This course introduces the key concepts of modern abstract analysis. The philosophy of this course is that modern analysis plays a fundamental role in mathematics itself and in applications to areas such as physics, computer science, economics and engineering.

Topics to be covered include elementary set theory, metric spaces, sequences, uniform convergence, continuity, the contraction mapping principle, integral equations and differential equations, topological spaces. The special topic for MATH6110 students has, for example, previously included an introduction to one or more of the following topics: axiomatic set theory; the construction of the natural numbers, integers, rationals and reals; the p-adic numbers; harmonic analysis; numerical analysis; the prime number theorem.

Note: This course is partially co-taught with the undergraduate course MATH2320 but will be assessed separately. Masters students will have a separate additional class each week on an additional special topic, and additional assessment item(s).

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Explain the fundamental concepts of real analysis and their role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of real analysis techniques.
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from real analysis.
4. Apply problem-solving using real analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.
5. Use deep knowledge and understanding of advanced real analysis to formulate responses to complex concrete and abstract problems.

Research-Led Teaching

Materials from current research will be incorporated into the class as appropriate.

Required Resources

Lecture notes can be downloaded from the course Wattle site. Textbook by Terence Tao, titled "Analysis II", is recommended and can be accessed from the ANU library. Other material will also be provided in Wattle, e.g. additional notes, alternative proofs, etc.

Recommended Resources

There are many excellent texts on the topics we are covering. Most have a title like ‘Introduction to analysis’ or ‘Introduction to metric spaces’ or similar. Finding a book you like and doing problems from it is a good way to learn the material.

Some suggested books are:

• W. Rudin, Principles of Mathematical Analysis, McGraw-Hill. (A bit difficult, but good.)
• J. Giles, Introduction to the Analysis of Metric Spaces, Australian Mathematical Society lecture series no. 3. (Almost exactly right for the metric space component.)
• G. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill. (At a higher level and deals with much more material, though the book will serve you well in later courses.)
• H. Royden, Real Analysis, Prentice-Hall, various editions. (Like Simmons, this does more than you need but is good for several later courses.) The latest edition is by Royden and Fitzpatrick; it includes more material than the Royden editions.
• T.W. Korner, A Companion to Analysis, American Mathematical Society. (A different kind of textbook, designed to help you think about the material.)

There are a variety of online platforms you will use to participate in your study program at ANU, across all of your courses. These could include videos for lectures and other instruction, two-way video conferencing for interactive learning, email and other messaging tools for communication, interactive web apps for formative and collaborative activities, print and/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

Staff Feedback

Students will be given feedback in the following forms in this course:

• written comments
• verbal comments
• feedback to whole class, groups, individuals, focus group etc

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). Feedback can also be provided to Course Conveners and teachers via the Student Experience of Learning & Teaching (SELT) feedback program. SELT surveys are confidential and also provide the Colleges and ANU Executive with opportunities to recognise excellent teaching, and opportunities for improvement.

Other Information

Use of Artificial Intelligence (AI): unless explicitly allowed in the assessment instructions, assessment tasks for this course should be completed without the assistance of generative AI (for example, without using ChatGPT).

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Regular course block. Metric spaces: we will build upon the material on analysis in Math 1115 and 1116, and study analysis on general metric spaces, focussing on questions such as convergence, completeness, compactness and continuity. Particular attention will be placed on the concept of uniform convergence and its consequences, illustrated through the study of power series and ordinary differential equations.	For the whole course: 3 lectures per week; 1 workshop in most weeks; Weekly in-lecture quizzes on Thursday; weekly workshop submissions, ; a mid-semester exam; and a final examination.
2	Add-on lecture block. These lectures, once a week on Monday at 11am, are for MATH6110 students, and any MATH2320 who wish to participate in the add-on. The lectures start from Week 2. In Week 12, there will be a two-hour lecture from 10am to 12pm. We will study constructions of the real number field relying only on introductory set theory as the starting point, will extend the previous study of Riemann integration to consider the question of which discontinuous functions are Riemann integrable, and study elements of Fourier analysis.	Assessment for the add-on block will include weekly in-class quizzes and two assignments.

Tutorial Registration

Workshops begin in Week 2. Workshop registration is via MyTimetable. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Learning Outcomes
Weekly in-class quizzes during Thursday lectures	10 %	1,2,3,4
Workshop submissions	12 %	1,2,3,4
Mid-semester Assessment	23 %	1,2,3,4
Final Exam	30 %	1,2,3,4
In-class quizzes in Add-on Lectures	12 %	3,5
Add-on Assignments	13 %	1,2,3,5

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines , which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Integrity Rule before the commencement of their course. Other key policies and guidelines include:

• Academic Integrity Policy and Procedure
• Student Assessment (Coursework) Policy and Procedure
• Extenuating Circumstances Application
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure
• Code of practice for teaching and learning

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Skills website. In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

This course is delivered in-person, on campus.

Important note: all announcements will be made in class. It is expected that students will attend all lectures and workshops.

Examination(s)

This course includes a mid-semester assessment and a final examination. The details and mode of delivery for the assessment and the exam will be communicated through the course Wattle site and the ANU examination timetable. The mid-semester assessment and the final exam will both be in-person. The mid-semester assessment is scheduled for Week 7 during Thursday lecture. There is a hurdle requirement on the final exam: to pass the course, a student must score at least 35% of the available marks on the final exam.

Assessment Task 1

Weekly in-class quizzes during Thursday lectures

The exact value of this assessment task is 10.5% for MATH6110 students. The quizzes will happen from 12:35-12:45pm each Thursday during the lectures, starting Week 2. The only exception is Week 7, where we will have a mid-semester assessment during the lecture. The quizzes will ask only basic questions, which test your understanding of the fundamental ideas in the course. Anyone struggling with questions in the quizzes should take this as a warning and discuss that with the convenors. There will be 10 quizzes altogether (Weeks 2-6 and 8-12), and we will drop the three lowest scores.

Assessment Task 2

Workshop submissions

Submissions of a written solution to one of workshop problem. These problems are typically longer and resemble more the questions you will see in the mid-semester assessment and the final exam. You will discuss these questions in groups and with your demonstrator during the workshop. You will be asked to write up a detailed solution of your own to one of the questions during each workshop, and the detailed write up is normally due one week after the workshop, i.e., if you have a workshop on Wednesday, the solution will be due the next Wednesday. You are welcome to discuss workshop problems with your classmates at any time during the semester. However, when writing up solutions, you must do so on your own.

Assessment Task 3

Mid-semester Assessment

The exact value of this assessment task is 22.5% for MATH6110 students. To be held in Week 7 during the Thursday lecture.

This is a summative assessment. You will be required to solve problems using knowledge from the first half of the course and submit personal solutions.

The date, time and mode of the assessment will be announced on the course Wattle site.

Assessment Task 4

Final Exam

To be held during the final examination period. Worth 30% of the final grade. Check the Course Wattle site for further information, and the Central ANU examination timetable to confirm the date, time and location of the exam.

There is a hurdle requirement on the final exam: to pass the course, a student must score at least 35% of the available marks on the final exam.

Assessment Task 5

In-class quizzes in Add-on Lectures

The add-on lectures held on Monday at 11am each week starting from Week 2 with a two-hour lecture at 10am in Week 12. Starting from Week 3 we will have weekly quizzes in Monday lecture from 11:45 to 11:55. There will be no add-on lecture and no add-on quiz in Week 8. As with Thursday quizzes, these quizzes will have only basic questions. There will be 9 quizzes in total and we will drop three lowest scores.

Assessment Task 6

Add-on Assignments

There will be two assignments, each worth 6.5% of the final grade for a total of 13%, related to the add-on lecture material. They will be due in Weeks 6 and 12. These will have several questions of varied difficulty. Problems for assignments will be published in Week 2 and Week 8, respectively. Students are welcome to discuss assignment problems with the classmates. However, when writing up solutions, they must do so on their own.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. The University’s students are an integral part of that community. The academic integrity principle commits all students to engage in academic work in ways that are consistent with, and actively support, academic integrity, and to uphold this commitment by behaving honestly, responsibly and ethically, and with respect and fairness, in scholarly practice.

The University expects all staff and students to be familiar with the academic integrity principle, the Academic Integrity Rule 2021, the Policy: Student Academic Integrity and Procedure: Student Academic Integrity, and to uphold high standards of academic integrity to ensure the quality and value of our qualifications.

The Academic Integrity Rule 2021 is a legal document that the University uses to promote academic integrity, and manage breaches of the academic integrity principle. The Policy and Procedure support the Rule by outlining overarching principles, responsibilities and processes. The Academic Integrity Rule 2021 commences on 1 December 2021 and applies to courses commencing on or after that date, as well as to research conduct occurring on or after that date. Prior to this, the Academic Misconduct Rule 2015 applies.

 

The University commits to assisting all students to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. All coursework students must complete the online Academic Integrity Module (Epigeum), and Higher Degree Research (HDR) students are required to complete research integrity training. The Academic Integrity website provides information about services available to assist students with their assignments, examinations and other learning activities, as well as understanding and upholding academic integrity.

Online Submission

Workshop submission will be online, via a process that will be detailed on the course Wattle page. You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. This course does not use Turnitin, having been granted an exemption.

Hardcopy Submission

It is expected that all assessment submission for this course will be online (other than for the in-person exams). For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Late submission is not permitted. Submission of assessment tasks without an extension after the due date is not permitted: a mark of 0 will be awarded.

Referencing Requirements

The Academic Skills website has information to assist you with your writing and assessments. The website includes information about Academic Integrity including referencing requirements for different disciplines. There is also information on Plagiarism and different ways to use source material. Any use of artificial intelligence must be properly referenced. Failure to properly cite use of Generative AI will be considered a breach of academic integrity.

Returning Assignments

Graded assignments will be returned electronically.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Resubmission of assignments is not permitted.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Accessibility for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents all ANU students