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It should be noted that these lists are provisional only and that any unit may be withdrawn due to resource constraints.

## MATH3063 Nonlinear ODEs with Applications

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2061.

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

Lecturer: Peter Kim .

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

This unit of study is an introduction to qualitative methods for 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 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.

## MATH3066 Algebra and Logic

Prerequisite: 6 credit points of Intermediate Mathematics.

Prohibitions: MATH3065, MATH3062

Lecturer: Oded Yacobi .

Assessment: One 2 hour exam (60%), two assignments (15% each), peer review of each assignment (5% each).

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times. Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Applications are presented, in particular the unsolvability of the celebrated classical construction problems of the Greeks. Quotient rings are introduced, culminating in a construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences. Axiomatics are placed in the context of reasoning within first order logic and set theory. The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including sketches of proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem and the undecidability of First Order Logic.

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2961 or MATH2962.

Prohibitions: MATH3901, MATH3001

Lecturer: Florica Cîrstea .

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

Lecturer: James Parkinson .

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 Nonlinear ODEs with Applications (Advanced)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2961, MATH2962

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

Lecturer: Martin Wechselberger .

Assessment: One two hour exam, class tests, assignments. (Weighting to be advised.)

This course serves as an introduction to the more modern theory of differential equations and dynamical systems. The emphasis is on obtaining a qualitative understanding of properties of a system. This course intertwines the study of the theory of ODEs with applications to systems modelling various phenomena. The more theoretical part includes a study of existence and uniqueness theorems for linear and nonlinear equations, asymptotic behaviour for nonlinear ODEs and elementary bifurcation theory. The applications in this unit will be drawn from mechanical, biological and chemical models, and other equations and systems from applied mathematics.

## MATH3061 Geometry and Topology

Prerequisite: 12 credit points of Intermediate Mathematics.

Prohibitions: MATH3006, MATH3001.

Lecturer: Boris Lishak and Andrew Mathas .

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.

## MATH3068 Analysis

Prerequisite: 12 credit points of Intermediate Mathematics.

Prohibitions: MATH3008, MATH2007, MATH2907, MATH2962

Lecturer: Daniel Hauer .

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

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.

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

Lecturer: Laurentiu Paunescu .

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

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.

## 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

Lecturer: Daniel Daners .

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.