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**An updated timetable is now available.** Some changes have been made to the week 4 timetable. This timetable does not have the correct rooms for Geometry and Groups that has a completely new list of rooms: Geometry and Groups Rooms.

There will be eight four-week courses offered at the summer school. Click on a course title for a more detailed description. Not all course combinations are possible. Refer to the course timetable section at the bottom of this page to determine compatible combinations.

Functional Analysis Dr Denis Potapov (UNSW) Functional analysis is a central pillar of modern analysis and its foundations will be covered in this course with an emphasis on the study of bounded linear maps between topological linear spaces. This provides the basic tools for the development of such areas as quantum mechanics, harmonic analysis and stochastic calculus. | |

Geometry and Groups Dr Stephan Tillmann (University of Sydney) According to Klein, geometry is the study of properties of a space which are invariant under a group of transformations. This motivates key ideas in areas such as group theory, algebraic topology, differential geometry and representation theory. This course is an introduction to geometry in the sense of Klein and Thurston and will provide a working knowledge of a variety of widely applicable concepts and tools. | |

Combinatorial Enumeration and Optimization A/Prof Ian Wanless (Monash University) and Dr Thomas Britz (UNSW) Combinatorial enumeration deals with clever ways of counting combinatorial objects such as permutations, partitions, graphs, words etc. We will concentrate on using generating functions to do the counting for us. Combinatorial optimisation deals with clever ways of arranging objects so that they fit together and includes strategies for solving scheduling problems, planning optimal travel routes, and designing cheap and yet robust communications networks. We will consider several classical (and beloved) optimization theorems. | |

Four Key Numerical Algorithms Dr William McLean (UNSW), A/Prof Robert Womersley (UNSW) and A/Prof Thanh Tran (UNSW) The theoretical basis, context and typical applications of four numerical algorithms from linear algebra, PDEs and optimization will be covered. Each algorithm has had a major impact on the development of computational mathematics and is in constant use. The course covers 1. Conjugate gradients; 2. Domain decomposition; 3. Interior point methods and semidefinite programming; 4. QR algorithm. | |

Optimization A/Prof Regina Burachik (University of South Australia) The need to optimize (understood as the search for a best option given the circumstances) is found everywhere from the natural and social sciences to engineering, business and economics. Optimizing requires a model, a mathematical theory for solving the model, and a computer code to implement the theory. This course focuses on the mathematical aspects of optimization. First, it gives basic tools of convex analysis, convex functions and separation theorems. Then it considers optimization problems, including convex (non-differentiable) and differentiable ones. We will analyse convergence properties of classical solution methods, such as steepest descent, Newton, and their variants, for unconstrained problems. Finally, we will study penalty, barrier, and exact penalty methods for constrained problems. | |

Modelling in Mathematical Biology Prof Graeme Pettet (Queensland University of Technology) Much of the art of modelling is associated with adopting appropriate techniques given the available data and the form of the analysis being sought. The formulation and solution of a series of key mathematical models that have had an impact in the biological sciences will be explored. Models formulated as ODEs and PDEs will be analysed with phase plane techniques and simulation using NetLogo. | |

Mathematical Foundations of Probability and Statistics A/Prof Ben Goldys (UNSW) Probabilistic arguments have been known for thousands of years - some of them can be found in the Old Testament and Talmud. In 1933 A. N. Kolmogorov demonstrated that measure theory and analysis provide the proper language to study probability in a rigorous way. Nowadays, the mathematical theory of probability is applied in the areas as diverse as statistics, mathematical finance, genetics and quantum field theory. It is also an important tool for mathematicians working in differential equations, differential geometry and graph theory. In this course we will introduce the basic concepts of the mathematical theory of probability and will discuss some of its applications. | |

Climate Statistics Prof John Boland (University of South Australia) Techniques used in the statistical analysis of climate variables will be covered in this course using case studies from renewable energy utilisation and water resource management. Tools will include time series analysis, probability forecasting, spatial-temporal analysis, wavelets and an examination of the coherence of models on varying time scales. We will also study artificial neural networks, ensemble forecasting and some methods from dynamical systems and finance. |

#### Timetable

**A draft timetable is now available.**

There will be three timetable groups at the summer school. **These groups have been changed (15/12/2011).** It is not possible to enrol in two courses from the same group. The groups are:

- Group A
- Geometry and Groups
- Optimization
- Climate Statistics

- Group B
- Combinatorial Enumeration and Optimization
- Mathematical Modelling in Biology
- Mathematical Foundations of Probability and Statistics

- Group C
- Functional Analysis
- Four Key Numerical Algorithms