The use of mathematical models has grown rapidly in recent years, owing to the advent of cheap and powerful computers, expanding from applications in the physical and earth sciences to the biological and environmental sciences, and now into industry and commerce. In this course we study the process of starting with an initial succinct non-mathematical description of a problem, formulate associated mathematical models, introduce new mathematical techniques and then determine and interpret solutions that are useful in a real life context. General computational and mathematical techniques and strategies will be introduced by examining specific scientific and industrial problems. The topics to be covered in this course include: Model type selection and formulation, Data analysis techniques (time/space and frequency domain), State Space and Transfer Function Models, Model Structure Identification, Testing and Sensitivity Analysis. Computations will be done using modern high level scientific computing environments such as SCILAB or PYTHON.The use of mathematical models has grown rapidly in recent years, owing to the advent of cheap and powerful computers, expanding from applications in the physical and earth sciences to the biological and environmental sciences, and now into industry and commerce.

In this course we study the process of starting with an initial succinct non-mathematical description of a problem, formulate associated mathematical models, introduce new mathematical techniques and then determine and interpret solutions that are useful in a real life context. General computational and mathematical techniques and strategies will be introduced by examining specific scientific and industrial problems. Computations will be done using modern high level scientific computing environments such as SCILAB or PYTHON.

Topics to be covered include Model type selection and formulation, Data analysis techniques (time/space and frequency domain), State Space and Transfer Function Models, Model Structure Identification, Testing and Sensitivity Analysis.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Exam (40%; LO 1, 2, 3, 4, 5)
• Three Assignments (15% each) demonstrating ability to apply techniques (15% each; LO 2, 3, 5)
• Tutorials demonstrating ability to use relevant software (15%; LO 3)

Requires MATH2305; or MATH2405; or 12 units of Group B Mathematics courses with a mark of 60 or better

This course presents the basic elements of scientific computing, in particular the methods for solving or approximating the solution of calculus and linear algebra problems associated with real world problems. Using a non-trivial model problem such as the heat equation, and sophisticated scientific computing and visualisation environments, students are introduced to the basic computational concepts of stability, accuracy and efficiency, as new numerical methods and techniques are introduce to solve progressively more challenging problems.

Honours Pathway Option (HPO):

To do this option, students must have completed MATH2405 or STAT2001 or a mark of 60% or more in MATH2305 or MATH1116. The HPO expands on the theoretical aspects of the underlying algorithms, and uses alternative assessment to assess these theoretical aspects.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• 10 tutorial responses demonstrating capacity to use relevant software (30%; LO 1, 2, 3)
• 5 written assignments demonstrating a problem-based understanding of algorithms (30%; LO 1, 2, 3)
• Final examination (40%; LO, 1, 2, 3)

Prerequisite course is MATH1116 or MATH2305 or MATH2405 or MATH2320 or STAT2001.

This course introduces the key concepts of modern real analysis. The philosophy of this course is that modern analysis play a fundamental role in mathematics itself and in applications to areas such as physics, computer science, economics and engineering. This course will have shared lectures with MATH2320 but will have different tutorials and assessment which will emphasise the application of techniques.
Topics to be covered include:
Review of the real number system, the foundations of calculus, elementary set theory; metric spaces, sequences, series and power series, uniform convergence, continuity, the contraction mapping principle; foundations of multidimensional calculus, applications to the calculus of variations, integral equations and differential equations.

Note: This is an Honours Pathway Course. It emphasises the sophisticated application of deep mathematical concepts.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of real analysis and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of real analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from real analysis
4. Apply problem-solving using real analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts

Assessment will be based on:

• Tutorials (15%; LO 1-4)
• Assignments (15%; LO 1-4)
• Final exam (70%; LO 1-4)

This course provides a mathematical introduction to fractal geometry and nonlinear dynamics, with applications to biological modelling and the geometry of real world images. What do models for the structure of ferns and complicated behaviour of the weather have in common? Both involve the iterative application of functions that map from a space to itself. Both can be treated from the classical geometrical point of view of Felix Klein. Invariants, such as fractal dimension, of important groups of transformations acting on two-dimensional spaces, pictures, and measures are explored. Deep mathematical ideas are explained in an intuitive and practical manner. Laboratory work includes projects related to digital imaging and biological modelling. A high point in the course is an introduction to fractal homeomorphisms: what they are and how to work with them in the laboratory.

Topics to be covered include:

Affine, projective and Mobius geometries, iterated function systems, metric spaces, elementary topology, the contraction mapping theorem, the collage theorem, orbits of points, local behaviour of transformations, code space and the shift transformation, Julia sets and the Mandelbrot set, superfractals, deterministic, Markov chain, and escape-time algorithms for constructing fractal sets. Regular and chaotic behaviour in nonlinear systems, characterization and measures of chaos, stability and bifurcations, routes to chaos, crises, Poincare sections, the relation of fractal structures to simple nonlinear systems.

Honours pathway option (HPO)

In the HPO option we will expand on the theoretical aspects of the underlying concepts. Alterative assessment in the assignments and exam will be used to assess these theoretical aspects.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the basic concepts and have a practical familiarity with fractal geometry and chaotic dynamics.
2. Be able to formulate and analyze fractal geometric models in biology and computer graphics.
3. (HPO only) Be able to prove the main theorems that underline projective IFS theory.

Assessment may be based on:

• Assignments (25%; LO 1-3)
• Notebooks (25%; LO 1-3)
• Exams (50%; LO 1-3)

12 units of A courses in Mathematics, including MATH1003 or MATH1013 or MATH1115; it is assumed that students will have some knowledge of differential equations and several variable calculus.

MATH2062

The course introduces stochastic processes with a view towards applications in fields such as finance, insurance, risk management, and operations research. The aim is to provide mathematics students with basic knowledge of stochastic processes where practical rather than theoretical aspects are emphasized.

Probability Modelling and Applications provides a sound foundation to progress to honours and post-graduate courses emphasizing the theory of mathematical finance and stochastic analysis.

The course contains sufficient material for students to feel comfortable with Markov chains, Poisson processes, and Brownian motion, and the conceptual formulation of topics in continuous time finance, insurance and risk management, where these processes are applied. Also the concept of martingales, which is fundamental for understanding the modern option pricing theory of Black and Scholes, is introduced.

Note: This is an HPC. It continues the development of sophisticated mathematical and probabilistic techniques and their application begun in STAT2001(HPC)

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Weekly Assignments (50%; LO 1-4)
• Final examination (50%; LO 1-4)

STAT2001 or MATH2007.

Just as there is a formula for solving a quadratic equation, there are similar formulae for solving the general cubic and quartic. Galois theory provides a solution to the corresponding problem for quintics --- there is no such formula in this case! Galois theory also enables us to prove (despite regular claims to the contrary) that there is no ruler and compass construction for trisecting an angle.

Topics to be covered include:

Galois Theory - fields, field extensions, normal extensions, separable extensions. Revision of group theory, abelian and soluble groups. Galois' Theorem. Solubility of equations by radicals. Finite fields. Cyclotomic fields.

Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Assignment 1 (20%; LO 1-4)
• Assignment 2 (20%; LO 1-4)
• Assignment 3 (20%; LO 1-4)
• Final exam (40%; LO 1-4)

36 lectures, tutorials by arrangement

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.
Topics to be covered will include:

• Topological Spaces - continuity, homeomorphisms, convergence, Hausdorff spaces, compactness, connectedness, path connectedness.
• Measure and Integration - Lebesgue outer measure, measurable sets and integration, Lebesgue integral and basic properties, convergence theorems, connection with Riemann integration, Fubini's theorem, approximation theorems for measurable sets, Lusin's theorem, Egorov's theorem, Lp spaces as Banach spaces.
• Hilbert Spaces - elementary properties such as Cauchy Schwartz inequality and polarization, nearest point, orthogonal complements, linear operators, Riesz duality, adjoint operator, basic properties or unitary, self adjoint and normal operators, review and discussion of these operators in the complex and real setting, applications to L2 spaces and integral operators, projection operators, orthonormal sets, Bessel's inequality, Fourier expansion, Parseval's equality, applications to Fourier series.
• Calculus in Euclidean Space - Inverse and implicit function theorems.

This is an Honours Pathway Course. It emphasises mathematical rigour and proof and develops modern analysis from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of advanced analysis such as topology and Lebeque integration and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of advanced analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanved analysis
4. Apply problem-solving using advanced analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.

Assessment will be based on:

• 6 or 7 assignments (total 70%; LO 1-4)
• Take home exam (30%; LO 1-4)

36 lectures, tutorials by arrangement

A mark of 60 or more in MATH2320.

MATH3021

This is a special topics course which introduces students to the key concepts and techniques of Differential Geometry. Possible topics include:

Surfaces in Euclidean space, general differentiable manifolds, tangent spaces and vector fields, differential forms, Riemannian manifolds, Gauss-Bonnet theorem.

Note: This is an Honours Pathway course. It emphasises mathematical rigour and proof and develops the fundamental ideas of differential geometry from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

4 written assignments involving problem-solving, proofs of theorems and extension of theory (25% each; LO 1, 2, 3)

36 lectures and tutorials by arrangement.

A mark of 60 or more in MATH2320 or a mark of 60 or more in MATH3116.

with MATH3027.

This is a special topics course which introduces students to the key concepts and techniques of:
First order logic
Axiomatisation of set theory
Model theory
Computability
Godel's Incompleteness Theorem.

Note: This is an HPC. It emphasises mathematical rigour and proof.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Assignments (100%; LO 1-3)

Offered in 2009 subject to staff availability and student demand.

A mark of 60 or more in MATH2021 or MATH2322.

with MATH3128.

Selected topics, normally fourth year honours and postgraduate courses, may be made available to undergraduate students. Interested students should consult the year coordinator or attend the honours course meeting on first Monday of each semester.

Note: This is an HPC. Students wishing to take topics in this course will be expected to have outstanding results in second year Honours Pathway Courses.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of a special topic in the mathematical sciences and its role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of specific techniques from the mathematical sciences.
3. Demonstrate capacity for mathematical reasoning through analysing, proving and explaining concepts from the mathematical sciences.

Assessment is expected to be based on:

• Three assignments (worth 10% each; LO 1-3)
• Final examination (70%; LO 1-3)

Please contact MATHSadmin@maths.anu.edu.au for consent to enrol in this course.

This course introduces students to key concepts and techniques in mathematical physics. Topics will be taken from contemporary research areas in mathematical physics. Possible topics include lie algebras, integrable models conformal field theory.

Note: This is an HPC. It emphasises mathematical rigour and proof for a range of advanced Mathematical Physics topics.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of a special topic in mathematical physics.
2. Demonstrate accurate and efficient use of specific mathematical physics techniques.
3. Demonstrate capacity for mathematical reasoning through analysing, proving and explaining concepts from mathematical physics.

Assessment is expected to be based on:

• Three assignments (100% in total; LO 1-3)

36 lectures and regular tutorials

This course will introduce students to current research trends in an area of computational mathematics. Possible topics: data mining algorithms, multiscale and multilevel techniques, the numerical solution of PDEs, optimisation, approximation in particular of high-dimensional functions, algorithms for the solution of linear systems of equations including iterative methods.

Note: This is an HPC. It emphasises mathematical rigour and proof for a range of advanced computational mathematics topics.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Three assignments (45% in total; LO 1-3)
• Final examination (55%; LO 1-3)

36 lectures and regular tutorials.

A solid background in mathematics is expected. Typically requires MATH3320 and MATH3511 and/or MATH3512, but consultation with course coordinators is essential.

The course is a natural continuation of MATH2307 Bioinformatics and Biological Modelling. However, MATH2307 is not a formal prerequisite for this course since it begins with a brief overview of the main concepts of MATH2307. Next, necessary concepts and techniques from the probability theory will be introduced. They will be applied to assessing the significance of the score of pairwise alignments of biological sequences. Further, we will give a brief introduction to maximum likelihood estimates and apply this general statistical theory to the Markov chains and hidden Markov models (HMMs) studied in MATH2307. Finally, the derivation of PAM and BLOSUM scoring matrices will be explained.

Note: This is an HPC. It emphasises mathematical rigour for a range of advanced Bioinformatics topics.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment is expected to be based on:

• Two assignments (15% each; LO 1-3)
• Final exam (70%; LO 1-3)

Offered in association with Fenner School.

This course presents the major model types used to represent environmental systems. Mathematical emphasis on how these models are constructed uses the theory of inverse problems while the practical emphasis uses systems methodology. The focus will be on hydrological systems and their basic processes, combined with the constraints imposed by the limitations of real observational data. Case studies and project assessment will cover catchment hydrology, soil physics, subsurface hydrology and stream transport.

Honours Pathway Option (HPO):

Students must have 12 units of Group B level Mathematics including MATH2405 or a mark of 60 or more in MATH2305 to choose this option. Students who choose this option will be expected to have a deeper understanding of the work and will be required to complete a major project worth 25 per cent in place of one of the 25 per cent assignments.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Two assignments (30%; LO 1, 2)
• Five lab reports (25%; LO 2, 3)
• Review (35%; LO 1, 2, 3, 4) or (HPO students only) Project (35%; LO 5)
• Two presentations (10%; LO 3, 4)

This course introduces important algorithms and techniques of scientific computing, focussing on the areas of linear algebra and optimisation, and presenting both theoretical and practical aspects of the algorithms. The course is highly relevant to students from disciplines such as science, engineering or economics where skills in numerical computations are important.

Honours Pathway Option (HPO):

To do this option, students must have completed MATH2405 or MATH2320 or MATH3116. HPO students will learn more about the theoretical aspects of the underlying algorithms, and will be assessed appropriately.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Use sophisticated scientific computing and visualisation environments to solve application problems
2. Analyse and intrepret results produced by computer algorithms
3. Explain the effects of errors in computation and how such errors affect solutions
4. Demonstrate the necessary analytical background for further studies leading to research in applied mathematics or related disciplines

Assessment will be based on:

• Five written assignments involving maths and programming problems (70%; LO 1, 2, 3, 4)
• Open-book examination (30%; LO 1, 2, 3, 4)

Prerequisite is MATH2305 or MATH2405 or MATH2320 or STAT2001 or MATH3116.

The need to protect information being transmitted electronically (such as the widespread use of electronic payment) has transformed the importance of cryptography. Most of the modern types of cryptosystems rely on (increasingly more sophisticated) number theory for their theoretical background. This course introduces elementary number theory, with an emphasis on those parts that have applications to cryptography, and shows how the theory can be applied to cryptography.
Number theory topics will be chosen from: the Euclidean algorithm, highest common factor, prime numbers, prime factorisation, primality testing, congruences, the Chinese remainder theorem, diophantine equations, sums of squares, Euler's function, Fermat's little theorem, quadratic residues, quadratic reciprocity, Pell's equation, continued fractions.

Cryptography topics will be chosen from: symmetric key cryptosystems, including classical examples and a brief discussion of modern systems such as DES and AES, public key systems such as RSA and discrete logarithm systems, cryptanalysis (code breaking) using some of the number theory developed.

Honours Pathway Option (HPO):

Students who take the HPO will complete extra work of a more theoretical nature. The assignments will be replaced by alternative assignments and the final exam will contain alternative questions requiring deeper conceptual understanding.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Solve problems in elementary number theory
2. Apply elementary number theory to cryptography
3. (HPO only) Develop a deeper conceptual understanding of the theoretical basis of number theory and cryptography

Assessment will be based on:

• Three assignments (10%; LO 1, 2, & 3 for HPO)
• Final examination (70%; LO 1, 2, & 3 for HPO)

36 lectures and 10 tutorials

Requires MATH2016; or MATH2302; or MATH2303; or MATH2301 with a mark of 60 or better; or MATH1116; or MATH1014 with a mark of 60 or better.

MATH3001, MATH3101 or MATH3401.

On satisfying the requirements of this course, students will have the knowledge and skills to:

• Five assignments (10% each; LO 1-4)
• Final exam (50%; LO 1-4)

This course introduces important algorithms and techniques of scientific computing, focussing on the areas of linear algebra and optimisation, and presenting both theoretical and practical aspects of the algorithms. The course is highly relevant to students from disciplines such as science, engineering or economics where skills in numerical computations are important.

Honours Pathway Option (HPO):

To do this option, students must have completed MATH2405 or MATH2320 or MATH3116. HPO students will learn more about the theoretical aspects of the underlying algorithms, and will be assessed appropriately.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Use sophisticated scientific computing and visualisation environments to solve application problems
2. Analyse and intrepret results produced by computer algorithms
3. Explain the effects of errors in computation and how such errors affect solutions
4. Demonstrate the necessary analytical background for further studies leading to research in applied mathematics or related disciplines

Assessment will be based on:

• Five written assignments involving maths and programming problems (70%; LO 1, 2, 3, 4)
• Open-book examination (30%; LO 1, 2, 3, 4)

Prerequisite is MATH2305 or MATH2405 or MATH2320 or STAT2001 or MATH3116.

This course provides an introduction to the theory of stochastic processes and its application in the mathematical finance area.

The course starts with background material on markets, modelling assumptions, types of securities and traders, arbitrage and maximisation of expected utility. Basic tools needed from measure and probability, conditional expectations, independent random variables and modes of convergence are explained. Discrete and continuous time stochastic processes including Markov, Gaussian and diffusion processes are introduced. Some key material on stochastic integration, the theory of martingales, the Ito formula, martingale representations and measure transformations are described. The well-known Black-Scholes option pricing formula based on geometric Brownian motion is derived. Pricing and hedging for standard vanilla options is presented. Hedge simulations are used to illustrate the basic principles of no-arbitrage pricing and risk-neutral valuation. Pricing for some other exotic options such as barrier options are discussed. The course goes on to explore the links between financial mathematics and quantitative finance. Results which show that the transition densities for diffusion processes satisfy certain partial differential equations are presented. The course concludes with treatment of some other quantitative methods including analytic approximations, Monte Carlo techniques, and tree or lattice methods.

Mathematics of Finance provides an accessible but mathematically rigorous introduction to financial mathematics and quantitative finance. The course provides a sound foundation for progress to honours and post-graduate courses in these or related areas.

Note: This is an Honours Pathway Course. It continues the development of sophisticated mathematical techniques and their application begun in MATH3029 or MATH3320.

On successful completion of this course, students will be able to:

Assessment will be based on:

• Assignments (50%; LO 1-4)
• Final examination (50%; LO 1-4)

MATH3029 OR MATH3320.

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Complex differentiability, conformal mapping; complex integration, Cauchy integral theorems, Taylor series representation, isolated singularities, residue theorem and applications to real integration. Topics chosen from: argument principle, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler.

Note: This is an HPC. It emphasises mathematical rigour and proof and develops the material from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of complex analysis and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of complex analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from complex analysis
4. Apply problem-solving using complex analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.

Assessment will be based on:

• Assignment 1 (30%; LO 1-4)
• Assignment 2 (30%; LO 1-4)
• Take home exam (40%; LO 1-4)

36 lectures, tutorials by arrangement

A mark of 60 or more in MATH3320.

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Measure theory- functions of bounded variation over R, absolute continuity and integration, examples of more general measures (Radon, Hausdorff, probability measures), Fubini-Tonelli theorem, Radon-Nikodym theorem.

Banach spaces and linear operators - classical function and sequence spaces, Hahn-Banach theorem, closed graph and open mapping theorems, and uniform boundedness principles, sequential version of Banach-Alaoglu theorem, spectrum of an operator and analysis of the compact self-adjoint case, Fredholm alternative theorem.

Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Three assignments (30% total; LO 1-4)
• Essay paper (20%; LO 1-4)
• Take home exam (50%; LO 1-4)

36 lectures, tutorials by arrangement

A mark of 60 or more in MATH3320.

with MATH3022.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of advanced algebra and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of advanced algebraic techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanced algebra
4. Apply problem-solving using advanced algebraic techniques applied to diverse situations in physics, engineering and other mathematical contexts

Assessment will be based on:

• Assignment 1 (20%; LO 1-4)
• Assignment 2 (20%; LO 1-4)
• Assignment 3 (20%; LO 1-4)
• Take home exam (40%; LO 1-4)

36 lectures plus totorial by arrangement.

Algebraic topology studies properties of topological spaces and maps between them by associating algebraic invariants (fundamental groups, homology groups, cohomology groups) to each space. This course gives a solid introduction to fundamental ideas and results that are employed nowadays in most areas of mathematics, theoretical physics and computer science. This course aims to understand some fundamental ideas in algebraic topology; to apply discrete, algebraic methods to solve topological problems; to develop some intuition for how algebraic topology relates to concrete topological problems.

Topics to be covered include:

Fundamental group and covering spaces; Brouwer fixed point theorem and Fundamental theorem of algebra; Homology theory and cohomology theory; Jordan-Brouwer separation theorem, Lefschetz fixed theorem; some additional topics (Orientation, Poincare duality, if time permits)

Honours Pathway Option

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Assignment 1 (20%: LO 1-4)
• Assignment 2 (20%; LO 1-4)
• Assignment 3 (20%; LO 1-4)
• Presentation (10%; LO 1-4)
• Take home exam (30%; LO 1-4)

36 lectures and 10 tutorials.

Requires a mark of 60 or more in both MATH3320 and MATH2322, and basic knowledge of abstract algebra, linear algebra, and point-set topology.

with MATH3060.

The course will discuss the three main classes of equations, elliptic, parabolic and hyperbolic. It is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered will include fundamental solutions, maximum principles, regularity (smoothness) of solutions, variational problems, Holder and Sobolev spaces.

Note: This is an Honours Pathway Course. It emphasises mathematical rigour and proof.

On satisfying the requirements of this course, students will have the knowledge and skills to:

4 written assignments involving problem-solving, proofs of theorems and extension of theory (25% each; LO 1, 2, 3)

A mark of 60 or more in MATH3320.

with MATH3127.

The main focus of the course will be supervised learning, primarily for classification. The emphasis will be on practical applications of the methodologies that are described, with the R system used for the computations. Attention will be given to

1) Generalisability and predictive accuracy, in the practical contexts in which methods are applied.

2) Low-dimensional visual representation of results, as an aid to diagnosis and insight.

3) Interpretability of model parameters, including potential for misinterpretation.

There will be very limited attention to regression methods with a continuous outcome variable. Relevant statistical theory will mostly be assumed and described rather than derived mathematically. There will be somewhat more attention to the mathematical derivation and description of algorithms.

Topic to be covered include:

• Basic statistical ideas - populations, distributions, samples and random samples
• Classification models and methods - including: linear discriminant analysis; trees; random forests; neural nets; boosting and bagging approaches; support vector machines.
• Linear regression approaches to classification, compared with linear discriminant analysis,
• The training/test approach to assessing accuracy, and cross-validation.
• Strategies in the (common) situation where source and target population differ, typically in time but in other respects also.
• Unsupervised models - kmeans, association rules, hierarchical clustering, model based clusters.
• Low-dimensional views of classification results - distance methods and ordination.
• Strategies for working with large data sets.
• Practical approaches to classification with real life data sets, using different methods to gain different insights into presentation.
• Privacy and security.
• Use of the R system for handling the calculations.

Note: This is an HPC, available as an HPC for students with outstanding results in mathematical and/or computing later year courses. Students will be required to do an indepth presentation of a current research topic, as well as demonstrate the use of advanced data mining techniques on data sets from numerous application areas.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• 3 Assignments (60%; LO 1-5)
• Presentation (30%; LO1-5)
• Commentary on other Presentations (10%; LO 1-5)