MATH3015 - Mathematics of Finance

Description: This course provides an introduction to the theory of stochastic processes and its application in the mathematical finance area.

The course starts with background material on markets, modelling assumptions, types of securities and traders, arbitrage and maximisation of expected utility. Basic tools needed from measure and probability, conditional expectations, independent random variables and modes of convergence are explained. Discrete and continuous time stochastic processes including Markov, Gaussian and diffusion processes are introduced. Some key material on stochastic integration, the theory of martingales, the Ito formula, martingale representations and measure transformations are described. The well-known Black-Scholes option pricing formula based on geometric Brownian motion is derived. Pricing and hedging for standard vanilla options is presented. Hedge simulations are used to illustrate the basic principles of no-arbitrage pricing and risk-neutral valuation. Pricing for some other exotic options such as barrier options are discussed. The course goes on to explore the links between financial mathematics and quantitative finance. Results which show that the transition densities for diffusion processes satisfy certain partial differential equations are presented. The course concludes with treatment of some other quantitative methods including analytic approximations, Monte Carlo techniques, and tree or lattice methods.

Mathematics of Finance provides an accessible but mathematically rigorous introduction to financial mathematics and quantitative finance. The course provides a sound foundation for progress to honours and post-graduate courses in these or related areas.

Note: This is an Honours Pathway Course. It continues the development of sophisticated mathematical techniques and their application begun in MATH3029 or MATH3320.

Pre-Requisites: MATH3029 Probability Modelling with Application

USYD: STAT2011/2911

Probability Modelling and Applications provides a sound foundation to progress to honours and post-graduate courses emphasizing the theory of mathematical finance and stochastic analysis.

The course contains sufficient material for students to feel comfortable with Markov chains, Poisson processes, and Brownian motion, and the conceptual formulation of topics in continuous time finance, insurance and risk management, where these processes are applied. Also the concept of martingales, which is fundamental for understanding the modern option pricing theory of Black and Scholes, is introduced.

Note: This is an HPC. It continues the development of sophisticated mathematical and probabilistic techniques and their application begun in STAT2001(HPC)

MATH3133 Environmental Mathematics

Description: In this course the major model types used to represent environmental systems are studied. Mathematical emphasis on how they are constructed will use the theory of inverse problems while the practical emphasis uses systems methodology. The focus will be on hydrological systems and their basic processes, combined with the constraints imposed by the limitations of real observational data. Case studies and project assessment will cover catchment hydrology, soil physics, subsurface hydrology and stream transport.

Pre-requisites: MATH2405 or MATH2305 Mathematical Methods 1 Honours

Description: This course provides an in depth exposition of the theory of differential equations and vector calculus. Applications will be related to problems mainly from the Physical Sciences.

Topics to be covered include:

Ordinary Differential Equations - Linear and non-linear first order differential equations; second order linear equations; initial and boundary value problems; Green's functions; power series solutions and special functions; systems of first and second order equations; normal modes of oscillation; nonlinear differential equations; stability of solutions; existence and uniqueness of solutions;

Advanced Vector Calculus - Curves and surfaces in three dimensions; parametric representations; curvilinear coordinate systems; Surface and volume integrals; use of Jacobians; gradient, divergence and curl; identities involving vector differential operators; the Laplacian; Green's and Stokes' theorems.

Note: This is an HPC, taught at a level requiring greater conceptual understanding than MATH2305.

MATH3512 Matrix Computations and Optimisation

Description: In this course, students will be introduced to important algorithms and techniques of scientific computing, focussing on the areas of linear algebra and optimisation. The course will present both theoretical and practical aspects of the algorithms. Students who have previously taken the course have come from areas such as science engineering and economics.

Honours Pathway Option:

Students must have completed MATH2405 or MATH2320 or Real Analysis (3rd year) to choose this option. In this option we will expand on the theoretical aspects of the underlying algorithms. Alternative assessment in the assignments and exam will be used to assess these theoretical aspects.

Pre-requisites: MATH2405 or MATH2306 Mathematical Methods 1 Honours

Description : See above

MATH3301 Number Theory and Cryptology

Description: This course is intended for students who want an introduction to elementry number theory, with an application to cryptography. Useful for mathematics engineering and computer science students.

Topics chosen from: the Euclidean algorithm, congruences, prime numbers, highest common factor, prime factorisation, diophantine equations, sums of squares, Chinese remainder theorem, Euler's function, continued fractions, Pell's equation, quadratic residues, quadratic reciprocity, cryptography and RSA codes.

Pre-requisites: MATH2016; or MATH2302; or MATH2303; or MATH2301 with a mark of 60 or better;

General Pre-requisite Description: This course is designed to show some of the interdependence of mathematics and computing, and is designed for students in both computer science and mathematics.

Topics to be covered include:

Foundations - Relations on sets, including equivalence, partial order relations and relational databases; properties of functions, permutations, arithmetic of integers modulo n.

Grammars and Automata - Phrase structure grammars, finite state automata, and the connections between the language accepted by an automaton, regular sets and regular grammars.

Graph Theory - Hamiltonian circuits, vertex colouring and the chromatic polynomial of a graph, planar graphs, applications including the travelling salesperson problem and scheduling problems.

Game Theory - Games of strategy as an application of graph theory, matrix games and solution of matrix games.

MATH3104 Groups and Rings Honours

Description: This course introduces the basic concepts of modern algebra such as groups and rings. The philosophy of this course is that modern algebraic notions play a fundamental role in mathematics itself and in applications to areas such as physics, computer science, economics and engineering. This course emphasizes the application of techniques.

Topics to be covered include:
Group Theory - permutation groups; abstract groups, subgroups, cyclic and dihedral groups; homomorphisms; cosets, Lagrange's Theorem, quotient groups, group actions; Sylow theory.

Ring Theory - rings and fields, polynomial rings, factorisation; homomorphisms, factor rings.

Linear algebra - unitary matrices, Hermitian matrices, canonical forms.
Note: This is an HPC. It emphasises the sophisticated application of deep mathemtical concepts.

Pre-requisites: A mark of 80 or more in MATH2305 and MATH2306 or a mark of 60 or more in MATH2405. Mathematical Honours 1

Description: This course provides an in depth exposition of the theory of differential equations and vector calculus. Applications will be related to problems mainly from the Physical Sciences.

Topics to be covered include:

Ordinary Differential Equations - Linear and non-linear first order differential equations; second order linear equations; initial and boundary value problems; Green's functions; power series solutions and special functions; systems of first and second order equations; normal modes of oscillation; nonlinear differential equations; stability of solutions; existence and uniqueness of solutions;

Advanced Vector Calculus - Curves and surfaces in three dimensions; parametric representations; curvilinear coordinate systems; Surface and volume integrals; use of Jacobians; gradient, divergence and curl; identities involving vector differential operators; the Laplacian; Green's and Stokes' theorems.

Note: This is an HPC, taught at a level requiring greater conceptual understanding than MATH2305.

ASTR3002 Galaxies and Cosmology

Description: Galaxies: classification and dynamics. Luminous matter and dark matter in galaxies. The expanding universe and cosmological models

Pre-requisites: Either one of MATH2305, MATH2405, ENGN2212, MATH2320, MATH2023 - Mathematical Honours 1

Description: see above

MATH3329 Relativity, Black Holes and Cosmology

Description: The theories of special and general relativity are presented with applications to black holes and cosmology. Topics to be covered include the following. Metrics and Riemannian tensors. The calculus of variations and Lagrangians. Spaces and space-times of special and general relativity. Photon and particle orbits. Model universes. The Schwarzschild metric and black holes. Gravitational lensing.

Pre-requisites: see above

Description: see above

MATH3325 Complex Honours Analysis

Description: This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Complex differentiability, conformal mapping; complex integration, Cauchy integral theorems, Taylor series representation, isolated singularities, residue theorem and applications to real integration. Topics chosen from: argument principle, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler.

Note: This is an HPC. It emphasizes mathematical rigour and proof and develops the material from an abstract viewpoint.

Pre-requisites: MATH3320 with 60+ mark. Analysis 2 Honours.

Description: This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered will include:

Topological Spaces - continuity, homeomorphisms, convergence, Hausdorff spaces, compactness, connectedness, path connectedness.

Measure and Integration - Lebesgue outer measure, measurable sets and integration, Lebesgue integral and basic properties, convergence theorems, connection with Riemann integration, Fubini's theorem, approximation theorems for measurable sets, Lusin's theorem, Egorov's theorem, Lp spaces as Banach spaces.

Hilbert Spaces - elementary properties such as Cauchy Schwartz inequality and polarization, nearest point, orthogonal complements, linear operators, Riesz duality, adjoint operator, basic properties or unitary, self adjoint and normal operators, review and discussion of these operators in the complex and real setting, applications to L2 spaces and integral operators, projection operators, orthonormal sets, Bessel's inequality, Fourier expansion, Parseval's equality, applications to Fourier series.

Calculus in Euclidean Space - Inverse and implicit function theorems.

Note: This is an HPC. It emphasises mathematical rigour and proof and develops modern analysis from an abstract viewpoint.

MATH3344 Algebraic Topology Honours

Description: Algebraic topology studies properties of topological spaces and maps between them by associating algebraic invariants (fundamental groups, homology groups, cohomology groups) to each space. This course gives a solid introduction to fundamental ideas and results that are employed nowadays in most areas of mathematics, theoretical physics and computer science. This course aims to understand some fundamental ideas in algebraic topology; to apply discrete, algebraic methods to solve topological problems; to develop some intuition for how algebraic topology relates to concrete topological problems.

Topics to be covered include:

Fundamental group and covering spaces; Brouwer fixed point theorem and Fundamental theorem of algebra; Homology theory and cohomology theory; Jordan-Brouwer separation theorem, Lefschetz fixed theorem; some additional topics (Orientation, Poincare duality, if time permits)

Note: This is an HPC. It builds upon the material of MATH3302 and MATH2322 and emphasises mathematical rigour and proof.

Pre-requisites: MATH3320 Analysis 2 Honours & MATH 2322 Algebra Honours 1

Description: See above for MATH3320. Algebra 1 is a foundational course in Mathematics, introducing some of the key concepts of modern algebra. The course leads on to other areas of algebra such as Galois Theory, Algebraic Topology and Algebraic Geometry. It also provides important tools for other areas such as theoretical computer science, physics and engineering.

Topics to be covered include:

Group Theory - permutation groups; abstract groups, subgroups, cyclic and dihedral groups; homomorphisms; cosets, Lagrange's Theorem, quotient groups; group actions; Sylow theory.

Ring Theory - rings and fields, polynomial rings, factorisation; homomorphisms, factor rings.

Linear algebra - real symmetric matrices and quadratic forms, Hermitian matrices, canonical forms.

Set Theory - cardinality.

Note: This is an HPC. It emphasises mathematical rigour and proof and develops modern algebra from an abstract viewpoint.

MATH3346 Data Mining Honours

Description: Topics to be covered include:
Basic statistical ideas - populations, distributions, samples and random samples
Classification models and methods - including: linear discriminant analysis; trees; random forests; neural nets; boosting and bagging approaches; support vector machines.
Linear regression approaches to classification, compared with linear discriminant analysis,
The training/test approach to assessing accuracy, and cross-validation.
Strategies in the (common) situation where source and target population differ, typically in time but in other respects also.
Unsupervised models - kmeans, association rules, hierarchical clustering, model based clusters.
Low-dimensional views of classification results - distance methods and ordination.
Strategies for working with large data sets.
Practical approaches to classification with real life data sets, using different methods to gain different insights into presentation.
Privacy and security.
Use of the R system for handling the calculations.
Note: This is an HPC, available as an HPC for students with outstanding results in mathematical and/or computing later year courses. Students will be required to do an indepth presentation of a current research topic, as well as demonstrate the use of advanced datamining techniques on data sets from numerous application areas.

Pre-requisites: entry is by invitation only.