services Adviser
Level 0, Kirk Wing, Hunter Courtyard
Phone: 04-463 6001
Email:
kaiwawao-maori@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/kaiwawao
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Victoria University of Wellington
Graduate Study in Mathematics
The Kaiwawao Mäori’s main objective is to encourage and assist students to participate and
succeed by providing support to all students of Mäori descent. If you have questions,
concerns or are unsure about whom to talk with / where to go, the Kaiwawao Mäori can help.
Manaaki Pihipihinga Programme
Room 109, 14 Kelburn Parade
Phone: 04-463 6015
Email:
manaaki-pihipihinga-programme@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/mentoring
This mentoring programme is for all Mäori and Pacific Nations students in the Faculties of
Commerce and Administration, and Humanities and Social Sciences. Mentors are successful
senior students who can assist you with course-related tasks.
Pacific Support Coordinator
Fa’afo’i Seiuli
Room 109b, 14 Kelburn Parade
Phone: 04-463 5842 or 027-563 5842
Email:
faafoi.seiuli@vuw.ac.nz
The Pacific Support Coordinator assists with the transition of Pacific students into university
life as well as helping them cope with academic studies – by making appointments with
services on a student's behalf, taking students to services that will help and by providing
information on scholarships.
Student Learning Support Service (SLSS)
Level 0, Kirk Wing, Hunter Courtyard
Phone: 04-463 5999
Email:
student-learning@vuw.ac.nz
Website: www.victoria.ac.nz/st_services/slss
Build confidence and maximise your academic success with support from SLSS. They offer
workshops and one to one tuition in such areas as essay writing, maths and stats, learning
strategies, study skills, and language skills.
SLSS offers regular seminars on topics of interest to postgraduate students, which have
included Writing a Research Proposal, Writing a Literature Review, Managing the Research
Process, What Makes a Good Argument, and Editing your Thesis.
Student Learning Support facilitates postgraduate writing workshops, helps set up and
maintain peer study/support groups and organises other workshops on request. Some
individual assistance is also available.
Vic OE (Overseas Exchange for Victoria students)
As a Victoria University student you have the chance to complete part of your degree at a
world-class institution overseas and studying towards your Victoria degree while paying
domestic fees.
Vic OE students are eligible for StudyLink loans and allowances. Victoria International will
provide some grant funding to all successful applicants.
Eligibility
If you are interested in applying for the Vic OE you must:

be enrolled as a full-time student at Victoria University of Wellington (at the time of
application)

have completed a year of full-time study by the date of your intended departure

have achieved a “B” average overall in your studies at Victoria

be able to demonstrate that you would be a good ambassador for Victoria.
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Postgraduate Prospectus 2009
Application Deadlines

16 January 2009 (for study in Trimester 2, 2009)

16 July 2009 (for study in Trimester 1, 2010)
For more information visit the website www.victoria.ac.nz/exchange
Exchange Destinations
Exchange agreements are in place between Victoria University and universities throughout
the UK and Europe (eg. University of Leeds, Royal Holloway and Université de Lyon III), Asia
(eg. Korea University, National University of Singapore), North America (eg; UCBerkeley,
Penn State, Dalhousie and UVic), and South America (Universidad de Chile, la Catolica de
Argentina and Universidad de Valparaiso).
Funding
Victoria International provides each successful Vic OE student with grant funding of $1000.
Students are also eligible for full Study Link student loans and allowances (if normally eligible
in New Zealand).
For more information contact the Victoria International office on Level 2, Rutherford House,
23 Lambton Quay.
More Information
In addition to the statutes published online at www.victoria.ac.nz/home/about_victoria/
policy.html and in the University Calendar students are advised to familiarise themselves
with the following publications available from the Faculty or School office or online:

PhD Handbook (www.victoria.ac.nz/home/publications/phd_handbook.pdf)

Human Ethics Committee Guidelines
20
Victoria University of Wellington
Graduate Study in Mathematics
FACULTY OF SCIENCE
Te Wahanga Putaiao
Faculty of Science Student Administration Office
Location:
Level 1, Cotton Building
Email:
science-faculty@vuw.ac.nz
Web:
http://www.victoria.ac.nz/science
Office hours: 8.30 am – 5.00 pm (Tuesday from 9.30 am)
Student Advisers can help with admission requirements, degree planning, changing
courses, transfer of credit from other tertiary institutions, and anything else that may crop up
during your time at Vic. They also deal with other aspects of student administration such as
enrolment, exams organisation and the maintenance of student records.
The advisers support students throughout their study. To ensure you get good continuity of
personal service, advisers manage a particular group of students, identified by the first letter
of your surname:
A-F
G-L
M-S
T-Z
Elisha Connell
Gina Mullis
Rachel Zhang
Celia Simpson
Johan Barnard
Shona de Sain
elisha.connell@vuw.ac.nz
gina.mullis@vuw.ac.nz
rachel.zhang@vuw.ac.nz
celia.simpson@vuw.ac.nz
Manager, Student and Academic Services
Associate Dean (Students)
463 5983
463 5982
463 7473
463 5981
tel 04-463 5980
tel 04-463 5092
Te Ropu Awhina Putaiao
The Faculty of Science provides support for Maori and Pacific students through Te Ropu
Awhina Putaiao. Senior Maori and Pacific students act as mentors and provide support,
especially for first-year science students. Students use the fully resourced whanau room in
the Cotton Building (CO 145A) for individual or group study, tutorials, meeting with mentors
and networking.
To find out more, contact
Liz Richardson (tel 04-463 5748, email: liz.richardson@vuw.ac.nz).
Liz is the Deputy Dean (Equity) for the Faculties of Science, Architecture & Design and
Engineering.
FACULTY OF HUMANITIES AND SOCIAL SCIENCES
Te Wahanga Aronui
Faculty of Humanities and Social Sciences Student Administration Office
Location:
Level 4, Murphy Building
Phone:
04-463 5745
Email:
fhss-student-admin@vuw.ac.nz
Web:
http://www.victoria.ac.nz/fhss
Office hours: 8.30 am – 5.00 pm (Tuesday from 9.30 am)
Student Advisers for BA(Hons)/MA students:
A-F
G-L
M-R
S-Z
Scott Webber
Michelle Butters
Janice Ikuia
Hillary Reid
Dr Kristina McGuiness-King
Dr Stuart Brock
scott.webber@vuw.ac.nz
michelle.butters@vuw.ac.nz
janice.ikuia@vuw.ac.nz
hillary.reid@vuw.ac.nz
Faculty Administration Manager
Associate Dean (Students)
463 5739
463 5740
463 5167
463 5742
tel 04-463 5192
tel 04-463 6970
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