Courses
The program, consisting of three 4-week courses and ten 2-week
courses, is outlined below. Click on the title of a course for more
detailed information, including prerequisites and assessment.
Due to the number of courses being offered, there will inevitably
be some clashes. The timetable will be organised to accommodate the
most popular combinations of courses. In your application you will
nominate several choices in order of preference.
At the discretion of their home university, Honours students may be
able take courses for credit towards their degree. Contact your Head of
Department or Honours coordinator for guidance on this matter.
Note that a 4 week course will be roughly equivalent to a regular one
semester course.
| Period 1 : 10 - 21 Jan 2005 |
Period 2 : 24 Jan - 4 Feb 2005 |
Measure Theory
Marty Ross |
Partial Differential Equations
Neil Trudinger, ANU, Nirmalendu Chaudhuri,
ANU &
Andrew Hassell, ANU |
Machine Learning
Alex Smola, ANU, SVN Vishwanathan, ANU,
Aapo Hyvarinen, Helsinki & Matthias Franz,
Max-Planck |
Commutative Algebra
Ruth Kantorovitz, Illinois/ANU |
Algebraic Geometry
Paul Norbury, Melbourne |
Combinatorial Matrices
Ian Wanless, ANU/CDU |
Combinatorial Geometry
Ben Burton, RMIT |
Bootstrap Methods and Edgeworth Expansion
Peter Hall, ANU |
Analysis of Survey Data
David Steel, Wollongong |
Fluid Mixing
Stephen Cox, Adelaide
& Jim Denier, Adelaide |
Stochastic Process Modelling
Daryl Daley, ANU |
Finite Volume Methods
Ian Turner, QUT |
Nonlinear Optimzation Methods
Rob Womersley, UNSW |
|
| Course: |
Measure Theory |
| Lecturer: |
Marty Ross |
| Duration: |
4 weeks, Periods 1 & 2, 10 Jan - 4 Feb 2005 |
| Content: |
Measure theory is the modern theory of integration, the method
of assigning a "size" to subsets of a universal set. It is more beautiful
and
more powerful (though also more technical) than the older theory of
Riemann integration. The course will be a reasonably standard introduction
to measure theory, with some emphasis upon geometric aspects. We will
cover most (but definitely not all) of the topics listed below,
subject to time and taste:
- General Measure Theory
(Outer measure; measurable sets; Borel and Radon measures;
the Caratheodory criterion for Borel measures)
-
Special Measures on Euclidean Space
(Lebesgue measure; Hausdorff measure; the Vitali Covering Theorem;
notions of dimension)
-
Integration
(Measurable functions; integration and convergence theorems;
the Area Formula; iterated integrals and Fubini's Theorem)
-
Functional Analysis
(Measures as linear functionals; Lp spaces; the Riesz Representation Theorem)
-
Further Topics
(Differentiation of measures; the Besicovitch Covering Theorem; the (Generalised)
Fundamental Theorem of Calculus; the Co-Area Formula)
|
| Hours: |
7 hours of lectures per week, with consultation as requested/required |
| Prerequisites: |
Familiarity with the fundamental concepts of analysis in Euclidean
Space (open and closed sets, continuity, completeness and compactness,
countability). Some corresponding familiarity with these notions
in general metric (or topological) spaces would be helpful but will not be
assumed. |
| Assessment: |
Problems assigned during lectures (50%), and a take-home exam
(50%) |
| Resources: |
At minimum, brief lecture notes will be provided.
We shall roughly follow the early
chapters of Measure Theory and Fine Properties of Functions
by Evans
and Gariepy (CRC, 1991), though the book is very terse (and
expensive!).
There are many good texts on measure theory; Real Analysis
by Royden
(3rd ed., Prentice Hall, 1988)
is good, and easy to find in libraries. (Texts which also
cover
probability will be less useful, as the language and
approach tends to be quite different.) |
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|
| Course: |
Partial Differential Equations |
| Lecturers: |
Neil Trudinger, ANU, Nirmalendu Chaudhuri,
ANU &
Andrew Hassell, ANU |
| Duration: |
4 weeks, Periods 1 & 2, 10 Jan - 4 Feb 2005 |
| Content: |
- 2nd order elliptic operators: existence, regularity, qualitative properties of solutions
- Sobolev spaces, variable coefficient theory
- Nonlinear PDE theory
- Hyperbolic PDE and propagation phenomena
|
| Prerequisites: |
Calculus and analysis in Rn, basic functional analysis.
Ideally students will be familiar with measure theory and Lp spaces, but it
could also be done in conjunction with the AMSI Measure Theory course. |
| Resources: |
Elliptic Partial Differential Equations of Second Order,
David Gilbarg & Neil S. Trudinger, Classics in
Mathematics, Springer-Verlag, 2001.
This book will be available for purchase at the Summer School.
Slides in PDF format:
|
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|
| Course: |
Machine Learning (joint with
Machine Learning Summer School) |
| Lecturers: |
Alex Smola, ANU, SVN Vishwanathan, ANU,
Aapo Hyvarinen, Helsinki & Matthias Franz, Max-Planck |
| Duration: |
4 weeks, Periods 1 & 2, 10 Jan - 4 Feb 2005 |
| Content: |
In Period 1, 10 - 21 January 2005,
Alex Smola and SVN Vishwanathan will present
an introductory course in Machine Learning aimed at
mathematicians. The topics will tentatively include:
- Introduction to Machine Learning and Probability Theory
- Density Estimation and Parzen Windows
- The Perceptron and Kernels
- Support Vector Classification
- Kernel Methods for Text Categorization and Biological Sequence Analysis
- Optimization
- Regression and Novelty Detection
- How to get good results in practice
In Period 2, 24 January - 4 February 2004, students will join the
Machine Learning Summer School
for the following two lecture series:
Follow the links for more details on these lectures.
|
| Hours: |
Approximately 6 - 7 hours per week |
| Prerequisites: |
Nothing beyond undergraduate knowledge in mathematics is expected. More specifically, we assume:
- Basic linear algebra (matrix inverse, eigenvector, eigenvalue, etc.)
- Some numerical mathematics (beenficial but not required), such as matrix factorization, conditioning, etc.
- Basic statistics and probability theory (Normal distribution, conditional distributions)
|
| Note: |
This offer is open to Honours, Masters and PhD students only. They do
not need to register for the Machine Learning Summer School if
they are participating in the Machine Learning component of the ICE-EM/AMSI
Summer School.
Staff members wishing to attend this course are asked to register
separately for the Machine
Learning Summer School.
|
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|
| Course: |
Commutative Algebra |
| Lecturer: |
Ruth Kantorovitz, Illinois/ANU |
| Duration: |
2 weeks, Period 1, 10 - 21 Jan 2005 |
| Content: |
Commutative algebra is the study of commutative rings. In addition to
being a beautiful subject in its own right, commutative algebra is
important as a foundation for algebraic geometry. The aim of this
course is to introduce the basic tools of commutative algebra which
will be used in the algebraic geometry course taught in the second
half of the summer school.
We will cover most of the following topics.
- the spectrum of a ring
- nilpotent and radicals
- Noetherian rings and
modules
- Hilbert basis theorem
- localization
- primary decomposition
-
Noether's normalization
|
| Hours: |
10 hours of lectures and 4 hours of tutorials |
| Prerequisites: |
Basic knowledge of abstract algebra, e.g. ideals and quotient rings,
prime and maximal ideals, polynomial rings,
Principal ideal domains (PIDs), fields, modules, homomorphisms. |
| Resources: |
- Introduction to commutative algebra, M. F. Atiyah and I.G. McDonald,
Addison-Wesley publishing
-
Undergraduate Commutative Algebra, Miles Reid, LMS Student Text 29
|
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|
| Course: |
Combinatorial Matrices |
| Lecturer: |
Ian Wanless, ANU/CDU |
| Duration: |
2 weeks, Period 1, 10 - 21 Jan 2005 |
| Content: |
In combinatorial matrices we are interested not so much in the
numerical value of entries but in how they are arranged, that is, the
pattern that they form. Examples include permutation matrices,
chessboards, Latin squares, Hadamard matrices, frequency squares,
orthogonal arrays and so on. This course will study some of the
properties of these matrices as well as introducing some important
combinatorial tools such as the principle of inclusion-exclusion,
Hall's marriage theorem and a matrix function called the permanent.
We will also meet some of the basic ideas of graph theory.
|
| Hours: |
5 hours of lectures and 2 hours of tutorials per week |
| Prerequisites: |
Minimal. Familiarity with very basic group theory will be
assumed, but is not a major part of the course. |
| Assessment: |
1 assignment (50%) and 1 exam (50%) |
| Resources: |
Slides in PDF format:
Other references:
-
Laywine, Charles F. and Mullen, Gary L.
Discrete mathematics using Latin squares.
Wiley, New York, 1998.
-
H. Minc,
Permanents,
Encyclopedia Math. Appl. 6,
Addison-Wesley, Reading, Mass., 1978.
-
van Lint, J. H.; Wilson, R. M.
A course in combinatorics.
Cambridge University Press, Cambridge, 1992.
|
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|
| Course: |
Bootstrap Methods and Edgeworth Expansion |
| Lecturer: |
Peter Hall, ANU |
| Duration: |
2 weeks, Period 1, 10 - 21 Jan 2005 |
| Content: |
Introduction to the bootstrap (a statistical method) and
aspects to theory for the bootstrap. |
| Hours: |
Ten to twelve hours of lectures. No tutorials are planned at this stage. |
| Prerequisites: |
Mathematical skills to the level of Third Year (Pass) or
Second Year (Advanced) courses. |
| Assessment: |
Exercises set during the class (there will be tutorials if the
students ask for them) and exam at the end of the course. |
| Resources: |
Slides in PDF format
Edgeworth Expansion:
Bootstrap methods:
|
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|
| Course: |
Fluid Mixing |
| Lecturers: |
Jim Denier, Adelaide
& Stephen Cox, Adelaide |
| Duration: |
2 weeks, Period 1, 10 - 21 Jan 2005 |
| Content: |
The mixing of fluids plays a vital role across a wide range of scales in
the smallest airways of the human lungs, fluid mixing is an essential part
of the breathing process that keeps us all alive; on the largest scales of planet Earth, fluid mixing is responsible for maintaining the overall
structure of its oceans and atmosphere.
In the economic sector, fluid mixing is central to many industries, both established and emerging, in Australia and overseas. One significant
area of application is the chemicals and plastics industry, which has an annual turnover of over $20 billion and employs over 77,000 people. Fluid
mixing underpins this industry. Frontier technologies such as biotechnology also crucially rely on fluid mixing, for example in
carrying out DNA analysis of tiny samples of bodily fluid.
This course describes some of the exciting developments in fluid mixing that have taken place over the past twenty years or so.
- Foundations - introduction to some fluid mechanics concepts, laminar versus turbulent mixing, Eulerian versus Lagrangian description of fluid motion (2 lectures)
- Mathematical models for mixing devices - Aref's blinking vortex flow, vortex in a pot, a circular stirring rod, the eccentric annular mixer (4 lectures)
- Simulation and exploration using MATLAB (1 tutorial/computer practical)
- Effects of diffusion on mixing - - why stir? diffusive tracers, particle return experiments (2 lectures)
- Measures of mixing - a dynamical systems approach, a statistical approach, the Poincare section, Lyapunov exponents (3 lectures)
- Further simulation and exploration using MATLAB (2 tutorials/computer practicals)
|
| Hours: |
Week 1: 6 hours of lectures, 1 hour tutorial (computer exploration) - Dr Jim Denier
Week 2: 5 hours of lectures, 2 hours of tutorials (computer exploration) - Dr Stephen Cox
|
| Prerequisites: |
Ordinary differential equations, partial differential equations.
Matlab will be used throughout this course. Examples will
be provided in matlab, and students will be expected to
adapt these examples in exploring various problems in fluid
mixing. No previous knowledge of fluid mechanics will be assumed.
|
| Assessment: |
A short written examination and a mini-project involving the exploration of fluid mixing using matlab.
Click here to download the assessment. |
| Resources: |
Full printed notes will be made available to all students as well as a CD containing notes, images from the course
and all the MATLAB 7 scripts used throughout the course.
Click here to download scripts for MATLAB 6.5.
|
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|
| Course: |
Finite Volume Methods (joint with APAC Summer School) |
| Lecturer: |
Ian Turner, QUT |
| Duration: |
2 weeks, Period 1, 10 - 21 Jan 2005 |
| Content: |
- Introduction to conservation equations
- Analytical solutions for the benchmark problems
- Introduction to the Finite Volume method
- Implementing boundary condition information into the model
- Treatment of nonlinearity using an inexact Newton method
- The treatment of advection within the framework of the FVM
Module 1: Introduction and the Generalised Transport Equation A
brief discussion of the conservation equations that describe fluid
motion. The notion of a generalised transport equation. Discussion
of boundary and initial conditions. Derivation of analytical
solutions for the benchmark problems considered in throughout the
lectures.
Module 2: Introduction to Finite Volume Methods (FVM) Basic
concepts and rules of the finite volume method. Application to the
Generalised one-dimensional Diffusion Equation. Cell-centred and
vertex-centred schemes. Treatment of diffusion coefficients at
control volume faces and source/sink terms. Incorporating boundary
condition information into the model. Inexact-Newton methods for
solving no n-linear transport equations.
Module 3: FVM for Advection-Diffusion Equations The treatment of
advection and the inclusion of these schemes within the framework
of the FVM. Monotonicity arguments, TVD schemes, upstream
averaging, other averaging methods. A brief discussion of flux
limiting.
Module 4: Specialised Topics FVM for radially symmetric problems.
Brief introduction to FVM in higher dimensions and implementation
details for arbitrary grids. |
| Hours: |
12 hours of lectures and 5 hours of practicals |
| Prerequisites: | Linear algebra, introduction to
differential/partial differential equations, introduction to
scientific computation and Matlab. |
| Assessment: |
Two assignments, each worth 50%, that cover the core material of the lectures. These assignments will
be due approximately 2 weeks after the completion of the course. The assessment will
take the form of a student written report and evidence of working software. Students
will receive a summary report with written comments and marks for each assignment. |
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|
| Course: |
Algebraic Geometry |
| Lecturer: |
Paul Norbury, Melbourne |
| Duration: |
2 weeks, Period 2, 24 Jan - 4 Feb 2005 |
| Content: |
Algebraic geometry is the study of the zero sets of polynomials. As
the name suggests, it combines algebra and geometry. In practice, one
might view it entirely algebraically, or entirely geometrically, with only
a partial awareness of the geometry or algebra. One aim of this course is
to give both views. Algebraic geometry is a fundamental tool in many areas
of mathematics, including differential geometry, number theory, integrable
systems and in physics, such as string theory.
- Commutative algebra
- Noetherian rings
- birational equivalence
- coordinate
ring of functions on a variety
- Zariski topology
- the Nullstellensatz
-
projective varieties
- genus of a curve
- singularities
More details:
http://www.ms.unimelb.edu.au/~pnorbury/algeom.html
|
| Hours: |
14 hours of lectures |
| Prerequisites: |
Algebra: fields, polynomial rings and Commutative Algebra
taught in Period 1, such as knowledge of ideals in a
commutative ring. It would be desirable also to know some topology,
geometry and complex analysis, such as the definition of a manifold and a
Riemann surface. |
| Resources: |
- Undergraduate Algebraic Geometry, Miles Reid, LMS Student Texts 12
- Algebraic Geometry, Daniel Bump, World Scientific Publishing
- Complex Algebraic Curves, Frances Kirwan, LMS Student Texts 23,
Cambridge University Press
|
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|
| Course: |
Combinatorial Geometry |
| Lecturer: |
Ben Burton, RMIT |
| Duration: |
2 weeks, Period 2, 24 Jan - 4 Feb 2005 |
| Content: |
This course will involve a study of polytopes and 3-manifold topology,
with a particular focus on combinatorics and algorithms. For the first
week we will concentrate on polytopes, covering the double description
method, shellability, the Dehn-Sommerville equations and the upper bound
theorem. In the second week we will move to 3-manifolds, examining
triangulations, normal surfaces, links with polytope theory and the use
of all of these in higher-level topological algorithms.
|
| Hours: |
5 hours of lectures and 2 hours of tutorials per week |
| Prerequisites: |
Some undergraduate linear algebra will be required, and a little
point set topology would be handy (though not necessary). |
| Assessment: |
A take-home exam to be sat at your home institution. |
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|
| Course: |
Analysis of Survey Data |
| Lecturer: |
David Steel, Wollongong |
| Duration: |
2 weeks, Period 2, 24 Jan - 4 Feb 2005 |
| Content: |
Most large-scale survey use sampling methods more complex than simple random sampling. Use of complex sampling methods affects the properties of standard statistical analysis methods based on assumptions of homogeneous relationships and independence between units in the population, possibly leading to large biases and inefficiencies. The objective of the course is for you to understand the principles and theory behind analysing data obtained from complex sample surveys. This knowledge and understanding will enable you to analyse data arising from a complex sample survey.
- Complex sampling methods including stratification and cluster sampling
- calculation of weights for use in survey estimation
- methods for accounting for stratification and cluster sampling in regression and other statistical analysis techniques
- theory underpinning methods used in common statistical software packages
|
| Hours: |
2hrs on January 24, 25, 28, 31 and February 1 and 4. Prof Steel will be available for email contact on other days. (We might programme an additional hour say on Feb 1 as tutorial time) |
| Prerequisites: |
Knowledge of basic statistical analysis methods, including multiple regression and chi-squared testing of independence, preferably at the level of a third year subject. |
| Assessment: |
6 exercise sheets corresponding to each 2hr session, worth a total of 20% and a take home exam worth 80%. Exercise sheets 1&2 due on Friday 28 January, exercise sheets 3&4 due Tuesday 1 February, exercise sheet 5 due on Friday 4 February and exercise 6 due by post on Wednesday 9 February. Exam will be distributed on Friday 4 February and returned by mail postmarked no later than 14 February. |
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|
| Course: |
Stochastic Process Modelling |
| Lecturer: |
Daryl Daley, ANU |
| Duration: |
2 weeks, Period 2, 24 Jan - 4 Feb 2005 |
| Content: |
Discrete-time random walks, Markov chains,
some elementary Markov modelling of continuous time
phenomena. |
| Prerequisites: |
Familiarity with probability theory and
manipulation (especially independence and dependence). |
| Assessment: |
Most likely a mixture of examples
to be completed and a short 'essay' |
| Resources: |
- Feller, W. (1968) Introduction to Probability Theory and
Applications, Vol.I. (Wiley)
- Bremaud, P. (1999). Markov chains: Gibbs fields, Monte Carlo
Simulation, and Queues. Springer, New York.
|
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|
| Course: |
Nonlinear Optimzation Methods |
| Lecturer: |
Rob Womersley, UNSW |
| Duration: |
2 weeks, Period 2, 24 Jan - 4 Feb 2005 |
| Content: |
Problem formulation, optimality conditions, duality,
numerical methods, including interior point and large
scale methods. It will include case studies from
distributing points on manifolds (sphere, torus).
|
| Hours: |
5 hours of lectures and 2 hours of labs per week. |
| Prerequisites: |
A knowledge of Matlab is desirable, but not essential. |
| Assessment: |
Take home exam and small project. |
| Resources: |
Lecture notes will be provided plus Matlab examples.
Jorge Nocedal and Stephen Wright, Numerical Optimization,
Springer, 1999.
Slides in PDF format:
Download sample codes
|
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