# Senior Mathematics and Statistics Handbook

## Pure Mathematics Units of Study

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It should be noted that these lists are provisional only and that any unit may be withdrawn due to resource constraints.

Note that MATH3964 and MATH3067 are offered only in even numbered years, while MATH3068 is offered only in odd numbered years.

## MATH3062 Algebra and Number Theory

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2068 or MATH2988 or MATH2968 recommended but not essential.

Prohibitions: MATH3002, MATH3902, MATH3962, MATH3009

Assessment: One two hour exam (70%), assignments (5%) and quizzes (25%).

The first half of the unit continues the study of elementary number theory, with an emphasis on the solution of Diophantine equations (for example, finding all integer squares which are one more than twice a square). Topics include the Law of Quadratic Reciprocity, representing an integer as the sum of two squares, and continued fractions. The second half of the unit introduces the abstract algebraic concepts which arise naturally in this context: rings, fields, irreducibles, and unique factorization. Polynomial rings, algebraic numbers, and constructible numbers are also discussed.

## MATH3063 Differential Equations and Biomathematics

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2061.

Prohibitions: MATH3003, MATH3923, MATH3020, MATH3920, MATH3963.

Assessment: One two hour exam (75%), assignments (15%) and quizzes (10%).

This unit of study 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 not be 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 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.

## MATH3065 Logic and Foundations

Prerequisite: 6 credit points of Intermediate Mathematics.

Prohibitions: MATH3005

Assessment: One two hour exam (60%), assignments (40%).

This unit is in two halves. The first half provides a working knowledge of the propositional and predicate calculi, discussing techniques of proof, consistency, models and completeness. The second half discusses notions of computability by means of Turing machines (simple abstract computers). (No knowledge of computer programming is assumed.) It is shown that there are some mathematical tasks (such as the halting problem) that cannot be carried out by any Turing machine. Results are applied to first-order Peano arithmetic, culminating in Gödel's Incompleteness Theorem: any statement that includes first-order Peano arithmetic contains true statements that cannot be proved in the system. A brief discussion is given of Zermelo-Fraenkel set theory (a candidate for the foundations of mathematics), which still succumbs to Gödel's Theorem.

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2961 or MATH2962.

Prohibitions: MATH3901, MATH3001

Assessment: One two hour exam (80%), assignments (20%).

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, Applications include the implicit function theorem, chaotic dynamical systems and an introduction to Hilbert spaces and abstract Fourier series.

## MATH3962 Rings, Fields and Galois Theory (Advanced)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2961.

Recommended prior study: MATH2968.

Prohibitions: MATH3902, MATH3002, MATH3062

Assessment: One two hour exam (70%), assignments (20%) and tutorial participation (10%).

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.

## MATH3963 Differential Equations and Biomathematics (Advanced)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2961.

Prohibitions: MATH3003, MATH3923, MATH3020, MATH3920, MATH3063.

Assessment: One two hour exam (60%), assignments (20%) and quizzes (20%).

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.

## Pure Mathematics – Semester 2 Units

Note that MATH3067 (Information & Coding Theory) and MATH3964 (Complex Analysis with Applications (Advanced)) are offered only in even numbered years, while MATH3068 (Analysis) is offered only in odd numbered years.

## MATH3061 Geometry and Topology

Prerequisite: 12 credit points of Intermediate Mathematics.

Prohibitions: MATH3006, MATH3001.

Assessment: One two hour exam (65%), quizzes (20%), assignments (15%).

The aim of the unit is to expand visual/geometric ways of thinking. The geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasizing the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). The classification of surfaces is given in several equivalent forms. The problem of colouring maps on surfaces is interpreted via graphs. The main geometric fact about knots is that every knot bounds a surface in 3-space. This is proved by a simple direct construction, and is then used to show that every knot is a sum of prime knots.

## MATH3067 Information and Coding Theory

Only offered in even numbered years.

Prerequisite: 12 credit points of Intermediate Mathematics.

Prohibitions: MATH3007, MATH3010

Assessment: One two hour exam (80%), assignments (10%) and tutorial participation (10%).

The related theories of information and coding provide the basis for reliable and efficient storage and transmission of digital data, including techniques for data compression, digital broadcasting and broadband internet connectivity. The first part of this unit is a general introduction to the ideas and applications of information theory, where the basic concept is that of entropy. This gives a theoretical measure of how much data can be compressed for storage or transmission. Information theory also addresses the important practical problem of making data immune to partial loss caused by transmission noise or physical damage to storage media. This leads to the second part of the unit, which deals with the theory of error-correcting codes. We develop the algebra behind the theory of linear and cyclic codes used in modern digital communication systems such as compact disk players and digital television.

## MATH3068 Analysis

Only offered in odd numbered years.

Prerequisite: 12 credit points of Intermediate Mathematics.

Prohibitions: MATH3008, MATH2007, MATH2907, MATH2962

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. The aim of the unit is to present enduring beautiful and practical results that continue to justify and inspire the study of analysis. This course will be useful not just to students of mathematics but also to engineers and scientists, and to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit has three parts: the foundations of calculus, the theory of Fourier series, and complex analysis.

The first part starts with a study of the limiting behaviour of sequences and series of numbers and of functions, and the relationship between limits, differentiation and integration. This is followed by a discussion of the construction and properties of elementary functions like the sine functions and the exponential functions. As a beautiful application we shall study study the Euler MacLaurin formula, a method of summation using Bernoulli polynomials.

In the second part we investigate Fourier series; these provide examples of infinite series of functions, as studied in the first part. The theory of Fourier series is an important tool in the study of periodic phenomena, such as wave motion. The theory is studied in detail, with proofs given for some of the most famous theorems, such as Dirichlet's theorem on pointwise convergence, Bessel's inequalities, Fejer's theorem and Parseval's identity. We shall use Fourier series to calculate some special values of the Riemann zeta function, and also to solve a boundary value problem.

The third part begins with the definition of complex numbers and functions of a complex variable. We shall study the topology of the complex plane, and also introduce general notions of topology. We shall study the basic properties of differentiation with respect to a complex variable, theory of power series, exponential functions and trigonometric functions of a complex variable, complex line integrals, Cauchy's theorem, Cauchy's integral formula, residues and calculations of integrals, Moreas' theorem, Weierstrass' theorem and the famous Riemann zeta function.

#### Recommended reference books

• A Friendly Introduction to Analysis, W. A. J. Kosmala, Pearson Prentice Hall International Edition.
• Fourier series and boundary value problems, by Ruel V. Churchill, James Ward Brown. Publisher, New York: McGraw-Hill.
• Complex variables and applications by Ruel V. Churchill, James Ward Brown. Publisher, New York: McGraw-Hill.

Assessment: One two hour exam, assignments and quizzes (100%).

## MATH3966 Modules and Group Representations (Advanced)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH3962.

Prohibitions: MATH3907, MATH3906

Assessment: One two hour exam (70%), assignments (20%) and tutorial participation (10%).

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.

Prerequisite: 12 credit points of Intermediate Mathematics, including MATH2961

Assumed knowledge: at least 6 credit points of advanced level senior or intermediate mathematics.

Prohibitions: MATH3903

Assessment: One two hour exam (75%), assignments (20%) and tutorial participation (5%).

This unit is an introduction to Differential Geometry, using ideas from calculus of several variables to develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. Differential geometry also plays an important part in both classical and modern theoretical physics. The initial aim is to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to present the calculus of differential forms as the natural setting for the key ideas of vector calculus, along with some applications.

## MATH3964 Complex Analysis with Applications (Advanced)

Only offered in even numbered years.

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2962.

Prohibitions: MATH3904, MATH3915.

Assessment: One two hour exam (70%), assignments (30%).

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.

## MATH3969 Measure Theory and Fourier Analysis (Advanced)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: at least 6 credit points of advanced level senior or intermediate mathematics.

Prohibitions: MATH3909

Assessment: One two hour exam (60%), assignments (10%) and quizzes (30%).

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.