ANU Extension Specialist Mathematics B

This course is designed for students who wish to study mathematics at a level beyond a standard double major in Mathematics at years 11 and 12. International students are not able to enrol in this course. Topics covered in year 11 will include: • Applications of the ‘Pigeon Hole Principle’, Review and further properties of the Fibonacci sequence, proofs of Binet’s formula, proof of Fermat’s Little Theorem, develop the algorithm for RSA public key cryptography, proof that RSA public key cryptography works. Properties of the real number system including completeness. • Countability of the sets of even integers, of odd integers, of all integers, of rational numbers, cardinality of the set of real numbers, power sets and cardinality, Russel’s Paradox, cardinality by geometric argument. Topics covered in year 12 will include: • The 5 Platonic solids, proof of Euler’s formula, proof that there are no other Platonic solids using Euler’s formula. A selection of topics from: Aperiodic Tilings of the Plane, spherical and hyperbolic geometries, visualising the fourth dimension, basic notions of topology, two dimensional surfaces, knots and links, fixed point theorems. • Iterative Dynamical Systems (population growth, logistic equation and extensions), generating fractals through iterative processes, self similarity in nature over a range of scales, Iterated Functions Systems, the Chaos Game, Julia and Mandelbrot sets, deterministic chaos, chaotic behaviour arising from deterministic systems, dimensions of fractals via scaling arguments. This is Part B of a 2 part course.