Postgraduate course prerequisites, etc
From MathsDept
Contents |
701
Research Issues in Mathematics Education
Research Skills in Mathematics Education
Research methodology for mathematics and statistics education, designed to meet the needs of students planning a Masters level dissertation 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.
702
The historical development, current trends, theories and practice of the mathematics and statistics curricula, and the interconnections between curriculum development and other mathematics education issues.
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.
703
Assessment in Mathematics Education
Theoretical Issues in Mathematics Education
The historical background, theories and recent research into the ways in which learners are assessed in mathematics and statistics education. This includes a focus on both theoretical and practical aspects of assessment in the mathematics classroom, and examination of the relationship between assessment and curriculum in the wider sense.
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. teaching and learning in mathematics.
705
Socio-political Issues in Mathematics Education
Social Issues in Mathematics Education
A selection of topics from cultural, social, historical and political issues arising in mathematics education. Critical examination of theories and current literature will be made, within a case-study approach.
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.
706
Technology in Mathematics Education
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.
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
712
An examination of a mathematical topic up to undergraduate level in the light of current research. The focus will be on investigating how that topic may be effectively learned at senior levels. Students taking this course should normally have studied mathematics or statistics at Stage II level.
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.
713
A study of the foundations of pure mathematics, formalizing the notions of mathematical proof and mathematical structure through Predicate Calculus and Model Theory. Includes an exploration of the limits of these formalizations (including Godel’s incompleteness theorems), and a study of Axiomatic Set Theory (including a discussion of consistency and independence).
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.
Prerequisite: MATHS 315 or PHIL 305
714
A broad introduction to aspects of elementary, analytical and computational number theory, including some or all of the following: primitive roots, quadratic residues, Diophantine equations, primality testing (and applications to cryptology), the two and four-squares theorems, arithmetical functions, Diophantine approximation, distribution of primes.
A broad introduction to various aspects of elementary, algebraic and computational number theory and its applications, including primality testing and cryptography.
Prerequisite: B+ pass in MATHS 328 or 320 B+ pass in MATHS 328 or 320
B+ in either MATHS 328 or MATHS 320 B+ in either MATHS 328 or MATHS 320
715
Theory and applications of combinatorial graphs (networks), block designs, and error-correcting codes. Topics include: graph connectivity, trees, colourings, embeddings, digraphs, matchings, incidence matrices, eigen value methods, Steiner systems, perfect and linear codes.
A study of combinatorial graphs (networks), designs and codes illustrating their application and importance in other branches of mathematics and computer science.
720
Groups, Fields and Galois Theory Groups, Fields and Galois Theory
Group Theory Group Theory
Fundamentals of group theory, including direct products, automorphisms of groups, group actions on sets, Sylow’s theorems, p-groups, free groups, group presentations, solubility, nilpotent groups, extensions and semi-direct products, plus other topics as time permits.
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
721
Rings, Modules, Algebras and Representations
Representations and Structure of Algebras and Groups
A sequel to the course MATHS 320, investigating the properties, extensions and applications of further algebraic structures (such as modules and other algebras), and the representation of algebras in terms of matrices.
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.
730
Concepts, examples and properties of measures of sets, with emphasis on the Lebesgue and Lebesgue-Stieltjes measures, the Lebesgue integral, measure spaces, the Fubini theorems, signed and complex measures, the Lebesgue-Radon-Nikodym theorem, the Vitali system, absolutely continuous functions, and the Fundamental Theorem of Calculus.
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
Recommended preparation: Strongly recommended: MATHS 333
731
Normed linear spaces, Banach spaces and Hilbert spaces, and some of the main developments in these areas. Highlights include: the Hahn-Banach theorem, the Banach-Steinhaus theorem, the Riesz Representation theorem, Fourier series, and the spectral theorem.
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 MATHS 333
Recommended preparation: MATHS 730 and MATHS 750
735
An introduction to differential geometry via the study of differentiable manifolds, tangent spaces and vector fields, differential forms, Stokes theorem, Frenet formulae, quadratic forms on surfaces, and the Gauss-Bonnet theorems.
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
Recommended preparation: Strongly recommended: MATHS 333 and MATHS 340
740
Analytic and harmonic functions, complex integration, hyperbolic geometry, conformal mappings, normal families, the Riemann mapping theorem, Mittag-Leffler and Weierstrass Theorems.
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.
Prerequisite: MATHS 332
Recommended preparation: Strongly recommended: MATHS 333 and MATHS 340
745
Chaos, fractals and bifurcation, and their commercial, medical and scientific applications. Discrete iterations, including the Julia and Mandelbrot sets, iterated function systems and higher-dimensional strange attractors. Quantum chaos and complexity theory are also discussed. This course may not be offered every year; further information may be obtained from the Department of Mathematics.
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.
750
Aspects of general, set-theoretic and algebraic topology including: properties and construction of topological spaces, continuous mappings, axioms of separation, countability, connectivity and compactness, metrization, covering spaces, the fundamental group, homology groups, fixed-point theorems, and applications.
Unlike most geometries, topology models objects which may be stretched. 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 might happen rather than precise details of when and where. The course covers aspects of general and algebraic topology.
Prerequisite: MATHS 332 or MATHS 353
Recommended preparation: Strongly recommended: MATHS 333
761
Ordinary Differential Equations and Dynamical Systems
Analytical and numerical techniques for determining the qualitative properties of solutions to nonlinear differential equations. Topics covered include: recurrent dynamics, symptotic stability, structural stability, the Smale horseshoe and chaos, local and global bifurcations.
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: MATHS 361 and either MATHS 362 or 363
Prerequisite: MATHS 340 and MATHS 361
763
Partial Differential Equations
Advanced Partial Differential Equations
A study of partial differential equations frequently arising in applications. This course studies Hilbert space and approximate methods for analytic and numerical solution of PDEs. Analytic methods include Green’s functions, boundary integral equations and variational formulations. Numerical methods include the Boundary Element Method and, in some years, the Finite Element Method.
A study of exact and approximate methods of solution for the linear partial differential equations that frequently arise in applications.
Prerequisite: MATHS 340 and MATHS 361
769
Advanced Mathematical Modelling
Applied Differential Equations
In this course we model systems taken from a variety of areas such as financial mathematics, fluid mechanics and population dynamics. Most of the systems studied are modelled with partial differential equations or stochastic differential equations and this makes the course a good applications-based setting for learning about PDEs and SDEs. 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.
Prerequisite: MATHS 340 and MATHS 361
770
Advanced techniques in numerical linear algebra, numerical ordinary and partial differential equations and numerical quadrature. The construction and analysis of algorithms for the solution of numerical problems. 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: MATHS 270 and one of MATHS 361, 362, 363
Prerequisite: MATHS 270 and one of MATHS 361, MATHS 340, MATHS 363
764
Mathematical Biology
Template:MATHS 764:2008 Description
783
787
This course can be taken either as a supervised reading paper or project or as a course on
Numerical methods for differential equations
This course is intended for students who are familiar with standard methods for solving ordinary differential equations, such as Runge– Kutta methods and linear multistep methods, or who have an interest in learning about these methods.
The content will be divided into three parts. First we will consolidate and formalise existing knowledge of the traditional methods. We will then introduce some new and more specialised topics, some of which are associated with so called “General linear methods”, which are generalisations of both Runge-Kutta and linear multistep methods. Finally, we will go more seriously into some of the new topics, with the actual selection based on interests that will have developed amongst members of the class.
Throughout the course the emphasis will be balanced between theoretical and practical considerations.
?? Template:MATHS BORIS:2007 Description
776
Honours dissertation in Mathematics or Applied Mathematics
Template:MATHS 776:2008 Prerequisite
Restriction: MATHS 791
Template:MATHS 776:2008 Description
792
Project in Mathematics 1
Deleted course
777
Project in Mathematics 1
Each of these courses involves participation in a research project or investigation in some topic from pure or applied mathematics, under the supervision of one or more staff members, and presentation, by the student, of the results in a seminar; further information may be obtained from the Department of Mathematics. Restriction: MATHS 792
