AMH2 Modern Asymptotics and Peerturbation Theory

Description: Differential equations model most natural phenomena we know. Yet their so-
lutions can be notoriously difficult to understand. In place of "exact"solutions,
a rich array of asymptotic methods have been developed to qualitatively un-
derstand the solutions. These are based on either the intrinsic variables or
an external parameter in the problem being large or small. The field is vast
and ranges from the fundamental theory of asymptotic expansions and pertur-
bation methods, developed by Poincare for the study of the solar system, to
modern advanced techniques which deal with cases where conventional asymp-
totics fails. This course will include asymptotics of boundary layers, WKB
and multiscale methods and modern techniques developed to model dendritic
growth (such as snowflakes), fluid flow (existence of solitary waves), and the
onset of chaos. The only background needed is the basic theory of differential
equations and complex analysis.

Pre-requisites: Available to Honours students only

AMH1 Computational Projects in applied Mathematics

Description: Lectures will be given on five projects over the semester, from which the
students will select three for detailed numerical study. The areas covered will
be relatively diverse in nature; for example, some of the following will be
included:
- Symplectic integration for Hamiltonian systems
- Data analysis using singular value decomposition
- Modelling the spread of disease
- Flux expulsion in MHD (magnetohydrodynamics)
- Solving PDEs with FFTs (fast fourier transforms)
- Models involving neural networks
- Numerical solution of stochastic ODEs
Students will be assessed by their reports on their chosen three projects. The
first project will be compulsory, so that the practical work can get underway
quickly. Following this there will be two groups of two projects, and students
will choose to do one project from each of these two groups. Each group of
projects will be introduced by a series of lectures, after which students will
select a project and work on it, in the following laboratory sessions and in their
own time. The programming language used will depend on the nature of the
individual projects and the programming experience of students attending.

Pre-requisites: Available to Honours students only

AMH8 Mathematical Physiology

Description: Physiological rhythms are central for life. Prominent examples are the beating
of the heart, the activity of neurons, or the release of hormones regulating
growth and metabolism. All these rhythms have in common, that they evolve
on at least two different time scales, i.e. there exist a quasi steady state of the
system on a slow time scale (e.g. the resting state of the heart) interspersed
by a dramatic change of the system on a fast time scale (e.g. the heartbeat
itself). Mathematical models of such systems are called slow/fast systems or
multiple scales problems.
In this unit we introduce a powerful mathematical technique suitable for
analysing such multiple scales problems called geometric singular perturba-
tion theory. This perturbation theory is based on the fact that the system
under study has a singular perturbation parameter and classical asymptotic
analysis breaks down. The method is very powerful and is based on dynamical
systems techniques like bifurcation theory and invariant manifold theory. We
will develop the basic mathematical tools to analyse physiological problems.
The class of physiological problems we study is electrical signaling in excitable
cells. We analyse the famous Hodgkin-Huxley model of the squid giant axon.
We use reduction techniques introduced by FitzHugh to obtain a qualitative
model of the neuron which can be fully analysed by geometric singular per-
turbation theory. This analysis will demonstrate how neurons can fire action
potentials and therefore communicate information. Another problem of inter-
est is electrical signaling in pancreatic cells which leads to secretion of insulin
due to bursting electrical activity.
The only background needed is the basic theory of differential equations and
bifurcation analysis (introduced in e.g. MATH3963).

Pre-requisites: MATH3963 Differential Equations and Biomathematics (Advanced)

Description: The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods of different nature. The theory has many applications and stimulates new developments in almost all areas of mathematics. The applications in this unit will be drawn from predator-prey systems, transmission of diseases, chemical reactions, beating of the heart and other equations and systems from mathematical biology. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, stability analysis, linearization, and hyperbolic critical points, and omega limit sets.

PMH7 Representations of the Symetric Groups

Description: Classical families of symmetric polynomials and relations between them,
Schur polynomials, Jacobi-Trudi determinants, Gessel-Viennot method;
Littlewood-Richardson coefficients, plactic monoid, combinatorial algorithms;
Double Schur polynomials, their vanishing properties, new classes of symmetric
functions.

Pre-Requistes: Available to Honours students only

PMH8 Partial Differential Equations

Description: The course is an introduction to the modern theory of partial differential
equations. The major types of equations will be studied including elliptic,
parabolic and if time permits also hyperbolic equations. We will look at
weak solutions, maximum principles, existence and uniqueness of solutions
to linear and non-linear equations. Assumed knowledge is the functional
analysis from first semester and some knowledge of the Lebesgue integral
and Fourier Transforms.

Pre-requistes: First Semester Honours - Functional Analysis

MSH5 Advance Time Series Analysis

Description: Course description: This is an advanced course in time series analysis and
forecasting.
Assumed knowledge: STAT 3011 Stochastic Processes and Time Series.
Contents: Review of linear time series models. Spectral theory of time
series; spectral density of an ARMA process; inference for the spectrum of
a stationary process. Spectral analysis in practice. Bivariate time series
analysis.

Pre-requistes: Assumed knowledge is STAT 3011 Stochastic Processes and Time Series

Description: Section I of this course will introduce the fundamental concepts of applied stochastic processes and Markov chains used in financial mathematics, mathematical statistics, applied mathematics and physics. Section II of the course establishes some methods of modeling and analysing situations which depend on time. Fitting ARMA models for certain time series are considered from both theoretical and practical points of view. Throughout the course we will use the S-PLUS (or R) statistical packages to give analyses and graphical displays.

MSH6 Aysmptotics and Extreme Value Theory

Description: This course will cover theory of stochastic di erential equations. We will
discuss properties of Brownian motion, geometric Brownian motion, de ne
stochastic integrals and solve SDEs. Applications to mathematical nance,
biology and engineering will be given.

Pre-requistes: Assumed knowledge: STAT 3911 Stochastic Processes and Time Series
(Advanced).

Description: This is an Advanced version of STAT3011. There will be 3 lectures in common with STAT3011. In addition to STAT3011 material, theory on branching processes and birth and death processes will be covered. There will be more advanced tutorial and assessment work associated with this unit.

MSH7 Applied Probability & Stochastic DE's

Description: This course has two sections. Section 1 covers applications of extreme
value theory (eg. weather, insurance and nance), extremes of iid sequences
(max-stable distributions, the extremal types theorem, domains
of attraction), exceedances (Poisson behaviour, generalised Pareto approximations),
extremes of dependent Gaussian sequences and extremes in
continuous time.
Section 2 deals with asymptotics: The -method, transformations, functional
statistics and the in
uence function, moments, M-estimates, robust
statistics.

Pre-requistes: Available to Honours students only

MATH3075/3975 Financial Mathematics (Normal and Advanced)

Description: This unit is an introduction to the mathematical theory of modern finance. Topics include: notion of arbitrage, pricing riskless securities, risky securities, utility theory, fundamental theorems of asset pricing, complete markets, introduction to options, binomial option pricing model, discrete random walks, Brownian motion, derivation of the Black-Scholes option pricing model, extensions and introduction to pricing exotic options, credit derivatives. A strong background in mathematical statistics and partial differential equations is an advantage, but is not essential. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. The lectures in the Normal unit are held concurrently with those of the corresponding Advanced unit. Note that students enrolled in MATH3075 and those enrolled in the advanced level unit MATH3975 attend the same lectures, but the assessment tasks for MATH3975 are more challenging than those for MATH3075.

Pre-requistes: 12 credit point of intermediate maths

MATH3078/3978 Partial Differential Equations and Waves (Normal and Advanced)

Description: This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics - the wave equation, the diffusion (heat) equation and Laplace's equation - are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of sound waves and water waves. MATH3978 As for MATH3078 PDEs & Waves but with more advanced problem solving and assessment tasks. Some additional topics may be included.

Pre-requistes: 12 credit point of intermediate maths

MATH3964 Complex Analysis with Applications(Advanced) (Not offered in 2009)

Description: This unit continues the study of functions of a complex variable and their applications introduced in the second year unit Real and Complex Analysis (MATH2962). It is aimed at highlighting certain topics from analytic function theory and the analytic theory of differential equations that have intrinsic beauty and wide applications. This part of the analysis of functions of a complex variable will form a very important background for students in applied and pure mathematics, physics, chemistry and engineering.

The course will begin with a revision of properties of holomorphic functions and Cauchy's theorem with added topics not covered in the second year course. This will be followed by meromorphic functions, entire functions, harmonic functions, elliptic functions, elliptic integrals, analytic differential equations, hypergeometric functions. The rest of the course will consist of selected topics from Greens functions, complex differential forms and Riemann surfaces.

Pre-requistes: MATH2962 Real and Complex Analysis

Description: Analysis is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. Starting off with an axiomatic description of the real number system, this first course in analysis concentrates on the limiting behaviour of infinite sequences and series on the real line and the complex plane. These concepts are then applied to sequences and series of functions, looking at point-wise and uniform convergence. Particular attention is given to power series leading into the theory of analytic functions and complex analysis. Topics in complex analysis include elementary functions on the complex plane, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

MATH3974 Fluid Dynamics (Advanced)

Description: This unit of study provides an introduction to fluid dynamics, starting with a description of the governing equations and the simplifications gained by using stream functions or potentials. It develops elementary theorems and tools, including Bernoulli's equation, the role of vorticity, the vorticity equation, Kelvin's circulation theorem, Helmholtz's theorem, and an introduction to the use of tensors. Topics covered include viscous flows, lubrication theory, boundary layers, potential theory, and complex variable methods for 2-D airfoils. The unit concludes with an introduction to hydrodynamic stability theory and the transition to turbulent flow.

Pre-requistes: MATH2961 Linear algebra & Vector Calculus & MATH2965 Introduction to Partial Differential Equations

Description: LA: This unit is an advanced version of MATH2061, with more emphasis on the underlying concepts and on mathematical rigour. Topics from linear algebra focus on the theory of vector spaces and linear transformations.
The connection between matrices and linear transformations is studied in detail. Determinants, introduced in first year, are revised and investigated further, as are eigenvalues and eigenvectors. The calculus component of the unit includes local maxima and minima, Lagrange multipliers, the inverse function theorem and Jacobians.
There is an informal treatment of multiple integrals: double integrals, change of variables, triple integrals, line and surface integrals, Green's theorem and Stokes' theorem.

PDE: This unit of study is essentially an Advanced version of MATH2065, the emphasis being on solutions of differential equations in applied mathematics. The theory of ordinary differential equations is developed for second order linear equations, including series solutions, special functions and Laplace transforms. Some use is made of computer programs such as Mathematica. Methods for PDEs (partial differential equations) and boundary-value problems include separation of variables, Fourier series and Fourier transforms.

MATH3966 Modules & Group Representations (Advanced)

Description: This unit deals first with generalized linear algebra, in which the field of scalars is replaced by an integral domain. In particular we investigate the structure of modules, which are the analogues of vector spaces in this setting, and which are of fundamental importance in modern pure mathematics. Applications of the theory include the solution over the integers of simultaneous equations with integer coefficients and analysis of the structure of finite abelian groups.

In the second half of this unit we focus on linear representations of groups. A group occurs naturally in many contexts as a symmetry group of a set or space. Representation theory provides techniques for analysing these symmetries. The component will deals with the decomposition of representation into simple constituents, the remarkable theory of characters, and orthogonality relations which these characters satisfy.

Pre-requistes: 12 credit point of intermediate maths. MATH3962 Rings, Fields and Galois Theory (Advanced)(assumed knowledge)

Description: This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize any polynomial into a product of linear factors by working over a "large enough" field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalizes the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions.

MATH3969 Measure Theory and Fourier Analysis (Advanced)

Description: Measure theory is the study of such fundamental ideas as length, area, volume, arc length and surface area. It is the basis for the integration theory used in advanced mathematics since it was developed by Henri Lebesgue in about 1900. Moreover, it is the basis for modern probability theory. The course starts by setting up measure theory and integration, establishing important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. This is then applied to Fourier Analysis, and results such as the Inversion Formula and Plancherel's Theorem are derived. Probability Theory is then discussed, with topics including independence, conditional probabilities, and the Law of Large Numbers.

Pre-requistes: 12 credit points of intermediate maths

STAT3013/3913 Statistical Inference (Normal/Advanced)

Description: Normal: In this course we will study basic topics in modern statistical inference. This will include traditional concepts of mathematical statistics: likelihood estimation, method of moments, properties of estimators, exponential families, decision-theory approach to hypothesis testing, likelihood ratio test as well as more recent approaches such as Bayes estimation, Empirical Bayes and nonparametric estimation. During the computer classes (using R software package) we will illustrate the various estimation techniques and give an introduction to computationally intensive methods like Monte Carlo, Gibbs sampling and EM-algorithm.

Advanced: This unit is essentially an Advanced version of STAT3013, with emphasis on the mathematical techniques underlying statistical inference together with proofs based on distribution theory. There will be 3 lectures per week in common with some material required only in this advanced course and some advanced material given in a separate advanced tutorial together with more advanced assessment work.

Pre-requistes: STAT2012 or STAT2912 Statistical Tests (Normal or Advanced)or STAT2003 or STAT2903 Statistical Models (Normal or Advanced)

Description: ST: This unit provides an introduction to the standard methods of statistical analysis of data: Tests of hypotheses and confidence intervals, including t-tests, analysis of variance, regression - least squares and robust methods, power of tests, non-parametric tests, non-parametric smoothing, tests for count data, goodness of fit, contingency tables. Graphical methods and diagnostic methods are used throughout with all analyses discussed in the context of computation with real data using an interactive statistical package.

SM: This unit provides an introduction to univariate techniques in data analysis and the most common statistical distributions that are used to model patterns of variability. Common discrete random models like the binomial, Poisson and geometric and continuous models including the normal and exponential will be studied. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit will have weekly computer classes where candidates will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

STAT3014/3914 Applied Statistics (Normal & Advanced)

Description: Normal: This unit has three distinct but related components: Multivariate analysis; sampling and surveys; and generalised linear models. The first component deals with multivariate data covering simple data reduction techniques like principal components analysis and core multivariate tests including Hotelling's T^2, Mahalanobis' distance and Multivariate Analysis of Variance (MANOVA). The sampling section includes sampling without replacement, stratified sampling, ratio estimation, and cluster sampling. The final section looks at the analysis of categorical data via generalized linear models. Logistic regression and log-linear models will be looked at in some detail along with special techniques for analyzing discrete data with special structure.

Advanced: This unit is an Advanced version of STAT3014. There will be 3 lectures per week in common with STAT3014. The unit will have extra lectures focusing on multivariate distribution theory developing results for the multivariate normal, partial correlation, the Wishart distribution and Hotellling's T2. There will also be more advanced tutorial and assessment work associated with this unit.

Pre-requisites: Normal- STAT(2012 or 2912 or 2004). Advanced - STAT2012 or STAT2912 or STAT2004.

Description: Normal: See above.

Advanced: This course will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using classical linear methods, together with concepts of collection of data and design of experiments. First we will consider linear models and regression methods with diagnostics for checking appropriateness of models. We will look briefly at robust regression methods here. Then we will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course we will use the R statistical package to give analyses and graphical displays.