Outline

Overview

Topology is the mathematical theory of the "shape of spaces". It gives a flexible framework in which the fabric of space is like rubber and thus enables the study of the general shape of a space. The spaces often arise indirectly: as the solution space of a set of equations; as the parameter space for a family of objects; as a point cloud from a data set; and so on. This leads to strong interactions between topology and a plethora of mathematical and scientific areas. The love of the study and use of topology is far reaching, including the use of topological techniques in the phases of matter and transition which received the 2016 Nobel Prize in Physics. This unit introduces you to a selection of topics in pure or applied topology. Topology receives strength from its areas of applications and imparts insights in return. A wide spectrum of methods is used, dividing topology into the areas of algebraic, computational, differential, geometric and set-theoretic topology. You will learn the methods, key results, and role in current mathematics of at least one of these areas, and gain an understanding of current research problems and open conjectures in the field.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	None
Corequisites ?	None
Prohibitions ?	None
Assumed knowledge ?	Familiarity with metric spaces (e.g., MATH4061 or equivalent) and algebraic topology (e.g., MATH4311 or equivalent). Please consult with the coordinator for further information
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Anne Thomas, anne.thomas@sydney.edu.au
Lecturer(s)	Anne Thomas, anne.thomas@sydney.edu.au

Type	Description	Weight	Due	Length
Oral exam ?	Final exam Oral exam through Zoom.	50%	Formal exam period	20 minutes (oral)
Outcomes assessed:
Assignment	Assignment 1 Report	25%	Week 06 Due date: 09 Sep 2022 at 23:59	5 questions
Outcomes assessed:
Assignment	Assignment 2 Report	25%	Week 12 Due date: 28 Oct 2022 at 23:59	5 questions
Outcomes assessed:

Assessment summary

Assignment 1: Work individually on about 5 questions regarding the content of the course from weeks 1-5. Type the answers into a pdf file to be submitted at the end of week 6. Detailed information for each assessment can be found on Canvas in week 4.
Assignment 2: Work individually on about 5 questions regarding the content of the course from weeks 1-10. Type the answers into a pdf file to be submitted at the beginning of week 12. Detailed information for each assessment can be found on Canvas in week 10.

Assessment criteria

High Distinction-85-100-

At HD level, a student demonstrates a flair for the subject as well as a detailed and comprehensive understanding of the unit material. A ‘High Distinction’ reflects exceptional achievement and is awarded to a student who demonstrates the ability to apply their subject knowledge and understanding to produce original solutions for novel or highly complex problems and/or comprehensive critical discussions of theoretical concepts.

Distinction-75-84

At DI level, a student demonstrates an aptitude for the subject and a well-developed understanding of the unit material. A ‘Distinction’ reflects excellent achievement and is awarded to a student who demonstrates an ability to apply their subject knowledge and understanding of the subject to produce good solutions for challenging problems and/or a reasonably well-developed critical analysis of theoretical concepts.

Credit-65-74

At CR level, a student demonstrates a good command and knowledge of the unit material.
A ‘Credit’ reflects solid achievement and is awarded to a student who has a broad general
understanding of the unit material and can solve routine problems and/or identify and
superficially discuss theoretical concepts.

Pass- 50-64-

At PS level, a student demonstrates proficiency in the unit material. A ‘Pass’ reflects satisfactory achievement and is awarded to a student who has threshold knowledge.

Fail-0-49-

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Without special circumstances which qualify for special consideration, late submissions will not be accepted.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Weekly schedule

WK	Topic	Learning activity
Week 01	Motivation and Foundations	Lecture (3 hr)
Week 02	Fundamental Group	Lecture (3 hr)
Week 03	Groups and Spaces	Lecture (3 hr)
Week 04	Covering Spaces	Lecture (3 hr)
Week 05	Covering Spaces	Lecture (3 hr)
Week 06	Homology	Lecture (3 hr)
Week 07	Homology	Lecture (3 hr)
Week 08	Cellular and Simplicial Homology	Lecture (3 hr)
Week 09	Cohomology	Lecture (3 hr)
Week 10	Duality	Lecture (3 hr)
Week 11	Duality	Lecture (3 hr)
Week 12	Further topics	Lecture (3 hr)
Weekly	Weekly tutorial	Tutorial (1 hr)

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Required readings

Algebraic Topology by Allen Hatcher https://www.math.cornell.edu/~hatcher/AT/AT.pdf

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. Demonstrate a coherent and advanced understanding of the key concepts of fundamental group, covering spaces, homology and cohomology.
LO2. Apply the fundamental principles and results of algebraic topology to solve given problems.
LO3. Distinguish and compare the properties of different types of topological spaces and maps between them.
LO4. Formulate topological problems in terms of algebraic invariants and determine the appropriate framework to solve them.
LO5. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
LO6. Devise computational solutions to complex problems in algebraic topology.
LO7. Compose correct proofs of unfamiliar general results in algebraic topology.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

This is the first time this unit has been offered.

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect

Current students

MATH5340: Topics in Topology

Semester 2, 2022 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect