Postgraduate Prospectus 2009 Graduate Study in Mathematics CONTENTS Teaching Staff and Graduate Studies Coordinator 2 Introduction 3 Purpose of this Document 3 Student Employment 3 VUW Graduate Awards 3 International Students 3 Enrolling For Graduate Study 3 Honours Degrees 4 The Prerequisite For Honours In Mathematics 4 Workload 4 Spreading The Course Over More Than One Year 4 List of Courses and Recommended Background 5 List of 400-Level Courses 6 Individual Study, Special Topics, and Projects 12 Substitution From Other Subjects 13 Honours in Logic & Computation 13 Graduate Diploma in Science (GDipSc) 13 Research Topics 14 Discrete Mathematics, Algebra, Logic, Theoretical Computer Science 14 Analysis, Topology and Geometry 14 Applied and Numerical Mathematics 14 Research Degrees 15 MSc Part 2, or MA 15 MSc in Stochastic Processes in Finance and Insurance (Parts 1 and 2) 15 PhD 15 General Information 16 Classes of degree 16 Examinations and assessment 16 Postgraduate research supervision 16 Funding 16 Postgraduate scholarships and prizes 16 Official School communications 16 School-provided facilities 17 Postgraduate Students Association 17 Te Rōpu Āwhina Pūtaiao 17 Student Services Group 17 Vic OE (Overseas Exchange for Victoria students) 19 Faculty of Science 21 Faculty of Humanities and Social Sciences 21 1
Transcript
Postgraduate Prospectus 2009 Graduate Study in Mathematics
CONTENTS
Teaching Staff and Graduate Studies Coordinator 2Introduction 3
Purpose of this Document 3Student Employment 3VUW Graduate Awards 3International Students 3Enrolling For Graduate Study 3Honours Degrees 4The Prerequisite For Honours In Mathematics 4Workload 4Spreading The Course Over More Than One Year 4List of Courses and Recommended Background 5List of 400-Level Courses 6Individual Study, Special Topics, and Projects 12Substitution From Other Subjects 13Honours in Logic & Computation 13Graduate Diploma in Science (GDipSc) 13
Research Topics 14Discrete Mathematics, Algebra, Logic, Theoretical Computer Science 14Analysis, Topology and Geometry 14Applied and Numerical Mathematics 14
Research Degrees 15MSc Part 2, or MA 15MSc in Stochastic Processes in Finance and Insurance (Parts 1 and 2) 15PhD 15
General Information 16Classes of degree 16Examinations and assessment 16Postgraduate research supervision 16Funding 16Postgraduate scholarships and prizes 16Official School communications 16School-provided facilities 17Postgraduate Students Association 17Te Rōpu Āwhina Pūtaiao 17Student Services Group 17Vic OE (Overseas Exchange for Victoria students) 19
Faculty of Science 21Faculty of Humanities and Social Sciences 21
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Postgraduate Prospectus 2009
MATHEMATICS
School: School of Mathematics, Statistics and Operations ResearchTe Kura Matai Tatauranga, Rangahau Punaha
Telephone: (04) 463-5341 from NZ, +64-4-463-5341 from overseas Fax: (04) 463-5045 from NZ, +64-4-463-5045 from overseasEmail: [email protected]: www.mcs.vuw.ac.nz/
STAFF ROOM CONTACT
Head of SchoolTo be confirmed
Deputy Head of SchoolTo be confirmed
Graduate Teaching StaffDr Chris Atkin Global Analysis 353 463 6739Dr Colin Bailey Mathematical Logic, General Algebra 354 463 5658Dr George Barmpalias Computability Theory 324 463 6744Dr Peter Donelan Singularity Theory, Invariant Theory, Robotics 441 463 5659Prof Rod Downey Computability, Complexity, Combinatorics, 324 463 5067
AlgebraProf Rob Goldblatt Mathematical Logic, General Algebra 438 463 5660(on leave for 2009)
Dr Noam Greenberg Computability Theory, Set Theory 436 463 6778Dr Byoung Du Kim Number Theory, Arithmetic Geometry 434 463 5665Dr Dillon Mayhew Matroid Theory, Graph theory, Combinatorics 435 463 5155
and ComplexityA/Prof Mark McGuinness Industrial Applied Maths, Modelling 323 463 5059(on leave for 2009)
Dr Hung Le Pham Functional Analysis tbaProf Matt Visser Black Holes, General Relativity, Cosmology 321 463 5115Prof Geoff Whittle Combinatorics, Matroids, Graph Theory 320 463 5650
Email: all staff can be reached at the address [email protected] firstname and lastname are as in the list above.
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.
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MATH Graduate Studies Coordinator for 2009Peter Donelan
INTRODUCTIONPurpose of this DocumentThis 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 EmploymentEach 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 AwardsThe 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 StudentsStudents 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 on-line through this website, or send an email to [email protected]
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 studyStudents 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.
Honours DegreesThe 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 is based on overall performance in the programme as a whole, and is not just a matter of totalling marks or grades awarded at different times. The assessment to be made is of “the candidate's quality of mind and command of the subject displayed over a range of material and tasks appropriate to the limited time specified for the course''.
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 MathematicsThe prerequisite for BA(Hons) or BSc(Hons) in Mathematics is 48 points in approved courses from 300-level MATH.
WorkloadEach 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 YearStudents 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.
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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 CRN Title Recommended WhenCode Background Taught
Enumeration of Patterns and Orders432 7673 Combinatorics 2: Matroids 214 or 314 2/3433 7674 Model Theory 309 or 409 2/3434 7675 Set Theory 2/3435 7676 Computability and Complexity 309 or equivalent 1/3436 7677 Algebra 1: Galois Theory 311 1/3437 7678 Algebra 2: Ideals, Varieties and Algorithms 311 2/3439 13578 Category Theory 311 1/3441 7680 Analysis 1: Measure Theory 312442 7681 Analysis 2: Functional Analysis 441452 591 Topology 1: General Topology 1/3453 593 Topology 2: 207, 311 2/3
Lie Groups, Lie Algebras & Their Representations461 7684 Differential Equations 301 1/3462 7685 Chaotic Dynamics 301 2/3464 10021 Differential Geometry 301 or equivalent 1/3465 10022 General Relativity and Cosmology 464 2/3482 6893 Special Topic: Elliptic Curves 206, 207 1/3483 8795 Special Topic: Knots, Polynomials & Complexity 2/3
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Postgraduate Prospectus 2009
List of 400-Level Courses
MATH 409 CRN 10004 MATHEMATICAL LOGIC 15 POINTS [1/3]
Coordinator: Dr Colin BaileyLectures/Tutorial: Mon, Tue, Thu, Fri 9-10, COLT122Textbook: Burris, Logic for Mathematics and Computer Science
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: 15 POINTS [1/3] ENUMERATION OF PATTERNS AND ORDERS
Coordinator: Dr Colin Bailey
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 15 POINTS [2/3]
Coordinator: Dr Dillon Mayhew
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 GreenbergRecommended 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.
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Graduate Study in Mathematics
MATH 434 CRN 7675 SET THEORY 15 POINTS [2/3]
Coordinator: 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 BarmpaliasRecommended 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 15 POINTS [1/3]
Coordinator: Dr BD KimReference: Ian Stewart, Galois Theory
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 15 POINTS [2/3] ALGORITHMS
Coordinator: Dr Peter DonelanRecommended 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 2009Coordinator: Prof Rob GoldblattUseful references: 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.
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Graduate Study in Mathematics
MATH 441 CRN 7680 ANALYSIS 1: MEASURE THEORY 15 POINTS
Coordinator: Dr Chris AtkinUseful references: Halmos: Measure Theory; Munroe: Introduction to Measure and
Integration; Saks: Theory of the Integral (chapters 1,2)Recommended MATH 312 (but nothing in MATH 312 is absolutely necessary; you background: 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 ANALYSIS 2: FUNCTIONAL ANALYSIS 15 POINTS
Coordinator: Dr Hung Le Pham Recommended background: MATH 441, or some equivalent introduction to integration.Useful reference: Simmons: Introduction to topology and modern analysis
Topics in functional, stochastic, or harmonic analysis, depending on circumstances.
MATH 452 CRN 591 TOPOLOGY 1: GENERAL TOPOLOGY 15 POINTS [1/3]
Coordinator: Dr Chris AtkinReferences: 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.
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Postgraduate Prospectus 2009
MATH 453 CRN 593 TOPOLOGY 2: LIE GROUPS, LIE ALGEBRAS 15 POINTS [2/3] AND THEIR REPRESENTATIONS
ONLY ONE OF MATH 453/MATH 462 OFFERED IN 2009Coordinator: Dr Peter DonelanUseful references: William Fulton and Joe Harris: Representation Theory, a First
Course; Brian Hall: Lie Groups, Lie Algebras, and RepresentationsHossein 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 2009Coordinator: Dr Mark McGuinnessTextbook: 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 2009Coordinator: Dr Peter DonelanTextbook: 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.
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Graduate Study in Mathematics
MATH 464 CRN 10021 DIFFERENTIAL GEOMETRY 15 POINTS [1/3]
Coordinator: Prof Matt VisserUseful references: Bernard Schutz, Geometrical Methods of Mathematical Physics
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: Prof Matt VisserUseful references: 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
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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:
Points CRNMATH 480 Special Topic 30 6891MATH 481 Special Topic 30 6892MATH 482 Special Topic 15 6893 for 1/3; or 9758 for 2/3MATH 483 Special Topic 15 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: Dr Noam GreenbergRecommended MATH 206, 207 are required. Particular 300-level courses are not background: prerequisites, but a level of mathematical maturity is expected.Recommended reading: Notes on the website:
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 pre-Dedekind 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 15 POINTS [2/3]
Coordinator: Professor Geoff WhittleTextbook: D. J. A. Welsh
Complexity, Knots, Colouring and Counting
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.
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 SubjectsUp 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 & ComputationThe 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.
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Postgraduate Prospectus 2009
RESEARCH TOPICSParticular fields for which supervision may be available are as follows.
Discrete Mathematics, Algebra and Number TheoryCurrent 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 GeometryThere 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 MathematicsDr McGuinness has research interests in mathematical modelling with differential equations, with applications in biomathematics, industrial processes, geophysical processes, and two-phase fluid flow in porous media. Dr Visser works in general relativity and quantum field theory, as well as in differential equations and modelling.
Mathematics EducationAssociate 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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Graduate Study in Mathematics
RESEARCH DEGREESMSc Part 2, or MAStudents 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
PhDThe 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
GENERAL INFORMATIONClasses of degreeHonours 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 assessmentPlease 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 supervisionAcademic 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-in-progress seminars organised from time to time by the School.
FundingThe 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, [email protected], for information.
Postgraduate scholarships and prizesStudents should check out the University’s Prizes and Scholarships database, accessible via: www.victoria.ac.nz/home/studying/scholarships_prizes.html
Official School communicationsOfficial 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.
Postgraduate Students AssociationRoom 202, 20 Kelburn Parade, Mon, Wed, Thurs 10am-1pmTel 0-4-463 6973, email [email protected], www.victoria.ac.nz/pgsaThe 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!
Te Rōpu Āwhina PūtaiaoAwhina 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, [email protected]). For further information check the Science Faculty website: www.victoria.ac.nz/science/Awhina
Student Services GroupStudent 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 Service14 Kelburn ParadePhone: 04-463 5896Email: [email protected]: www.victoria.ac.nz/st_services/accommodationIf 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 ParadePhone: 04-463 5393Email: [email protected]: www.victoria.ac.nz/st_services/careersContact 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
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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 Service2 Wai-te-ata RoadPhone: 04-463 5310Email: [email protected] Website: www.victoria.ac.nz/st_services/counsellingCounsellors 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 ECEsPhone: 04-463 5151Email: [email protected]: www.victoria.ac.nz/st_services/crechesThe 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 BuildingPhone: 04-463 6070Email: [email protected]: www.victoria.ac.nz/st_services/disabilityAt 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 Advice14 Kelburn ParadePhone: 04-463 6644 for information, 04-463 7474 for an appointmentEmail: [email protected] Website: www.victoria.ac.nz/st_services/finadviceFinance 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 Service4 Wai-te-ata RoadPhone: 04-463 5308 or 04-463 7474Email: [email protected] Website: www.victoria.ac.nz/st_services/healthThe 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.
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 ProgrammeRoom 109, 14 Kelburn ParadePhone: 04-463 6015Email: [email protected]: www.victoria.ac.nz/st_services/mentoringThis 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 SeiuliRoom 109b, 14 Kelburn ParadePhone: 04-463 5842 or 027-563 5842Email: [email protected]
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 CourtyardPhone: 04-463 5999Email: [email protected]: www.victoria.ac.nz/st_services/slssBuild 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.
EligibilityIf 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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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/exchangeExchange DestinationsExchange 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).
FundingVictoria 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 InformationIn 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
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Graduate Study in Mathematics
FACULTY OF SCIENCETe Wahanga Putaiao
Faculty of Science Student Administration OfficeLocation: Level 1, Cotton BuildingEmail: [email protected]: http://www.victoria.ac.nz/scienceOffice 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:
Johan Barnard Manager, Student and Academic Services tel 04-463 5980Shona de Sain Associate Dean (Students) tel 04-463 5092
Te Ropu Awhina PutaiaoThe 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: [email protected]).Liz is the Deputy Dean (Equity) for the Faculties of Science, Architecture & Design and Engineering.
FACULTY OF HUMANITIES AND SOCIAL SCIENCESTe Wahanga Aronui
Faculty of Humanities and Social Sciences Student Administration OfficeLocation: Level 4, Murphy BuildingPhone: 04-463 5745Email: [email protected]: http://www.victoria.ac.nz/fhssOffice hours: 8.30 am – 5.00 pm (Tuesday from 9.30 am)
