MATH3061 Geometry and Topology Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures andone 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions:MATH3001, MATH3006 Assessment: One 2 hour exam, tutorial tests, assignments. 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, emphasising 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 proven by a simple direct construction, and this fact is used to show that every knot is a sum of prime knots. MATH3961 Metric Spaces (Advanced) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics units Prohibitions: MATH3901, MATH3001 Assumed knowledge: MATH2961 or MATH2962 Assessment: 2 hour exam,assignments, quizzes 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. MATH3062 Algebra and Number Theory Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3962, MATH3902, MATH3002, MATH3009 Assessment: One 2 hour exam, quizzes and assignments Note: Students are advised to take MATH(2068 or 2968) before attempting this unit. 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 factorisation. Polynomial rings, algebraic numbers and constructible numbers are also discussed. Textbooks Walters, RFC. Number Theory: an Introduction. Carslaw Publications. Niven, I. Zuckerman, HS. Montgomery, HL. An Introduction to the Theory of Numbers. Wiley. Herstein, IN. Topics in Algebra. Blaisdell. Childs, LN. A Concrete Introduction to Higher Algebra. Springer. MATH3962 Rings, Fields and Galois Theory (Adv) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3062, MATH3902, MATH3002 Assumed knowledge: MATH2961 Assessment: One 2 hour exam, assignments and quizzes Note: Students are advised to take MATH2968 before attempting this unit . 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. Textbooks I.H. Herstein, Abstract algebra, second edition, MacMillian, 1990. S. Lang Algebra, third edition, Springer-Verlag, Graduate texts in Mathematics, 2002. I.N. Stewart, Galois Theory, Chapman and Hall, 1973. MATH3063 Differential Equations and Biomaths Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3020, MATH3920, MATH3003, MATH3923, MATH3963 Assumed knowledge: MATH2061 Assessment: One 2 hour exam, assignments, quizzes 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. MATH3963 Differential Equations & Biomaths (Adv) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3020, MATH3920, MATH3003, MATH3923, MATH3063 Assumed knowledge: MATH2961 Assessment: One 2 hour exam, assignments, quizzes 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, linearisation, and hyperbolic critical points, and omega limit sets. MATH3964 Complex Analysis with Applications (Adv) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3904, MATH3915 Assumed knowledge: MATH2962 Assessment: One 2 hour exam, assignments and quizzes 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 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. MATH3065 Logic and Foundations Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 6 credit points of Intermediate Mathematics Prohibitions: MATH3005 Assessment: One 2 hour exam, tutorial tests, assignments. 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 Godel'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 Godel's Theorem. MATH3966 Modules and Group Representations (Adv) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3906, MATH3907 Assumed knowledge: MATH3962 Assessment: One 2 hour exam, assignments and quizzes 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. MATH3067 Information and Coding Theory Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions:MATH3007, MATH3010 Assessment: One 2 hour exam, tutorial tests, assignments. 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. MATH3969 Measure Theory & Fourier Analysis (Adv) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorials per week. Prerequisites: 12 credit points Intermediate Mathematics Prohibitions: MATH3909 Assumed knowledge: At least 6 credit points of Advanced Mathematics units of study at Intermediate or Senior level Assessment: One 2 hour exam, assignments, quizzes 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. MATH3974 Fluid Dynamics (Advanced) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with average grade of at least Credit Prohibitions: MATH3914 Assumed knowledge: MATH2961, MATH2965 Assessment: One 2 hour exam 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. MATH3075 Financial Mathematics Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions:MATH3975, MATH 3015, MATH3933 Assessment: Two class quizzes and one 2 hour exam 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. MATH3975 Financial Mathematics (Advanced) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions:MATH3933, MATH3015, MATH3075 Assessment: Two class quizzes and one 2 hour exam 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. Students enrolled in this unit at the Advanced level will be expected to undertake more challenging assessment tasks. The lectures in the Advanced unit are held concurrently with those of the corresponding Normal unit. MATH3076 Mathematical Computing Credit points: 6 Teacher/Coordinator: Dr D J Ivers Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour laboratory per week. Prerequisites: 12 credit points of Intermediate Mathematics and one of MATH(1001 or 1003 or 1901 or 1903 or 1906 or 1907) Prohibitions: MATH3976, MATH3016, MATH3916 Assessment: One 2 hour exam, assignments, quizzes This unit of study provides an introduction to Fortran 95 programming and numerical methods. Topics covered include computer arithmetic and computational errors, systems of linear equations, interpolation and approximation, solution of nonlinear equations, quadrature, initial value problems for ordinary differential equations and boundary value problems. MATH3976 Mathematical Computing (Advanced) Credit points: 6 Teacher/Coordinator: Dr D J Ivers Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics and one of MATH(1903 or 1907) or Credit in MATH1003 Prohibitions: MATH3076, MATH3016, MATH3916 Assessment: One 2 hour exam, assignments, quizzes See entry for MATH3076 Mathematical Computing. MATH3977 Lagrangian & Hamiltonian Dynamics (Adv) Credit points:6 Teacher/Coordinator: Dr. Leon Poladian Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions: MATH2904, MATH2004, MATH3917 Assessment: One 2 hour exam and assignments and/or quizzes This unit provides a comprehensive treatment of dynamical systems using the mathematically sophisticated framework of Lagrange and Hamilton.This formulation of classical mechanics generalizes elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamical theory from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear very simple. The unit will also explore connections between geometry and different physical theories beyond classical mechanics. Students will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. Hamilton-Jacobi theory will be used to elegantly solve problems ranging from geodesics (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity. MATH3078 PDEs and Waves Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3978, MATH3018, MATH3921 Assumed knowledge: MATH(2061/2961) and MATH(2065/2965) Assessment: One 2 hour exam, one lecture quiz 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. Textbooks Powers, DL. Boundary Value Problems. Harcourt-Brace 4th Edition. 1999. MATH3978 PDEs and Waves (Advanced) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions:MATH3078, MATH3018, MATH3921 Assumed knowledge: MATH(2061/2961) and MATH(2065/2965) Assessment: One 2 hour exam, one lecture quiz As for MATH3078 PDEs & Waves but with more advanced problem solving and assessment tasks. Some additional topics may be included. Textbooks Powers, DL. Boundary Value Problems. Harcourt-Brace 4th Edition. MATH3061 Geometry and Topology Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions:MATH3001, MATH3006 Assessment: One 2 hour exam, tutorial tests, assignments. 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, emphasising 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 proven by a simple direct construction, and this fact is used to show that every knot is a sum of prime knots. MATH3961 Metric Spaces (Advanced) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics units Prohibitions: MATH3901, MATH3001 Assumed knowledge: MATH2961 or MATH2962 Assessment: 2 hour exam, assignments, quizzes 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. MATH3062 Algebra and Number Theory Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3962, MATH3902, MATH3002, MATH3009 Assessment: One 2 hour exam, quizzes and assignments Note: Students are advised to take MATH(2068 or 2968) before attempting this unit. 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 factorisation. Polynomial rings, algebraic numbers and constructible numbers are also discussed. Textbooks Walters, RFC. Number Theory: an Introduction. Carslaw Publications. Niven, I. Zuckerman, HS. Montgomery, HL. An Introduction to the Theory of Numbers. Wiley. Herstein, IN. Topics in Algebra. Blaisdell. Childs, LN. A Concrete Introduction to Higher Algebra. Springer. MATH3962 Rings, Fields and Galois Theory (Adv) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3062, MATH3902, MATH3002 Assumed knowledge: MATH2961 Assessment: One 2 hour exam, assignments and quizzes Note: Students are advised to take MATH2968 before attempting this unit. 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. Textbooks I.H. Herstein, Abstract algebra, second edition, MacMillian, 1990. S. Lang Algebra, third edition, Springer-Verlag, Graduate texts in Mathematics, 2002. I.N. Stewart, Galois Theory, Chapman and Hall, 1973. MATH3063 Differential Equations and Biomaths Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3020, MATH3920, MATH3003, MATH3923, MATH3963 Assumed knowledge: MATH2061 Assessment: One 2 hour exam, assignments, quizzes 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. MATH3963 Differential Equations & Biomaths (Adv) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3020, MATH3920, MATH3003, MATH3923, MATH3063 Assumed knowledge: MATH2961 Assessment: One 2 hour exam, assignments, quizzes 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, linearisation, and hyperbolic critical points, and omega limit sets. MATH3964 Complex Analysis with Applications (Adv) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3904, MATH3915 Assumed knowledge: MATH2962 Assessment: One 2 hour exam, assignments and quizzes 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 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. MATH3065 Logic and Foundations Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 6 credit points of Intermediate Mathematics Prohibitions: MATH3005 Assessment: One 2 hour exam, tutorial tests, assignments. 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 Godel'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 Godel's Theorem. MATH3966 Modules and Group Representations (Adv) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3906, MATH3907 Assumed knowledge: MATH3962 Assessment: One 2 hour exam, assignments and quizzes 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. MATH3067 Information and Coding Theory Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions:MATH3007, MATH3010 Assessment: One 2 hour exam, tutorial tests, assignments. 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. MATH3969 Measure Theory & Fourier Analysis (Adv) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorials per week. Prerequisites: 12 credit points Intermediate Mathematics Prohibitions: MATH3909 Assumed knowledge: At least 6 credit points of Advanced Mathematics units of study at Intermediate or Senior level Assessment: One 2 hour exam, assignments, quizzes 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. MATH3974 Fluid Dynamics (Advanced) Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with average grade of at least Credit Prohibitions: MATH3914 Assumed knowledge: MATH2961, MATH2965 Assessment: One 2 hour exam 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. MATH3075 Financial Mathematics Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions:MATH3975, MATH 3015, MATH3933 Assessment: Two class quizzes and one 2 hour exam 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. MATH3975 Financial Mathematics (Advanced) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions:MATH3933, MATH3015, MATH3075 Assessment: Two class quizzes and one 2 hour exam 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. Students enrolled in this unit at the Advanced level will be expected to undertake more challenging assessment tasks. The lectures in the Advanced unit are held concurrently with those of the corresponding Normal unit. MATH3076 Mathematical Computing Credit points: 6 Teacher/Coordinator: Dr D J Ivers Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour laboratory per week. Prerequisites: 12 credit points of Intermediate Mathematics and one of MATH(1001 or 1003 or 1901 or 1903 or 1906 or 1907) Prohibitions: MATH3976, MATH3016, MATH3916 Assessment: One 2 hour exam, assignments, quizzes This unit of study provides an introduction to Fortran 95 programming and numerical methods. Topics covered include computer arithmetic and computational errors, systems of linear equations, interpolation and approximation, solution of nonlinear equations, quadrature, initial value problems for ordinary differential equations and boundary value problems. MATH3976 Mathematical Computing (Advanced) Credit points: 6 Teacher/Coordinator: Dr D J Ivers Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics and one of MATH(1903 or 1907) or Credit in MATH1003 Prohibitions: MATH3076, MATH3016, MATH3916 Assessment: One 2 hour exam, assignments, quizzes See entry for MATH3076 Mathematical Computing. MATH3977 Lagrangian & Hamiltonian Dynamics (Adv) Credit points:6 Teacher/Coordinator: Dr. Leon Poladian Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions: MATH2904, MATH2004, MATH3917 Assessment: One 2 hour exam and assignments and/or quizzes This unit provides a comprehensive treatment of dynamical systems using the mathematically sophisticated framework of Lagrange and Hamilton.This formulation of classical mechanics generalizes elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamical theory from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear very simple. The unit will also explore connections between geometry and different physical theories beyond classical mechanics. Students will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. Hamilton-Jacobi theory will be used to elegantly solve problems ranging from geodesics (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity. MATH3078 PDEs and Waves Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3978, MATH3018, MATH3921 Assumed knowledge: MATH(2061/2961) and MATH(2065/2965) Assessment: One 2 hour exam, one lecture quiz 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. Textbooks Powers, DL. Boundary Value Problems. Harcourt-Brace 4th Edition. 1999. MATH3978 PDEs and Waves (Advanced) Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions:MATH3078, MATH3018, MATH3921 Assumed knowledge: MATH(2061/2961) and MATH(2065/2965) Assessment: One 2 hour exam, one lecture quiz As for MATH3078 PDEs & Waves but with more advanced problem solving and assessment tasks. Some additional topics may be included. Textbooks Powers, DL. Boundary Value Problems. Harcourt-Brace 4th Edition. 1999. |