Access Grid Semester 1 2009

Offered by the University of Sydney

Course: Commutative Algebra, Pure Maths Honours

Lecturer: David Easdown

Summary: In this course we study commutative rings, the natural framework for developing tools of enormously wide application in
higher mathematics, including algebraic geometry, number theory and extension theory. We elaborate on a selection of topics from the first seven chapters of the classic text by Atiyah and Macdonald (see references), providing an introduction to the subject and a platform for
further study and applications.

The course is divided into four parts: (1) overview and introduction to commutative ring theory; (2) introduction to theory of modules,
tensor products and exactness properties; (3) study of rings and modules of fractions and properties of localisation; (4) chain conditions,
study of Noetherian rings, primary decompositions and seminal theorems such as the Jordan-Holder Theorem, the Hilbert Basis Theorem and the Hilbert Nullstellensatz. Given time and interests of course participants, we may explore several applications possibly together as a class or through individual or group projects.

Assessment: by assignments, project work and take-home examination.

References:

(1) Course notes by David Easdown

(2) Introduction to Commutative Algebra, by M.F. Atiyah and
I.G. Macdonald

Timetable: Monday, Tuesday and Wednesday 3-4pm

Offered by the University of South Australia

Course: Financial Time Series

Lecturer: John Bolland - john.bolland@unisa.edu.au

Time: Tuesdays 11am-1pm (Adelaide Time)

Summary: We will be first studying the components of a time series in general terms, and then progressing to the particular methods necessary for analysing time series in the financial realm. We will also endeavour to investigate where these methods may be applicable to other fields. The students will be called upon to work independently on project work.

The components of a time series model
Additive and multiplicative models
Multiple regression analyses
Spectral decomposition
Box-Jenkins models
Forecasting techiniques
Smoothing of time series
GARCH and other volatility models
Stochastic Differential Equations

Learning objectives and Graduate Qualities

On completion of this course, students should be able to:

understand the techniques of time series analysis and be able to apple them to not financial time series but also other applications (GQ 1,2,3,4,6)
prepare a summary of the scientific literature in a particular topic of interest (1, 2, 4, 6, 7)
create and interpret statistical and time series questions and develop strategies for testing these questions (GQ 1, 2, 3, 4,6)
perform calculations and interpret the results of a range of testing and estimation procedures, paying particular attention to the underlying assumptions(GQ 1,2,3,6).
use software packages to analyse data (GQ 1,2,3,4,6)
write a short summary report of investigation.(GC 4, 6, 7)
present the results of an investigation (GC 6).

Offered by LaTrobe University

Course: Topology and Dynamics (MAT4TD)

Lecturer: John Banks - J.Banks@latrobe.edu.au

Time: Wednesdays 1-3pm

Summary:
We develop some definitions and results of very general application in point set topology and use them to explore the theory of (discrete) topological dynamics. The point set topology is developed simultaneously with the topological dynamics and applications of the topology to the dynamics being added progressively as we proceed.
(a) The point set topology stream culminates in proofs of major results including Tychonoff's
theorem (via the Alexander lemma) and Urysohn's metrization theorem.
(b) The topological dynamics stream focusses on the way in which open sets evolve dynamically
and the relevance of these ideas to definitions of chaos. Symbolic dynamics and inverse limits
will also be explored.
(NOTE: This is an updated description designed to be more informative than the one currently
appearing in the university handbook.)

Prerequisite knowledge:

(a) It is assumed students have already encountered the theory of point set topology and metric spaces including general definitions of:
- Open, closed and dense sets and the closure of a set,
- Continuity and convergence,
- Hausdorff spaces,
- Connectedness and path connectedness,
- Compactness,
- Basis for a topological space,
- Finite product of topological spaces.
and the standard theorems relating these concepts to one another.
(b) Although some previous experience of these ideas is assumed, they will be revised as further
definitions and results in point set topology are gradually introduced.
(c) No previous study of dynamics is assumed.
(d) Host prerequisite subject: MAT3TA, Topology and Analysis
(e) Host prerequisite subject URL: Go to http://udb-iasprd.latrobe.edu.au/udb1subprd_
public/publicview$.startup and type in unit code MAT3TA.

Assessment:

- Exam/assignment/class work breakdown
Exam 0 %
Assignment 40 %
Class work 0 %
Mini-Project 60 %
- Assignment due dates: March 25 and May 13.
- Mini-project due June 11.
- Approximate exam date: N/A

Required student resources

- Comprehensive unit text will be provided on-line (and as hard copy if requested).
- No mandatory software requirements, but access to LATeX is desirable.
- Electronic whiteboard will be used for student presentations.

Offered by Wollongong University

Course: MATH971 Applied Non-Linear Differential Equations

Lecturer: Dr M.L Nelson - mnelson@uow.edu.au

Summary:
This course provides an introduction to applied non-linear ordinary differential equations. This course is applied mathematics. There will be no technical lemmas or abstract definitions!

Topics to be covered include (but are not limited to):
- First-order differential equation: Graphical insights, steady-state solutions and their stability, steady-state diagrams and bifurcations.
- Singularity theory with a distinguished parameter: singularity theory and bifurcation points, constructing static bifurcation diagrams.
- Systems of two first-order differential equations
- steady-state solutions and their stability: local and Liapunov.
- the absence of periodic solutions: Bendixon's Criteria and Dulac's Test.
- periodic behaviour: the Hopf bifurcation Theorem, sub-critical and super-critical hopf bifurcations.
- bifurcations and steady-state diagrams: singularity theory and bifurcation points.
- degenerate Hopf bifurcations: the double Hopf bifurcation, the Bautin bifurcation, the double-zero eigenvalue bifurcation.

Prerequisite knowledge: No knowledge of applied mathematics is assumed. Little knowledge above first year calculus is
required. It will be assumed that you have used Maple previously (but if not, you can pick it up). If you don't like Maple you are free to use an equivalent package.

Assessment
- Exam/assignment/class work breakdown
Your final mark in MATH971 will be determined as follows. Two marks will be calculated
using scheme one (S1) and scheme two (S2).
Scheme S1 S2
Exam 60 % 40 %
Assignments 40 % 60 %
Your final mark will be the higher of the marks calculated using schemes one and two. Scaling of marks is not standard procedure in this subject. Note that you are not required to 'pass' each individual component to receive a pass grade in MATH971. However, you would seriously jeopardise your chances of passing this subject if you do not aim to be successful in every component of the assessment.

Assignment due dates will be given when assignments are handed out. Generally, students are given two weeks
to complete assignments.

Approximate exam date Between 15th June and 26th June

All lecture notes will be made available on the course web-page.
- Software (local access) You will need access to Maple or an equivalent package.