Mathematics of Finance MATH3015 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. Requisite Statement: MATH3029 OR MATH3320 Fractional Geometry and Chaotic Dynamics MATH3062 Description: This course provides a mathematical introduction to fractal geometry and nonlinear dynamics with focus on biological modelling and the geometry of real world images. What do models for the structure of ferns and complicated behaviour of the weather have in common? Both involve the iterative application of functions that map from a space to itself. Both can be treated from the classical geometrical point of view of Felix Klein. Invariants, such as fractal dimension, of important groups of transformations acting on two-dimensional spaces, pictures, and measures are explored. Deep mathematical ideas are explained in an intuitive and practical manner. Laboratory work includes projects related to digital imaging and biological modelling. A high point in the course is an introduction to fractal homeomorphisms: what they are and how to work with them in the laboratory. Topics to be covered include: Affine, projective and Mandouml; bius geometries, iterated function systems, metric spaces, elementary topology, the contraction mapping theorem, the collage theorem, orbits of points, local behaviour of transformations, code space and the shift transformation, Julia sets and the Mandelbrot set, superfractals, deterministic, Markov chain, and escape-time algorithms for constructing fractal sets. Regular and chaotic behaviour in nonlinear systems, characterization and measures of chaos, stability and bifurcations, routes to chaos, crises, Poincare sections, the relation of fractal structures to simple nonlinear systems. Honours pathway option (HPO) The HPO option we will expand on the theoretical aspects of the underlying concepts. Alterative assessment in the assignments and exam will be used to assess these theoretical aspects.
Applied Algebra 1 Honours: Groups, Rings and Advanced Linear Algebra MATH3104Description:
Outcomes: On satisfying the requirements of this course, students will have the knowledge and skills to: 1. Explain the fundamental concepts of advanced algebra such as groups and rings and their role in modern mathematics and applied contexts2. Demonstrate accurate and efficient use of advanced algebraic techniques 3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanced algebra 4. Apply problem-solving using advanced algebraic techniques applied to diverse situations in physics, engineering and other mathematical contexts
Environmental Mathematics MATH3133Description:
Academic Contact: Professor Tony Jakeman and Dr Barry Croke Complex Analysis Honours MATH3228Description: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. Outcomes: On satisfying the requirements of this course, students will have the knowledge and skills to: 1. Explain the fundamental concepts of complex analysis and their role in modern mathematics and applied contexts Requisite Statement: A mark of 60 or more in MATH3320 Academic Contact: MATHSadmin@maths.anu.edu.au Number Theory and Cryptography MATH3301Description: he need to protect information being transmitted electronically
(such as the widespread use of electronic payment) has transformed the
importance of cryptography. Most of the modern types of cryptosystems
rely on (increasingly more sophisticated) number theory for their
theoretical background. This course introduces elementary number
theory, with an emphasis on those parts that have applications to
cryptography, and shows how the theory can be applied to cryptography. Cryptography topics will be chosen from: symmetric key cryptosystems, including classical examples and a brief discussion of modern systems such as DES and AES, public key systems such as RSA and discrete logarithm systems, cryptanalysis (code breaking) using some of the number theory developed. Honours Pathway Option (HPO): Students who take the HPO will complete extra work of a more theoretical nature. The assignments will be replaced by alternative assignments and the final exam will contain alternative questions requiring deeper conceptual understanding. Outcomes:
Analysis 3 Honours: Functional analysis, Spectral theory and Applications MATH3325Description: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: Measure theory- functions of bounded variation over R, absolute continuity and integration, examples of more general measures (Radon, Hausdorff, probability measures), Fubini-Tonelli theorem, Radon-Nikodym theorem. Banach spaces and linear operators - classical function and sequence spaces, Hahn-Banach theorem, closed graph and open mapping theorems, and uniform boundedness principles, sequential version of Banach-Alaoglu theorem, spectrum of an operator and analysis of the compact self-adjoint case, Fredholm alternative theorem. Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint. Outcome:On satisfying the requirements of this course, students will have the knowledge and skills to: 1. Explain the fundamental concepts of functional analysis and their role in modern mathematics and applied contexts2. Demonstrate accurate and efficient use of functional analysis techniques 3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis 4. Apply problem-solving using functional analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts
Relativity, Black Holes and Cosmology MATH3329Description:
Theory of Partial Differential Equations Honours MATH3341Description:The course will discuss the three main classes of equations, elliptic, parabolic and hyperbolic. It 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 fundamental solutions, maximum principles, regularity (smoothness) of solutions, variational problems, Holder and Sobolev spaces. Note: This is an Honours Pathway Course. It emphasises mathematical rigour and proof. Outcomes: On satisfying the requirements of this course, students will have the knowledge and skills to: 1. Explain the concepts and language of partial differential equations and their role in modern mathematics and applied contexts2. Analyse and solve complex problems using partial differential equations as functional and analytical tools 3. Apply problem-solving with partial differential equations to diverse situations in physics, engineering and other mathematical contexts
Algebraic Topology Honours MATH3344Description: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. 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)
Academic Contact: Bryan Wang |