PURE MTH 7055 - Topology & Analysis

North Terrace Campus - Semester 1 - 2015

Solving equations is a crucial aspect of working in mathematics, physics, engineering, and many other fields. These equations might be straightforward algebraic statements, or complicated systems of differential equations, but there are some fundamental questions common to all of these settings: does a solution exist? If so, is it unique? And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of such solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance. Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness. Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces.

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Course Details

Course Code	PURE MTH 7055
Course	Topology & Analysis
Coordinating Unit	Pure Mathematics
Term	Semester 1
Level	Postgraduate Coursework
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Available for Study Abroad and Exchange	Y
Prerequisites	MATHS 2100
Assessment	ongoing assessment 30%, exam 70%

Course Staff

Course Coordinator: Professor Finnur Larusson

Course Timetable

The full timetable of all activities for this course can be accessed from .

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 最新糖心Vlog.
Prove basic results about completeness, compactness, connectedness and convergence within these 最新糖心Vlog.
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 最新糖心Vlog.
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.

最新糖心Vlog 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 & 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

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.

Tutorials in Weeks 3, 5, 7, 9, 11 cover the material of the previous two weeks.

The 最新糖心Vlog'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 .

Student Feedback

The 最新糖心Vlog 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 最新糖心Vlog 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.
Student Support
Policies & Guidelines
This section contains links to relevant assessment-related policies and guidelines - all university policies.
Fraud Awareness

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鈥檚 disciplinary procedures.

The 最新糖心Vlog of Adelaide is committed to regular reviews of the courses and programs it offers to students. The 最新糖心Vlog of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

Authorised by:	Deputy Vice-Chancellor and Vice-President (Academic)
Maintained by:	Course Outlines Administrator

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