PURE MTH 4038 - Pure Mathematics Topic A - Honours
North Terrace Campus - Semester 1 - 2020
General Course Information
Course Code PURE MTH 4038 Course Pure Mathematics Topic A - Honours Coordinating Unit School of Mathematical Sciences Term Semester 1 Level Undergraduate Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Y Restrictions Honours students only Course Description Please contact the School of Mathematical Sciences for further details.
Course Coordinator: Dr Guo Chuan Thiang
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesIn 2020, the topic of this course is Functional Analysis
Motivated by the development of calculus of variations, integral equations, approximation theory and quantum physics in the early twentieth century, functional analysis has grown into a broad field of modern analysis. Broadly speaking, it is the study of infinite dimensional linear topological spaces as well as properties of linear maps among these spaces. Of particular importance are linear operators on Hilbert spaces, as they play a fundamental role in quantum mechanics, partial differential equations, signal processing, ergodic theory, dynamics, and many other branches of mathematics, physics and engineering. Besides these applications, the subject of functional analysis also has important connections to geometry, topology and number theory.
1. Fundamentals of Hilbert spaces (4 lectures).
2. Bounded linear operators on Hilbert spaces and some important theorems in abstract functional analysis (7 lectures).
3. Projections, unitaries, self-adjoint and normal operators (5 lectures).
4. Compact operators and its spectral decompositions (4 lectures).
5. Spectral theory of self-adjoint and normal operators (6 lectures).
6. Advanced topics which may be added to the core topics (sample: quantum mechanics, unbounded operators, operator algebras, representation theory, Fourier analysis, Fredholm index and Toeplitz operators) (4 lectures).
On successful completion of this course, students should
1. Understand properties of Hilbert spaces and their bounded linear operators; know how to apply these properties;
2. Be able to identify and work on key examples involving Hilbert space analysis;
3. Understand the concept of the spectrum of an operator, and compute the spectrum of specific examples;
4. Be able to state and prove the spectral theorem for compact and for self-adjoint operators;
5. Be aware of concrete applications of functional analysis in specialised topics and connections to other areas of mathematics.
The course requires knowledge of point-set topology, basic measure theory, and a firm background in linear algebra. Good knowledge of real analysis and some exposure to complex analysis would be desirable. It is recommended that students have taken Topology and Analysis III and/or Integration & Analysis III, previously and are familiar with basic group theory.
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s) Deep discipline knowledge
- informed and infused by cutting edge research, scaffolded throughout their program of studies
- acquired from personal interaction with research active educators, from year 1
- accredited or validated against national or international standards (for relevant programs)
all Critical thinking and problem solving
- steeped in research methods and rigor
- based on empirical evidence and the scientific approach to knowledge development
- demonstrated through appropriate and relevant assessment
Required ResourcesCourse lecture notes will be provided.
Recommended ResourcesSome standard functional analysis textbooks are:
J. Conway, A course in functional analysis (Good introduction at advanced undergraduate/beginning graduate level)
M. Reed and B. Simon, Methods of modern mathematical physics. I, Functional analysis (Good foundation for mathematical quantum theory)
W. Rudin, Functional analysis (Quite abstract, graduate level, deals with general topological vector spaces and Banach spaces)
K. Yosida, Functional analysis
G. Folland: Real analysis: Modern techniques and their applications
Online LearningThis course will have an active MyUni website.
Learning & Teaching Activities
Learning & Teaching ModesThe lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.The following table is a guide to the workload for each component of the course.
Activity Quantity Workload hours Lecture 30 90 Assignments 6 66 Total 156
Learning Activities Summary1) Introduction to of smooth manifolds, smooth maps, and vector fields (10 Lectures)
2) Riemannian geometry (12 lectures)
3) Lie groups and their Lie algebras, subgroups, homomorphisms (8 lectures)
Specific Course RequirementsNone.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 70% all Homework assignment Formative and summative One week after assigned 30% all
Due to the current COVID-19 situation modified arrangements have been made to assessments to facilitate remote learning and teaching. Assessment details provided here reflect recent updates.
1) The total assignment weightage will increased from 30% to 50%, with a total of six take-home assignments as before. The due dates are projected to be: 20 March, 6 April, 27 April, 15 May, 29 May, 12 June.
2) There will be three timed and graded quizzes administered in MyUni, contributing 15% to the final grade. This will comprise mainly multiple choice, fill-in-the-blank, and true/false questions. A sample quiz will be made available, so you can get used to the mechanism.
3) There will be a final exam scheduled during a 2-3 hour slot in the usual exam weeks (20-30 June), administered online within MyUni, with weightage 35%.
Assessment Related RequirementsAn aggregate score of 50% is required to pass the course.
Assessment DetailThere will be a total of 6 homework assignments, given out at intervals of about two weeks.
SubmissionHomework assignments must be given to the lecturer in person or emailed as a pdf. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M11 (Honours Mark Scheme) Grade Grade reflects following criteria for allocation of grade Reported on Official Transcript Fail A mark between 1-49 F Third Class A mark between 50-59 3 Second Class Div B A mark between 60-69 2B Second Class Div A A mark between 70-79 2A First Class A mark between 80-100 1 Result Pending An interim result RP Continuing Continuing CN
Further details of the grades/results can be obtained from Examinations.
Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.
Final results for this course will be made available through Access Adelaide.
The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.
SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.
- Academic Support with Maths
- Academic Support with writing and speaking skills
- Student Life Counselling Support - Personal counselling for issues affecting study
- International Student Support
- AUU Student Care - Advocacy, confidential counselling, welfare support and advice
- Students with a Disability - Alternative academic arrangements
- Reasonable Adjustments to Teaching & Assessment for Students with a Disability Policy
- LinkedIn Learning
Policies & Guidelines
This section contains links to relevant assessment-related policies and guidelines - all university policies.
- Academic Credit Arrangement Policy
- Academic Honesty Policy
- Academic Progress by Coursework Students Policy
- Assessment for Coursework Programs
- Copyright Compliance Policy
- Coursework Academic Programs Policy
- Elder Conservatorium of Music Noise Management Plan
- Intellectual Property Policy
- IT Acceptable Use and Security Policy
- Modified Arrangements for Coursework Assessment
- Student Experience of Learning and Teaching Policy
- Student Grievance Resolution Process
Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's student’s disciplinary procedures.
The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.