PURE MTH 7055 - Topology & Analysis

North Terrace Campus - Semester 1 - 2014

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.

• General Course Information
Course Details
Course Code PURE MTH 7055 Topology & Analysis Pure Mathematics Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week MATHS 1012 MATHS 2100 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.
Course Staff

Course Coordinator: Dr Paul McCann

Course Timetable

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

• Learning Outcomes
Course Learning Outcomes
1. Demonstrate an understanding of the concepts of metric spaces and topological spaces, and their role in mathematics.
2. Demonstrate familiarity with a range of examples of these structures.
3. Prove basic results about completeness, compactness, connectedness and convergence within these structures.
4. Use the Banach Fixed Point Theorem to demonstrate the existence and uniqueness of solutions to classes of equations.
5. Prove basic results about the properties of bounded operators on Hilbert spaces.
6. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
7. Demonstrate skills in communicating mathematics orally and in writing.

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
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1,2,5,6
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 3,4,5,6
Skills of a high order in interpersonal understanding, teamwork and communication. 2,5,6
A proficiency in the appropriate use of contemporary technologies. 1,2,6
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 1,3,6
A commitment to the highest standards of professional endeavour and the ability to take a leadership role in the community. 5,6
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. 5,6
• Learning 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''
Online Learning

Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
• Learning & Teaching Activities
Learning & Teaching Modes
The lecturer will present 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 lecture 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 solution to fortnightly tutorial problems. Fortnightly assignments helps increase  understanding of the theory and its applications, and timely feedback through these problems allows students to gauge their progress.

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

 Week 1 Review, Metrics Introduction: review of sets, relations and functions. Metric spaces and examples. Week 2 Metric Spaces Cauchy sequences, convergence in metric spaces. Open sets, closed sets. Week 3 Metric Spaces Closure, completeness, subsets of complete spaces. Banach Fixed Point Theorem. Week 4 Metric Spaces Examples of Fixed Point Theorem. Completing a metric space. Continuous functions. Week 5 Metric Spaces Compactness, equivalence with sequential compactness, and consequences. Week 6 Metric Spaces Properties of continuous functions on compact sets. Review of metric spaces and introduction to topological spaces. Week 7 Topology Topological spaces, examples. Ordering topologies. Equivalent axiom systems for topological spaces. Convergence in Hausdorff and non-Hausdorff spaces. Week 8 Topology Compactness. Zariski topology, subspaces. Connectedness and path connectedness. Week 9 Topology Homeomorphism, topological invariants, generating topologies, weak topology. Week 10 Topology Product topology, quotient topology, summary. Hilbert spaces. Week 11 Hilbert Spaces Completion of square summable sequences. Orthonormal sets, orthonormal bases, separable spaces. Decomposition via orthogonal complement. Week 12 Hilbert Spaces Hilbert dimension. Uniqueness of "Hilbert Space". Fourier Series. Bounded operators on Hilbert space. Examples, and spectra: point spectrum and continuous spectrum.
Tutorials in Weeks 3, 5, 7, 9, 11 cover the material of the previous two weeks.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

Assessment Summary

Examination Summative Examination period 70% All
Homework assignments Formative and summative Weeks 2, 4, 6, 8, 10, 12 24% All
Tutorial exercises Formative Weeks 3, 5, 7, 9, 11 6% All

Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
Assessment Detail

Assignment 1 Week 1 Week 2 4%
Tutorial exercises 1 Week 2 Week 3 see below
Assignment 2 Week 3 Week 4 4%
Tutorial exercises 2 Week 4 Week 5
Assignment 3 Week 5 Week 6 4%
Tutorial exercises 3 Week 6 Week 7
Assignment 4 Week 7 Week 8 4%
Tutorial exercises 4 Week 8 Week 9
Assignment 5 Week 9 Week 10 4%
Tutorial exercises 5 Week 10 Week 11
Assignment 6 Week 11 Week 12 4%

It is expected that each student will present at least once in the tutorials. Tutorial presentations will be worth 6%. 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.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
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 Access Adelaide.

• 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.

• Student Support
• Policies & Guidelines
• 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 studentâ€™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.

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