This chapter contains descriptions of units in the Pure Mathematics program, arranged by semester. Students who wish to take an advanced unit of study and who have not previously undertaken advanced level work in second year should speak to one of the coordinators and be prepared to devote extra time to the unit to compensate.
It should be noted that these lists are provisional only and that any unit may be withdrawn due to resource constraints.
Topology began as a branch of geometry, but has for some time been firmly established as one of the basic disciplines of modern pure mathematics. In particular, the ideas of topology have shed light on the foundations of analysis (calculus).
Topology is sometimes called "rubber-sheet" geometry, because it studies those properties of figures which remain unchanged if the figures are stretched, twisted, bent or otherwise "elastically" deformed. Such elastic deformations are continuous functions (since points which start out close together end up close together after the deformation). So the ideas involved are really about continuity. The first part of this unit deals with topology from this point of view. We begin with metric spaces, and move on to more general topological spaces, where we can dispense with our usual idea of "distance", and define continuity in terms of open sets.
Topics covered include: open sets, closed sets, homeomorphisms, compactness, connectedness, finite topologies and relative topologies.
The second part of the unit deals with surfaces, and their topological properties. We will discuss some strange surfaces, like the Möbius strip and the Klein bottle, and we will see that, topologically speaking, a donut is the same as a teacup! The unit concludes with the classification theorem for surfaces.
An impressive feature of twentieth century mathematics has been the recognition of the power of abstraction. The algebra that has evolved from this approach now provides a unifying thread which interweaves most areas of mathematics including geometry, analysis, topology and number theory.
This unit is concerned primarily with algebraic systems such as rings and fields, which are generalizations of familiar examples such as polynomials and real numbers. We generalize familiar notions of divisibility, greatest common divisors and primality from the integers to other rings. Homomorphisms, quotient structures and the construction of finite fields are also discussed.
This unit is an introduction to the theory of systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will be, not on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. The aim is to develop a good geometrical intuition into the behaviour of solutions to such systems.
Some background in linear algebra, and familiarity with concepts such as limits and continuity, will be assumed.
The unit is mainly concerned with a general notion of computability, studied by means of Turing machines (simple abstract computers). In particular, we will look at some problems which cannot be solved by any computer. (Note: no experience with computing is required.) In the second part of the unit, the results from the first part are applied to mathematics itself. The conclusion is that there is no systematic way of discovering all mathematical truths.
List of Topics:
Turing machines, computable functions, universal Turing machines,
insolubility of the halting problem.
Recursive and recursively enumerable sets. Effective computability
and Church's thesis. Gödel numbers.
Logical systems, first-order Peano arithmetic, Gödel's
incompleteness theorem.
Cryptography is the branch of mathematics that provides the techniques which enable confidential information to be transmitted over public networks. This unit introduces the student to cryptography, with an emphasis on the cryptographic primitives that are in most common use today. Following a review of classical cryptosystems, modern symmetric cryptosystems (chiefly DES) and non-symmetric cryptosystems (chiefly RSA) will be studied. In the second part of the unit, these cryptographic primitives will be used to construct protocols for realising digital signatures, data integrity, identification, authentication and key distribution. An important feature of the course will be weekly exercises in practical cryptography using the Computer Algebra system Magma. The Code Book by Simon Singh, published in 1999, will give interested students a feel for the type of material covered.
Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory.
Topics covered include: metric spaces, convergence, completeness and the contraction mapping theorem; metric topology, open and closed subsets; topological spaces, subspaces, product spaces; continuous mappings and homeomorphisms; compact spaces; connected spaces; Hausdorff spaces and normal spaces.
In this unit the tools of modern algebra are developed as an introduction to Galois Theory, which deals with the solution of polynomial equations in one variable. The same tools provide an analysis of the classical problem of determining whether certain geometrical constructions, such as the trisection of a given angle, can be performed using only ruler and compasses. The unit begins with the definitions and basic properties of rings, homomorphisms and ideals, continues with an investigation of factorization in principal ideal domains such as the Gaussian integers and and the ring of polynomials over a field, and concludes with a study of algebraic field extensions and their automorphisms.
No specific topics from 1st and 2nd year units are assumed, apart from some linear algebra.
Differential Geometry is an important branch of mathematics in which one uses Calculus to study geometric objects, such as curves, surfaces and higher-dimensional objects. It also has close connections with classical and modern physics. This unit covers elementary properties of curves and surfaces in three-dimensional space, following Do Carmo's book, leading to the celebrated Gauss-Bonnet Theorem. If time allows, we will either introduce the language of differential forms or develop some global theory of differential geometry.
This unit continues the study of functions of a complex variable introduced in the second year units Vector Calculus and Complex Variables (advanced and normal) assuming some knowledge of algebra (for example, that covered in the second year unit MATH 2008 Introduction to Modern Algebra). It will be advantageous for students to also take either Metrics Spaces or Topology if they intend to do this unit.
The unit begins with a review of power series and convergence, and then develops the main approaches to the notion of complex analytic function, namely: power series, complex differentiability, the Cauchy-Riemann equations, the Cauchy Integral Formula and differential forms (Morera's theorem). We shall then consider the Residue theorem, isolated singularities and the groups of holomorphic transformations of the three standard simply-connected domains. Other topics such as ordinary differential equations and modular forms will be included if time permits.
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. This unit of study is an introduction to the subject covering a broad range of theoretical and applied methods. In particular, it covers some elementary methods to solve certain classes of equations. It then covers more theoretical aspect like existence and uniqueness theorems, stability of equilibria and orbits, linearization, hyperbolic critical points and the principle of linearized stability and instability for systems of first order equations. Special topics include the Bendixson negative criterion, limit sets and limit cycles, the Poincaré-Bendixson theorem, Lyapunov functions and Lyapunov stability. Finally, power series solutions lead to an introduction to perturbation methods such as the Lindstedt-Poincaré method. All results and techniques will be illustrated by suitable examples from applications in areas like physics, biomathematics and chemistry.
Over the last 100 years or so, transformations (bijective mappings from the Euclidean plane to itself) have come to play an increasingly important role in geometry. In this unit, various groups of transformations are studied in some detail. Isometries, affine transformations and symmetry groups (including the famous frieze groups) are all discussed. The basic approach is via vectors and matrices, emphasizing the interplay between geometry and linear algebra. Each provides insight into the other. The final section of the course concentrates on transformations of the projective plane. The underlying theme of the unit is the classification and characterisation of the different groups of transformations of both types of plane.
The unit provides a general introduction to the theory of error-correcting codes. After studying general error correcting block codes, we concentrate on linear codes, with the aim of constructing efficient codes which can be practically implemented. We are led to the study of cyclic codes – a special case of linear codes, with nice algebraic properties. The unit concludes with the construction of classes of cyclic codes that are used in modern digital communication systems, including the code used in the compact disc player to correct errors caused by dust and scratches.
The aim of the unit is to present some of the beautiful and practical results which continue to justify and inspire the study of analysis.
The unit includes a review of sequence, series, power series and Fourier series. It introduces the notions of asymptotic and uniform convergence. Among topics studied are the Bernoulli numbers, Bernoulli polynomials, the Euler-Maclaurin summation formula, the Riemann zeta function and Stirling's approximation for factorials.
The unit is an introduction to elementary number theory, with an emphasis on the solution of Diophantine equations (that is, finding integer solutions to such equations as x2+y2 = z2, x2–21y2 = 1. Three main tools are developed: (i) the theory of divisibility and congruence, (ii) rational approximation (continued fractions) and (iii) the law of quadratic reciprocity.
This unit is a general introduction to the ideas and applications of information theory. The basic concept here is that of entropy. This idea of entropy goes back more than a century to the work of Boltzmann in statistical mechanics. Interest in the concept was enormously increased by the work of Claude Shannon in the late 1940's. Shannon showed that entropy was a basic property of any (discrete) probability space, and established a fundamental relation between the entropy of a randomly varying signal and the maximum rate at which the signal could be transmitted through a communication line.
Shannon's theory also has practical importance in the design of codes for data compression, where the entropy of a data source is directly related to the possible degree of compression of the data.
There is another interpretation of entropy in terms of the financial value of information to a gambler. This has implications for the design of gambling strategies and investment portfolios.
The unit will cover both of these areas of application. Prerequisites will be kept to a minimum. Some previous exposure to probability theory is likely to be useful, although the basic concepts will be reviewed in the unit. Otherwise students can consult one of the many introductory texts in this area.
Lecturer: A/Prof D. Taylor
Prerequisite: MATH 3902 (or credit in MATH 3002) and 12 credit
points of intermediate mathematics units.
This unit deals 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, analysis of the structure of finite Abelian groups, and techniques for obtaining canonical forms for matrices.
Students will be assumed to be familiar with the basic concepts of ring theory.
The purpose of this unit is to give an introduction to some modern ideas in the study of nonlinear dynamical systems.
We concentrate largely on one-dimensional discrete systems. The dynamics of the apparently simple systems we study turn out to be remarkably complicated. We show how seemingly elementary nonlinear maps, such as the quadratic map mx(1–x) give rise to fractal sets. This leads into a discussion of concepts like topological conjugacy, symbolic dynamics, chaos theory, the Sarkovskii Theorem and, in particular, bifurcations of maps. We also study how period doubling bifurcations can lead to chaos; homeomorphisms of the circle and the rotation number. A more general discussion of the important topic of bifurcation theory is also given.
Integration is a very useful tool in many areas of mathematics. Lebesgue's theory of integration is the one used in most modern analysis, providing very general and natural conditions under which integrals are defined. The theory is based on measure theory, which is a generalisation of the ideas of area and volume. Measure theory is also the foundation of probability theory, and is important for understanding many different subjects from quantum physics to financial mathematics. In this unit, measure theory is applied to the study of Fourier series and integrals. The first part deals with measure, outer measure, construction of measure and Lebesgue measure. The second part covers measurable functions, integration theory, Fatou's lemma, dominated convergence theorem. The third part deals with product measure, convolution, Fourier transform and Fourier inversion. The additional topics expectation, Radon-Nikodym derivative, and conditional probability may be covered, if time permits.
Public Key Cryptography (PKC) enables two parties to communicate securely over a public communications network, without them having first to exchange a secret key. PKC provides secure communications over the Internet, over mobile phone networks and in many other situations. This course draws on ideas from algebra, number theory and geometry to provide the student with a thorough grounding in the mathematical basis of the most popular PKCs. Specifically, the unit treats PKCs based on the difficulty of integer factorisation (RSA), the discrete logarithm problem in a finite field (Diffie-Hellman, ElGamal) and the discrete logarithm problem in the group of rational points of an elliptic curve over a finite field. Attacks on these cryptosystems will be treated in some depth. \endblurb