Printable course descriptions 2008
From MathsDept
2008 Calendar Course Descriptions
MATHS 91F
Foundation Mathematics 1 This first Mathematics course for students enrolled in the Tertiary Foundation Certificate programme aims to promote an understanding of number skills, including an introduction to algebra. Students will learn how to use simple technology and develop their problem solving abilities.
MATHS 92F
Foundation Mathematics 2
This second Mathematics course for students enrolled in the Tertiary Foundation Certificate programme aims to use the skills learnt in MATHS 91F to develop an understanding of functions in their tabular, algebraic and graphical representations. This course prepares students for either MATHS 101 or 102.
MATHS 93F
Foundation Mathematics 3 This first Extension Mathematics course for students enrolled in the Tertiary Foundation Certificate programme aims to promote an understanding of numerical and algebraic skills at a deeper level than MATHS 91F. Students will learn how to use simple technology and develop their problem solving abilities.
MATHS 94F
Foundation Mathematics 4
This second Extension Mathematics course for students enrolled in the Tertiary Foundation Certificate programme aims to use the skills learnt in MATHS 93F to develop an understanding of functions, including differential functions, in their tabular, algebraic and graphical representations. This course prepares students for success in either MATHS 101 or 102.
MATHS 101/MATHS 101G
Mathematics in Society Students will encounter the role mathematics plays in understanding and guiding human activity. The teaching is thematic and students experience how fundamental mathematical ideas occur in modelling diverse features of our society such as the environment (eg, air pollution) and medicine (eg, burns, drug dosages).
Recommended preparation: For students who have not studied mathematics at NCEA Level 3 (or equivalent) or have no formal mathematical background.
MATHS 102
Functioning in Mathematics This introduction to calculus focuses on the development of mathematical skills and concepts leading up to calculus, through active participation in problems using functions to model real life contexts. Prepares students for further study, for instance, MATHS 108, MATHS 150.
Recommended preparation: For students who have achieved fewer than 12 credits in Calculus or Statistics at NCEA Level 3, or who have achieved at least 18 credits in Mathematics at NCEA Level 2 (or equivalent).
MATHS 108
General Mathematics 1 The main gateway to mathematics for students, including those taking this subject as part of other majors. Selected topics in algebra and calculus and their applications including: sets, real numbers, integers; linear algebra including matrices, linear functions, linear equations; functions, equations and inequalities; limits and continuity; differential calculus of one and two variables; integral calculus of one variable. These are studied in general settings using applications from science, commerce and information systems.
Prerequisites: MATHS 102 or at least 12 credits in NCEA Level 3 Calculus or at least 18 credits in NCEA level 3 Statistics (or equivalent).
MATHS 150
Advancing Mathematics 1 The main pathway for well prepared students who are planning further courses in the mathematical sciences, including mathematics, statistics, computer science, physics, economics, or finance. It gives an introduction to the use of careful mathematical language and reasoning in the context of the calculus and linear algebra. MATHS 150 is a lot of fun, and is the recommended preparation for MATHS 250.
Prerequisites: B+ in MATHS 102, or at least 18 credits in Calculus at NCEA Level 3, including at least 6 credits at merit or excellence (or equivalent).
MATHS 153
Accelerated Mathematics
A version of MATHS 150 for high achieving Year 13 students. Enrolment, restricted to students still at high school, requires the consent of the Department.
Prerequisites: See course website.
MATHS 162
Modelling and Computation In this introduction to mathematical modelling and scientific computing, students will learn how to formulate mathematical models and how to solve them using numerical and other methods. This is a core course for students who wish to advance in Applied Mathematics.
Corerequisites: Concurrent enrolment in MATHS 108 or MATHS 150
MATHS 190
Great Ideas Shaping Our World
Mathematics contains many powerful and beautiful ideas that have shaped the way we understand our world. This course explores some of the grand successes of mathematical thinking. No formal mathematics background is required, just curiosity about topics such as infinity, paradoxes, cryptography, knots and fractals.
Prerequisites: n/a
MATHS 202
Tutoring in Mathematics This is a mainly practical course in which selected students learn tutoring skills that are put to use in MATHS 102 tutorials. In a small interactive class, students learn to mark, to question strategically and to facilitate learning. The theory and issues of mathematics education as a research field is also introduced.
Selection: Students are selected for the course on the basis of their academic record, and the results of a personal interview with the lecturer to determine their interest in the course and their communication skills.
Prerequisites: 30 points from courses in Mathematics and Departmental consent required.
MATHS 208
General Mathematics 2 This sequel to MATHS 108 features applications from the theory of multi-variable calculus, linear algebra and differential equations to real-life problems in statistics, economics, finance, computer science, and operations research. Matlab is used to develop analytical and numerical methods of solving problems.
Prerequisite: 15 points from ENGSCI 111, PHYSICS 111, MATHS 108, 130, 150, 151, 153
MATHS 250
Advancing Mathematics 2 This preparation for advanced courses in mathematics is intended for all students who plan to progress further in mathematics. Covers material from multivariable calculus and linear algebra that has many applications in science, engineering and commerce. The emphasis is on both the results and the ideas underpinning these.
Prerequisites: 15 points from ENGSCI 111, MATHS 150, 153, 208, PHYSICS 111, or a B+ pass in MATHS 108
MATHS 253
Advancing Mathematics 3 The standard sequel to MATHS 250. It covers topics in linear algebra and multi-variable calculus. It is a foundation for a large number of Stage III courses in mathematics and statistics, and for many advanced courses in physics and other applied sciences. All students intending to advance in mathematics should take this course.
Prerequisites: 15 points from MATHS 152, 250, PHYSICS 112, 210, or an A- pass in MATHS 208
COMPSCI 225
An introduction to logic, principles of counting, mathematical induction, recursion, relations and functions, graphs and trees, and algorithms. This course is suited to students who are interested in the foundations of computer science, mathematics and logic.
MATHS 255
Principles of Mathematics An introduction to mathematical thinking and proofs. The emphasis of the course is that it is not only important to find the right answer to a problem but also to be able to convince others (and ourselves) that the answer is correct. MATH255 is an essential course for all students advancing in pure mathematics and many areas of applied mathematics and statistics.
Prerequisite: 15 points from MATHS 152, 250, PHYSICS 112, 210, or an A- pass in MATHS 208
MATHS 260
Differential Equations The study of differential equations is central to mathematical modelling of systems that change. This course develops methods for understanding the behaviour of solutions to ordinary differential equations. Qualitative and elementary numerical methods for obtaining information about solutions are discussed, as well as some analytical techniques for finding exact solutions in certain cases. Some applications of differential equations to scientific modelling are discussed. A core course for applied mathematics. ==Prerequisite:
Prerequisites: 15 points from MATHS 150, 153, 208, 250, PHYSICS 111, or at least an A- in MATHS 108
MATHS 270
Numerical Computation Many mathematical models occurring in Science and Engineering cannot be solved exactly using algebra and calculus. This course introduces students to computer-based methods that can be used to find approximate solutions to these problems. The methods covered in the course are powerful yet easy to learn. This is a core course for students who wish to advance in Applied Mathematics.
Prerequisite: MATHS 108 or 150 or equivalent, and a computing course such as COMPSCI 101 or MATHS 162 or equivalent
MATHS 302
Teaching and Learning Mathematics For people interested in thinking about the social, cultural, political, economic, historical, technological and theoretical ideas that influence mathematics education. Students who want to understand the forces that shaped their own mathematics education or who are interested in teaching will enjoy this course. Students will develop the ability to communicate ideas in essay form.
Recommended preparation: at least 45 points from courses in Mathematics or Statistics.
MATHS 315
Mathematical Logic Logic addresses the foundations of mathematical reasoning. It models the process of mathematical proof by providing a setting and the rules of deduction. Builds a basic understanding of first order predicate logic, it introduces model theory and demonstrates how models of a first order system relate to mathematical structures. The course is vital for anyone studying high level computer science or mathematical logic.
Prerequisites: COMPSCI 225 or MATHS 255 or PHIL 222
MATHS 320
Algebraic Structures This is a framework for a unified treatment of many different mathematical structures. It concentrates on the fundamental notions of groups, rings and fields. The abstract descriptions are accompanied by numerous concrete examples. Applications abound:symmetries, geometry, coding theory, cryptography and many more. This is a vital course for those planning graduate study in pure mathematics.
Prerequisites: MATHS 255 or 328, or an A- pass in 253
MATHS 326
Combinatorial Computing Combinatorics is a branch of mathematics that studies collections of objects that satisfy specified criteria. An important part of combinatorics is graph theory, which is now connected to other disciplines including bioinformatics, electrical engineering, molecular chemistry and social science. The use of combinatorics in solving counting and construction problems is covered using topics that include algorithmic graph theory, codes and incidence structures, and combinatorial complexity.
Prerequisites: COMPSCI 225 or MATHS 255
MATHS 328
Algebra and Applications The goal of this course is to show the power of algebra and number theory in the real world. It concentrates on concrete objects like polynomial rings, finite fields, groups of points on elliptic curves, studies their elementary properties and shows their exceptional applicability to various problems in information technology including cryptography, secret sharing, and reliable transmission of information through an unreliable channel.
Prerequisites: MATHS 255, or B+ pass in COMPSCI 225 and one of MATHS 208, 250, 253
MATHS 332
Real Analysis A standard course for every student intending to advance in pure mathematics. It develops the foundational mathematics underlying calculus, it introduces a rigorous approach to continuous mathematics and fosters an understanding of the special thinking and arguments involved in this area. The main focus is analysis in one real variable with the topics including real fields, limits and continuity, Riemann integration and power series.
Prerequisites: MATHS 253 and 255, or 253 and a B+ in MATHS 260
MATHS 333
Analysis in Higher Dimensions By selecting the important properties of distance many different mathematical contexts are studied simultaneously in the framework of metric and normed spaces. Examines carefully the ways in which the derivative generalises to higher dimensional situations. These concepts lead to precise studies of continuity, fixed points and the solution of differential equations. A recommended course for all students planning to advance in pure mathematics.
Prerequisites: MATHS 332
Strongly recommended preparation: MATHS 253 and 255
MATHS 340
Real and Complex Calculus Calculus plays a fundamental role in mathematics, answering deep theoretical problems and allowing us to solve very practical problems. Extends the ideas of calculus to two and higher dimensions, showing how to calculate integrals and derivatives in higher dimensions and exploring special relationships between integrals of different dimensions. It also extends calculus to complex variables.
Prerequisites: MATHS 253
MATHS 361
Partial Differential Equations Partial differential equations are used to model many important phenomena in the real world (such as heat flow and wave motion). An introductory course on methods of solution for linear partial differential equations in one, two and three dimensions.
Prerequisites: MATHS 260 and 253 or equivalent, or PHYSICS 211
MATHS 362
Methods in Applied Mathematics Techniques such as variational methods, Green's functions, and perturbation analysis are a crucial part of the applied mathematician's toolbox. This course will cover a selection of such advanced topics in detail, and is suitable for those students intending to advance in applied mathematics or physics.
Prerequisites: Either MATHS 260 and 253 or equivalent, or PHYSICS 211
MATHS 363
Advanced Modelling and Computation Much of modern research in applied mathematics, physics and engineering relies heavily on the construction and numerical solution of mathematical models. This course covers the theory and practice of such computational approaches, including the study of numerical linear algebra and differential equations, and bifurcations in ordinary differential equations. Matlab is used extensively.
MATHS 701
Research Skills in Mathematics Education Prepares students for postgraduate study in mathematics and statistics education. Its emphasis is on workshops in the key research skills required by students working at this level. It will cover a range of research issues and techniques.
MATHS 702
Mathematics Curriculum Considers such issues as the historical development of mathematics and statistics curricula, current New Zealand and international trends, the relationship between curriculum and assessment, and the politics of curriculum development.
MATHS 703
Theoretical Issues in Mathematics Education An analysis of theoretical perspectives that inform research in mathematics education, with a focus on learning theories, both social and psychological, and their implications for teaching and learning in mathematics.
MATHS 705
Social Issues in Mathematics Education An examination of cultural, social, and language issues that arise in mathematics and statistics education. It will cover literature and theory on topics including language and learning, gender and equity issues, and cultural aspects of mathematics and statistics learning.
MATHS 706
Technology and Mathematics Education The use of computers and calculators in mathematics education, with a focus on both theoretical and practical aspects of the use of computers in the mathematics classroom. The pedagogical implications of computers for the present and the future of mathematics education are discussed.
MATHS 712
Mathematics and Learning Recent theoretical perspectives on the teaching and learning of school and university mathematics are linked to the learning of either calculus or algebra. The focus is on the mathematics content, applications, and effective learning at school and university. The topics will be 2007 Calculus, 2008 Algebra and 2009 Calculus.
STATS 708
Topics in Statistical Education Covers a wide range of research in statistics education at the school and tertiary level. An examination of the issues involved in statistics education in the curriculum, teaching, learning, technology and assessment areas.
MATHS 713
Logic and Set Theory A study of the foundations of pure mathematics, formalising the notions of a “mathematical proof” and “mathematical structure” through predicate calculus and model theory. Explores the limits of these formalisations such as those posed by Gödel’s Incompleteness theorems, and it includes a study of axiomatic set theory.
MATHS 714
Number Theory A broad introduction to various aspects of elementary, algebraic and computational number theory and its applications, including primality testing and cryptography.
MATHS 715
Graph Theory and Combinatorics Theory and applications of combinatorial graphs (networks), block designs, and error-correcting codes. Topics include graph connectivity, trees, colourings, embeddings, digraphs, matchings, incidence matrices, eigenvalue methods, Steiner systems, perfect and linear codes.
MATHS 720
Group Theory A study of groups focusing on basic structural properties, presentations, automorphisms and actions on sets, illustrating their fundamental role in the study of symmetry (for example in crystal structures in chemistry and physics), topological spaces, and manifolds.
MATHS 721
Representations and Structure of Algebras and Groups
Representation theory studies properties of abstract groups and algebras by representing their elements as linear transformations of vector spaces or matrices, thus reducing many problems about the structures to linear algebra, a well-understood theory.
MATHS 730
Measure Theory and Integration Presenting the modern elegant theory of integration as developed by Riemann and Lebesgue, it includes powerful theorems for the interchange of integrals and limits so allowing very general functions to be integrated, and illustrates how the subject is both an essential tool for analysis and a critical foundation for the theory of probability.
MATHS 731
Functional Analysis Provides the mathematical foundations behind some of the techniques used in applied mathematics and mathematical physics in particular. For example, many phenomena in physics can be described by the solution of a partial differential equation (e.g. the Heat equation, the Wave equation and Schrödinger's equation etc). This course presents some of the fundamental ideas that under-pin the modern treatment of these topics.
MATHS 735
Analysis on Manifolds and Differential Geometry Studies surfaces and their generalisations, smooth manifolds, and the interaction between geometry, analysis and topology; it is a central tool in many areas of mathematics, physics and engineering. Topics include Stokes’ theorem on manifolds and the celebrated Gauss Bonnet theorem.
MATHS 740
Complex Analysis Studies analytic functions and their properties, which often reflect the topology of the region on which the functions are defined. This relationship yields powerful conceptual and computational tools and results, including the uniformisation theorem of Riemann which is a cornerstone in conformal geometry. The concepts and objects have applications in many branches of mathematics, physics, and engineering.
MATHS 745
Chaos, Fractals and Bifurcations Presents the classical fractals of computer science and art such as Julia and Mandelbrot sets, iterated function systems and higher-dimensional strange attractors, and illustrates applications of chaos, fractals and bifurcation to areas including commerce, medicine, biological and physical sciences.
MATHS 750
Topology Unlike most geometries, topology models objects which may be stretched non-uniformly. Its ideas have applications in other branches of mathematics as well as physics, chemistry, economics and beyond. Its results give a general picture of what is possible rather than precise details of when and where. The course covers aspects of general and algebraic topology.
MATHS 761
Dynamical Systems Mathematical models of systems that change are frequently written in the form of nonlinear differential equations, but it is usually not possible to write down explicit solutions to these equations. This course covers analytical and numerical techniques that are useful for determining the qualitative properties of solutions to nonlinear differential equations.
MATHS 763
Advanced Partial Differential Equations A study of exact and approximate methods of solution for the linear partial differential equations that frequently arise in applications.
MATHS 764
Mathematical Biology A course introducing central concepts in mathematical biology, with emphasis on modelling gene dynamics and physiological systems. The physiology part will cover enzyme kinetics, transport across biological membranes, and action potentials. (N.B. this course has been given in Semester 1 2007 under the catalogue number MATHS 786).
MATHS 769
Applied Differential Equations Systems taken from a variety of areas such as financial mathematics, fluid mechanics and population dynamics can be modelled with partial differential equations and stochastic differential equations. This course uses such applications as the context to learn about these two important classes of differential equations.
MATHS 770
Advanced Numerical Analysis This course covers the use, implementation and analysis of efficient and reliable numerical algorithms for solving several classes of mathematical problems. The course assumes students have done an undergraduate course in numerical methods and can use Matlab or other high-level computational language.
MATHS ??? Special topics
Each of these courses (MATHS 781 - 784) deals with some special topic(s) from pure mathematics. Not all of them are offered every year; further information may be obtained from the Department of Mathematics.
This course can be taken either as a supervised reading paper or project or as a course on Lie Groups and Lie Algebras.
Lie Groups and Lie Algebras
This course on Lie Groups and Lie Algebras will be given by Dr. Tom ter Elst.
Symmetries and invariants play a fundamental role in mathematics. Especially important are symmetries that depend continuously on various parameters. These form Lie groups. Closely related are structures called Lie algebras. Historically these structures have played an absolutely pivotal role in many areas which range from the theory of differential equations, and their solutions, to the classification of elementary particles. Strongly recommended for students advancing in theoretical physics and pure mathematics.
