## Courses - Faculty of Science

# Mathematics

### Stage I

Functioning in Mathematics

An introduction to calculus that builds mathematical skills and develops conceptual thinking. MATHS 102 works as a refresher course for those who haven’t studied Mathematics for some time, a confidence builder for those lacking Mathematical confidence and a preparation course for further study in Mathematics.

*Restriction: MATHS 102 may not be taken concurrently with any other Mathematics course, except MATHS 190 and may not be taken after ENGSCI 111 or any Mathematics course at Stage I or above, except MATHS 190/190G*

General Mathematics 1

A general entry to Mathematics for commerce and the social sciences, following Year 13 Mathematics. MATHS 108 covers selected topics in algebra and calculus and their applications, including: linear functions, linear equations and matrices; functions, equations and inequalities; limits and continuity; differential calculus of one and two variables; integral calculus of one variable. Recommended preparation: It is recommended that NCEA students have a rank score of at least 210 and a merit or excellence in the Differentiation Standard 91578. *Prerequisite: MATHS 102 or 110 or at least 13 credits in Mathematics at NCEA Level 3 including the Differentiation Standard 91578, or D in CIE A2 Mathematics or C in CIE AS Mathematics or 3 out of 7 in IB Mathematics: Analysis and Approaches (SL or HL)*

*Restriction: ENGGEN 150, ENGSCI 111, MATHS 120, 130, 208, 250*

Mathematics for Natural Sciences

A general entry to Mathematics for the natural sciences, following Year 13 Mathematics. Covers selected topics in algebra and calculus and their application to chemistry, biology and other natural sciences. Recommended Preparation: It is recommended that NCEA students have a rank score of at least 210 and a merit or excellence in the Differentiation Standard 91578. *Prerequisite: MATHS 102 or 108 or at least 13 credits in Mathematics at NCEA Level 3, or D or better in Cambridge A2 Mathematics, C or better in AS Mathematics, pass in IB Mathematics: Analysis and Approaches (SL or HL)*

*Restriction: ENGGEN 150, ENGSCI 111, MATHS 208, 250. More than 15 points from MATHS 120 and 130*

Algebra

A foundation for further mathematics courses, essential for students intending to major in Mathematics, Applied Mathematics, Statistics, Physics, or who want a strong mathematical component to their degree. Develops skills and knowledge in linear algebra, together with an introduction to mathematical language and reasoning, including complex numbers, induction and combinatorics. Recommended preparation: Merit or excellence in the Differentiation Standard 91578 at NCEA Level 3. *Prerequisite: MATHS 208, or B- or higher in MATHS 108, or A- or higher in MATHS 110, or A+ in MATHS 102, or at least 18 credits in Mathematics at NCEA Level 3 including at least 9 credits at merit or excellence, or B in CIE A2 Mathematics, or 5 out of 7 in IB Mathematics: Analysis and Approaches (SL or HL)*

Calculus

A foundation for further mathematics courses, essential for students intending to major in Mathematics, Applied Mathematics, Statistics, Physics, or who want a strong mathematical component to their degree. Develops skills and knowledge in calculus of functions of a single variable. Recommended preparation: Merit or excellence in the Differentiation Standard 91578 at NCEA Level 3. *Prerequisite: MATHS 208, or B- or higher in MATHS 108, or A- or higher in MATHS 110, or A+ in MATHS 102, or at least 18 credits in Mathematics at NCEA Level 3 including at least 9 credits at merit or excellence, or B in CIE A2 Mathematics, or 5 out of 7 in IB Mathematics: Analysis and Approaches (SL or HL)*

Computational Mathematics

An introduction to computational mathematics and programming in MATLAB. The course will introduce some basic concepts in computational mathematics and give applications that include cryptography, difference equations, stochastic modelling, graph theory and Markov chains.

*Corequisite: ENGGEN 150 or ENGSCI 111 or MATHS 108 or 120* *
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Restriction: MATHS 199*

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.

*Restriction: MATHS 190 may not be taken after any Mathematics course at Stage III*

Advancing in Mathematics

An introduction to University level mathematics, for high-achieving students currently at high school. The numerical computing environment MATLAB is used to study beautiful mathematics from algebra, analysis, applied mathematics and combinatorics. Students will learn to write mathematical proofs and create mathematical models to find solutions to real-world problems.

*Prerequisite: Departmental approval*

### Stage II

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.

*Prerequisite: 15 points from MATHS 108, ENGSCI 111, ENGGEN 150, or MATHS 120 and MATHS 130, or a B- or higher in MATHS 110* *
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Restriction: Cannot be taken, concurrently with, or after MATHS 250, 253*

Algebra and Calculus 2

Designed for all students who plan to progress further in mathematics, this course follows directly from MATHS 120 and 130. Covering topics from multivariable calculus and linear algebra, which have many applications in science, engineering and commerce. Students will learn mathematical results and procedures as well as the underpinning ideas and mathematical proofs.

*Prerequisite: MATHS 120 and 130, or ENGGEN 150 or ENGSCI 111*

Algebra and Calculus 3

A sequel to MATHS 250, further developing and bringing together linear algebra and calculus. Students will learn about quadratic forms, projections, spectral decomposition, methods of multicriteria optimisation, double, triple and line integrals, Green’s theorem and applications.

*Prerequisite: MATHS 250*

Fundamental Concepts of Mathematics

Fundamentals of mathematics important to many branches of the subject and its applications. Topics include equivalence relations, elementary number theory, counting techniques, elementary probability, geometry, symmetry and metric spaces. This is an essential course for all students advancing beyond Stage II in pure mathematics, and highly suitable for other students in the mathematical sciences.

*Corequisite: MATHS 250*

Differential Equations

The study of differential equations is central to mathematical modelling of systems that change. 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: MATHS 208 or 250 or ENGSCI 211 or a concurrent enrolment in MATHS 250*

Numerical Computation

Many mathematical models occurring in Science and Engineering cannot be solved exactly using algebra and calculus. Students are introduced to computer-based methods that can be used to find approximate solutions to these problems. The methods covered in the course are powerful yet simple to use. This is a core course for students who wish to advance in Applied Mathematics.

*Prerequisite: MATHS 120 and 130, or 15 points from ENGGEN 150, ENGSCI 111, MATHS 108, 110 and 15 points from COMPSCI 101, 105, 130, INFOSYS 110, 120, MATHS 162, 199*

### Stage III

Perspectives in Mathematics Education

For people interested in thinking about the social, cultural, political, economic, historical, technological and theoretical ideas that influence mathematics education, who want to understand the forces that shaped their own mathematics education, or who are interested in teaching. Students will develop their ability to communicate ideas in essay form. Recommended preparation: At least 45 points from courses in Mathematics or Statistics.

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, introduces model theory and demonstrates how models of a first order system relate to mathematical structures. The course is recommended for anyone studying high level computer science or mathematical logic.

*Prerequisite: B+ or higher in COMPSCI 225 or MATHS 254 or PHIL 222*

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 course is recommended for those planning graduate study in pure mathematics.

*Prerequisite: MATHS 250, 254*

Combinatorics

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.

*Prerequisite: MATHS 250 or 254, and a B+ or higher in COMPSCI 225*

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.

*Prerequisite: MATHS 250 and 254, or a B+ or higher in COMPSCI 225 and 15 points from MATHS 250, 253*

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.

*Prerequisite: MATHS 250, 254*

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.

*Prerequisite: MATHS 332 or a B or higher in MATHS 254*

Algebraic Geometry

Algebraic geometry is a branch of mathematics studying zeros of polynomials. The fundamental objects in algebraic geometry are algebraic varieties i.e., solution sets of systems of polynomial equations.

*Prerequisite: MATHS 332, and at least one of MATHS 320, 328 and Departmental approval* *
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Restriction: MATHS 734*

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. Recommended preparation: MATHS 253.

*Prerequisite: MATHS 250*

Complex Analysis

Functions of one complex variable, including Cauchy’s integral formula, the index formula, Laurent series and the residue theorem. Many applications are given including a three line proof of the fundamental theorem of algebra. Complex analysis is used extensively in engineering, physics and mathematics. Strongly recommended: MATHS 333.

*Prerequisite: MATHS 332 and Departmental approval* *
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Restriction: MATHS 740*

Topology

Aspects of point-set, set-theoretic and algebraic topology including: properties and construction of topological spaces, continuous functions, axioms of separation, countability, connectivity and compactness, metrisation, covering spaces, the fundamental group and homology theory. Recommended preparation: MATHS 333.

*Prerequisite: MATHS 332 and Departmental approval* *
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Restriction: MATHS 750*

Partial Differential Equations

Partial differential equations (PDEs) are used to model many important applications of phenomena in the real world such as electric fields, diffusion and wave propagation. Covers: linear PDEs and analytical methods for their solution, weak solutions. Recommended preparation: MATHS 253.

*Prerequisite: MATHS 250, 260*

Methods in Applied Mathematics

Covers a selection of techniques including the calculus of variations, asymptotic methods and models based on conservation laws. These methods are fundamental in the analysis of traffic flow, shocks, fluid flow, as well as in control theory, and the course is recommended for students intending to advance in Applied Mathematics. Recommended preparation: MATHS 253, 361.

*Prerequisite:MATHS 250, 260*

Advanced Modelling and Computation

In real-world situations, the interesting and important variables are often not directly observable. To address this problem, mathematical models and quantities that are observable are usually employed to carry out inference on the variables of interest. This course is an introduction to fitting of models to (noisy) observational data and how to compute estimates for the interesting variables. Numerical methods for partial differential equations, which are commonly used as models for the observations, will also be covered.

*Prerequisite: MATHS 260 and 270*

Directed Study

*To complete this course students must enrol in MATHS 382 A and B, or MATHS 382*

Directed Study

Directed study on a topic or topics approved by the Academic Head or nominee.

*To complete this course students must enrol in MATHS 386 A and B, or MATHS 386*

Capstone: Mathematics

An exploration of the role of mathematics in society and culture, and the activities performed by mathematicians as teachers, critics, and innovators. Students will develop their skills in communication, critical thinking, teaching, and creative problem solving.

*Prerequisite: MATHS 250 and 30 points at Stage III in Mathematics*

### Postgraduate 700 Level Courses

Introduction to Research in Mathematics Education

What is Mathematics Education research, and how can it inform practice? This course introduces a range of skills and methods for conducting and critically consuming research in mathematics education. Students will explore issues and techniques in Mathematics Education research as they design their own research studies to inform their teaching and learning practice.

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Mathematical Processes in the Curriculum

Historically, mathematics curricula have emphasised the what of mathematics (content), at the expense of considering the how. This course uses hands-on experiences and research literature to explore how to teach, learn and do mathematics through processes such as communication, modelling, problem solving, and proving.

What Can Be More Practical Than a Good Theory?

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.

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Contemporary Issues in Mathematics Education

This course explores contemporary topics in mathematics education research and their impact on teaching and learning. Students will investigate and critically examine research and scholarly literature, and consider the implications of current knowledge for their own practice.

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Technology and Mathematics Education

Practical and theoretical perspectives on ways that technology can enhance teaching and learning of mathematics. Students will consider and critically examine affordances, constraints and obstacles in the use of technology.

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Special Topic

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Special Topic

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Special Topic

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Directed Study in Mathematics Education

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

Directed Study in Mathematics Education

*Prerequisite: MATHS 302 or significant teaching experience or department approval* *
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To complete this course students must enrol in MATHS 711 A and B, or MATHS 711*

Teaching and Learning in Algebra

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. Students taking this course should normally have studied mathematics or statistics at 200 level.

*Prerequisite: MATHS 302 or significant teaching experience or department approval*

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. It includes a study of axiomatic set theory.

*Prerequisite: MATHS 315 or PHIL 305*

Number Theory

A broad introduction to various aspects of elementary, algebraic and computational number theory and its applications, including primality testing and cryptography.

*Prerequisite: B+ in MATHS 328 or 320*

Graph Theory and Combinatorics

A study of combinatorial graphs (networks), designs and codes illustrating their application and importance in other branches of mathematics and computer science.

*Prerequisite: 15 points from MATHS 320, 326, 328 with a B or higher*

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.

*Prerequisite: MATHS 320*

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.

*Prerequisite: MATHS 320*

Lie Groups and Lie Algebras

Symmetries and invariants play a fundamental role in mathematics. Especially important in their study are the Lie groups and the related structures called Lie algebras. These structures have played a pivotal role in many areas, from the theory of differential equations to the classification of elementary particles. Strongly recommended for students advancing in theoretical physics and pure mathematics. Recommended preparation: MATHS 333.

*Prerequisite: MATHS 320 and 332*

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. Strongly recommended: MATHS 333.

*Prerequisite: MATHS 332*

Functional Analysis

Provides the mathematical foundations behind some of the techniques used in applied mathematics and mathematical physics; it explores how many phenomena in physics can be described by the solution of a partial differential equation, for example the heat equation, the wave equation and Schrödinger's equation. Recommended preparation: MATHS 730 and 750.

*Prerequisite: MATHS 332 and 333*

Algebraic Geometry

Algebraic geometry is a branch of mathematics studying zeros of polynomials. The fundamental objects in algebraic geometry are algebraic varieties i.e., solution sets of systems of polynomial equations.

*Prerequisite: MATHS 332 and at least one of MATHS 320, 328* *
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Restriction: MATHS 334*

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. Strongly recommended: MATHS 333 and 340.

*Prerequisite: MATHS 332*

Complex Analysis

An introduction to functions of one complex variable, including Cauchy's integral formula, the index formula, Laurent series and the residue theorem. Many applications are given including a three line proof of the fundamental theorem of algebra. Complex analysis is used extensively in engineering, physics and mathematics. Strongly recommended: MATHS 333.

*Prerequisite: MATHS 332* *
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Restriction: MATHS 341*

Topology

Aspects of point-set, set-theoretic and algebraic topology including: properties and construction of topological spaces, continuous functions, axioms of separation, countability, connectivity and compactness, metrization, covering spaces, the fundamental group and homology theory. Strongly recommended: MATHS 333.

*Prerequisite: MATHS 332* *
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Restriction: MATHS 350*

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.

*Prerequisite: B- in both MATHS 340 and 361*

Nonlinear Partial Differential Equations

A study of exact and numerical methods for non-linear partial differential equations. The focus will be on the kinds of phenomena which only occur for non-linear partial differential equations, such as blow up, shock waves, solitons and special travelling wave solutions.

*Prerequisite: B- in both MATHS 340 and 361*

Advanced Partial Differential Equations

A study of exact and approximate methods of solution for the linear partial differential equations that frequently arise in applications.

*Prerequisite: B- in both MATHS 340 and 361*

Mathematical Biology

A course introducing central concepts in mathematical biology, with emphasis on modelling of physiological systems and gene dynamics.

*Prerequisite: B- in both MATHS 340 and 361*

Mathematical Modelling

Advanced topics in mathematical modelling, including selected topics in a range of application areas, principally taken from the physical and biological sciences.

*Prerequisite: At least B- or better in both MATHS 340 and 361*

Inverse Problems

Covers the mathematical and statistical theory and modelling of unstable problems that are commonly encountered in mathematics and applied sciences.

*Prerequisite: At least B- in both MATHS 340 and 363, or PHYSICS 701*

Stochastic Differential and Difference Equations

Differential and difference equations are often used as preliminary models for real world phenomena. The practically relevant models that can explain observations are, however, often the stochastic extensions of differential and difference equations. This course considers stochastic differential and difference equations and applications such as estimation and forecasting. Recommended preparation: MATHS 363.

*Prerequisite: B- in both MATHS 340 and 361*

Advanced Numerical Analysis

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.

*Prerequisite: B- in MATHS 270, 340 and 361*

Honours Research Project - Level 9

*Restriction: MATHS 791* *
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To complete this course students must enrol in MATHS 776 A and B, or MATHS 776*

Project in Mathematics 1 - Level 9

A supervised investigation or research project including seminar presentation in pure or applied mathematics.

Dissertation in Mathematics Education - Level 9

*To complete this course students must enrol in MATHS 785 A and B, or MATHS 785*

Special Topic: Inverse Problems and Stochastic Differential Equations

Covers deterministic inverse problems: Hilbert spaces and linear operator theory, singular value decomposition and pseudoinverses, Tikhonov regularisation, nonlinear problems and iterative methods, continuous time processes, stochastic differential equations, random walks and Wiener processes, Itô calculus, and applications of SDE's.

*Prerequisite: B- or higher in MATHS 340 and 361* *
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Restriction: MATHS 769, 766*

Project in Mathematics 2 - Level 9

An investigation into a topic from pure or applied mathematics, under the supervision of one or more staff members.

Project in Mathematics 3 - Level 9

An investigation into a topic from pure or applied mathematics, under the supervision of one or more staff members.

MSc Thesis in Applied Mathematics - Level 9

*To complete this course students must enrol in MATHS 795 A and B*

Masters Thesis Mathematics - Level 9

*To complete this course students must enrol in MATHS 796 A and B*