## PURE MTH 4102 - Topology and Analysis - Honours

### North Terrace Campus - Semester 1 - 2022

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 4102 Topology and Analysis - Honours School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week MATHS 2100 Honours students only 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: Associate Professor Thomas Leistner

##### 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 differential equations.
5. Demonstrate an understanding of the concepts of Hilbert spaces and Banach spaces, and their role in mathematics.
6. Demonstrate familiarity with a range of examples of these structures.
7. Prove basic results about Hilbert spaces and Banach spaces and operators between such spaces.
8. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
9. 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)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1,2,3,4,5,6,7

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

8

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

9

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

8,9

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

8
• Learning Resources
None.
##### Recommended Resources
You are not expected to buy any textbook. If you wish to read a textbook along with your lecture notes, you can have a look at some of the following books.
• 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 course videos will be available on MyUni, guiding the students through 30 lectures worth of material. Students are expected to actively
engage with the material as they follow along. This is complemented by a weekly face-to-face workshop hour for active learning guided by the lecturer and a weekly tutorial. Students will be expected to participate in tutorials each week, in which students will solve problems together in small groups. It will be important for each student to participate actively in the online discussion board. The expectation is that students will ask questions, as well as answer each others questions. Interaction and discussion of any difficulties that arise during the lectures is encouraged. Short online quizzes will be given weekly to help develop understanding.  Four homework assignments help develop understanding of the theory and its applications, and timely feedback allows students to gauge their progress.  A group project with a written report further develops research skills, teamwork skills, and communication skills.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Lectures, workshops and quizzes 30 84 Tutorials 11 22 Assignments 4 24 Group project 1 26 Total 156
##### Learning Activities Summary
 Weeks 1-5 Metric spaces Metric spaces, examples, convergent sequences, open and closed sets, Cauchy sequences, complete metric spaces, continuous maps, the Banach fixed point theorem, motivation and examples, Picard's existence and uniqueness theorem for solutions of differential equations, compactness, uniform continuity, the Heine-Borel theorem, the Arzela-Ascoli theorem. Week 6-8 Topology Topological spaces, examples, Hausdorff spaces, compact spaces, continuous maps, homeomorphisms, connected and path connected spaces. Week 9-12 Hilbert and Banach spaces Normed vector spaces, Banach spaces, examples, bounded linear maps, bounded linear functionals, dual spaces, inner products, Cauchy-Schwarz inequality, parallellogram law, orthogonality, Hilbert spaces, examples, orthogonal projections, Riesz representation theorem, 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:

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 50% All
Homework assignments Formative and summative Weeks 3, 6, 9, 12 20% All
Quizzes Formative and summative Each week 10% All
Group project report Formative and summative Set in Week 6. Complete draft due for feedback from lecturer in Week 11. Final submission in Week 13. 15% All
Tutorial participation Formative and summative Each week 5% All

##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
##### Assessment Detail
Weekly quizzes each week each week 10%
Assignment 1 Week 1 Week 3 5%
Assignment 2 Week 4 Week 6 5%
Assignment 3 Week 6 Week 9 5%
Assignment 4 Week 10 Week 12 5%
Active tutorial participation each week each week 5%
Group project report Week 6 Week 13 15%
Final Examination Exam period Exam period 50%
There is a quiz due each week, each weighted equally.
A complete draft of the group project report is due to be handed in to the lecturer for feedback no later than Monday Week 11.
##### Submission
Homework assignments must be submitted on MyUni. It will be assumed that the students have read and accepted the Academic Honesty Statement on MyUni.

Assignments will be returned within two weeks. Students may apply to be excused from or obtain an extension for 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:

M11 (Honours Mark Scheme)
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.

• 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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