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.