PURE MTH 3002 - Topology and Analysis III
North Terrace Campus - Semester 1 - 2015
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General Course Information
Course Details
Course Code PURE MTH 3002 Course Topology and Analysis III Coordinating Unit Pure Mathematics Term Semester 1 Level Undergraduate Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Prerequisites MATHS 2100 Assumed Knowledge MATHS 2100 Assessment ongoing assessment 30%, exams 70% Course Staff
Course Coordinator: Professor Finnur Larusson
Course Timetable
The full timetable of all activities for this course can be accessed from .
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Learning Outcomes
Course Learning Outcomes
- Demonstrate an understanding of the concepts of metric spaces and topological spaces, and their role in mathematics.
- Demonstrate familiarity with a range of examples of these structures.
- Prove basic results about completeness, compactness, connectedness and convergence within these structures.
- Use the Banach fixed point theorem to demonstrate the existence and uniqueness of solutions to differential equations.
- Demonstrate an understanding of the concepts of Hilbert spaces and Banach spaces, and their role in mathematics.
- Demonstrate familiarity with a range of examples of these structures.
- Prove basic results about Hilbert spaces and Banach spaces and operators between such spaces.
- Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
- Demonstrate skills in communicating mathematics orally and in writing.
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) Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. 1,2,3,4,5,6,7 The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 3,4,7,8 An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 8 Skills of a high order in interpersonal understanding, teamwork and communication. 9 A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all A commitment to the highest standards of professional endeavour and the ability to take a leadership role in the community. all -
Learning Resources
Required Resources
None.Recommended Resources
- Cohen, Graham, "A course in modern analysis and its applications"
- Simmons, George F., "Introduction to topology and modern analysis''
- Apostol, Tom M., "Mathematical analysis''
- Kreyszig, Erwin, "Introductory functional analysis with applications''
- Sutherland, Wilson A., "Introduction to metric and topological spaces''
- Munkres, James, "Topology"
- Larusson, Finnur, "Lectures on real analysis" (the last two chapters)
Online Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni. -
Learning & Teaching Activities
Learning & Teaching Modes
The lecturer will guide the students through the course material in 30 lectures. Students are expected to actively engage with the material during the lectures, and interaction and discussion of any difficulties that arise during the lectures is encouraged. Students are expected to attend all lectures, but (where possible) the lectures will be recorded to help cover absences and for revision purposes. Students will be expected to present solutions to fortnightly tutorial problems. Fortnightly assignments help develop understanding of the theory and its applications, and timely feedback allows students to gauge their progress.Workload
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Lectures 30 90 Tutorials 5 25 Assignments 6 42 Total 157 Learning Activities Summary
Tutorials in Weeks 3, 5, 7, 9, 11 cover the material of the previous two weeks.Lecture Schedule Week 1 Metric spaces Metric spaces, examples, convergent sequences, open and closed sets. Week 2 Metric spaces Open and closed sets (cont.), Cauchy sequences, complete metric spaces. Week 3 Metric spaces Continuous maps, the Banach fixed point theorem, motivation and examples. Week 4 Metric spaces Picard's existence and uniqueness theorem for solutions of differential equations. Week 5 Metric spaces Compactness, uniform continuity, the Heine-Borel theorem, the Arzela-Ascoli theorem. Week 6 Topology Topological spaces, examples, Hausdorff spaces, compact spaces. Week 7 Topology Continuous maps, homeomorphisms, connected and path connected spaces. Week 8 Topology Connected and path connected spaces (cont.). Normed vector spaces, Banach spaces, examples. Week 9 Hilbert and Banach spaces Bounded linear maps, bounded linear functionals, dual spaces. Week 10 Hilbert and Banach spaces Inner products, Cauchy-Schwarz inequality, parallellogram law, orthogonality. Week 11 Hilbert and Banach spaces Hilbert spaces, examples, orthogonal projections, Riesz representation theorem. Week 12 Hilbert and Banach spaces Adjoint operators, structure theorem for separable Hilbert spaces. -
Assessment
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 Summary
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 70% All Homework assignments Formative and summative Weeks 2, 4, 6, 8, 10, 12 25% All Tutorial exercises Formative Weeks 3, 5, 7, 9, 11 5% All Assessment Related Requirements
An aggregate score of 50% is required to pass the course.Assessment Detail
Assessment task Set Due Weighting Assignment 1 Week 1 Week 2 4 1/6% Tutorial exercises 1 Week 2 Week 3 see below Assignment 2 Week 3 Week 4 4 1/6% Tutorial exercises 2 Week 4 Week 5 Assignment 3 Week 5 Week 6 4 1/6% Tutorial exercises 3 Week 6 Week 7 Assignment 4 Week 7 Week 8 4 1/6% Tutorial exercises 4 Week 8 Week 9 Assignment 5 Week 9 Week 10 4 1/6% Tutorial exercises 5 Week 10 Week 11 Assignment 6 Week 11 Week 12 4 1/6%
Each student will present at least once in the tutorials. Tutorial presentations will be worth 5%. This may have to be adjusted depending on enrolment.Submission
Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.Course Grading
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description FNS Fail No Submission F 1-49 Fail P 50-64 Pass C 65-74 Credit D 75-84 Distinction HD 85-100 High Distinction CN Continuing NFE No Formal Examination RP Result Pending 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 .
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Student Feedback
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.
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Student Support
- Academic Integrity for Students
- Academic Support with Maths
- Academic Support with writing and study skills
- Careers Services
- Library Services for Students
- LinkedIn Learning
- Student Life Counselling Support - Personal counselling for issues affecting study
- Students with a Disability - Alternative academic arrangements
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Policies & Guidelines
This section contains links to relevant assessment-related policies and guidelines - all university policies.
- Academic Credit Arrangements Policy
- Academic Integrity Policy
- Academic Progress by Coursework Students Policy
- Assessment for Coursework Programs Policy
- Copyright Compliance Policy
- Coursework Academic Programs Policy
- Intellectual Property Policy
- IT Acceptable Use and Security Policy
- Modified Arrangements for Coursework Assessment Policy
- Reasonable Adjustments to Learning, Teaching & Assessment for Students with a Disability Policy
- Student Experience of Learning and Teaching Policy
- Student Grievance Resolution Process
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Fraud Awareness
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