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Unit outline_

MATH3061: Geometry and Topology

Semester 2, 2024 [Normal day] - Camperdown/Darlington, Sydney

The aim of the unit is to expand visual/geometric ways of thinking. The Geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasising the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The Topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). Topics include the classification of surfaces, map colouring, decomposition of knots and knot invariants.

Unit details and rules

Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites
? 
12 credit points of MATH2XXX
Corequisites
? 
None
Prohibitions
? 
MATH3001 or MATH3006
Assumed knowledge
? 

Theory and methods of linear transformations and vector spaces, for example MATH2061, MATH2961 or MATH2022

Available to study abroad and exchange students

Yes

Teaching staff

Coordinator Stephan Tillmann, stephan.tillmann@sydney.edu.au
Lecturer(s) Stephan Tillmann, stephan.tillmann@sydney.edu.au
The census date for this unit availability is 2 September 2024
Type Description Weight Due Length
Supervised exam
? 
Final exam
Final exam
60% Formal exam period 2 hours
Outcomes assessed: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12
Assignment Geometry assignment
Geometry assignment
10% Week 05
Due date: 30 Aug 2024 at 23:59

Closing date: 09 Sep 2024
1 week
Outcomes assessed: LO1 LO2 LO3 LO4
Online task Quiz 1 (Geometry)
Online quiz
10% Week 07
Due date: 11 Sep 2024 at 23:59

Closing date: 11 Sep 2024
50 minutes
Outcomes assessed: LO1 LO6 LO5 LO4 LO3 LO2
Assignment Topology assignment
Topology assignment
10% Week 11
Due date: 18 Oct 2024 at 23:59

Closing date: 28 Oct 2024
1 week
Outcomes assessed: LO7 LO8 LO9
Online task Quiz 2 (Topology)
Online quiz
10% Week 13
Due date: 30 Oct 2024 at 23:59

Closing date: 30 Oct 2024
50 minutes
Outcomes assessed: LO7 LO11 LO9 LO8

Assessment summary

Detailed information for each assessment can be found on Canvas.

Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

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

For more information see sydney.edu.au/students/guide-to-grades.

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.

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.

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.

Support for students

The Support for Students Policy 2023 reflects the University’s commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy 2023. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

WK Topic Learning activity Learning outcomes
Week 01 1. Introduction; 2. Linear algebra review; 3. The Euclidean plane Lecture (3 hr) LO1 LO2
Week 02 1. Transformations; 2. Isometries; 3. Translations; 4. Reflections Lecture (3 hr) LO1 LO2 LO3
1. Introduction; 2. Linear algebra review; 3. The Euclidean plane Tutorial (1 hr) LO1 LO2
Week 03 1. Fixed points; 2. Rotations; 3. Glide-reflections; 4. Classification of isometries Lecture (3 hr) LO1 LO2 LO3
1. Transformations; 2. Isometries; 3. Translations; 4. Reflections Tutorial (1 hr) LO1 LO2 LO3
Week 04 1. Conjugation; 2. Parity; 3. Affine transformations Lecture (3 hr) LO3 LO4
1. Fixed points; 2. Rotations; 3. Glide-reflections; 4. Classification of isometries Tutorial (1 hr) LO1 LO2 LO3
Week 05 1. The derivative of an isometry; 2. The projective plane Lecture (3 hr) LO2 LO4 LO5
1. Conjugation; 2. Parity; 3. Affine transformations Tutorial (1 hr) LO3 LO4
Week 06 1. Collineations; 2. Conics Lecture (3 hr) LO5 LO6
1. The derivative of an isometry; 2. The projective plane; 3. Collineations Tutorial (1 hr) LO2 LO4 LO5 LO6
Week 07 1. Graphs, subdivision; 2. Trees; 3. Eulerian circuits Lecture (3 hr) LO7
1. Conics ; 2. Graphs, subdivision; 3. Trees; 3. Eulerian circuits Tutorial (1 hr) LO5 LO6 LO7
Week 08 1. Disc; 2. Annulus; 3. Torus; 4. Möbius band; 5. Klein bottle; 6. Sphere; 7. Projective planes; 8. Homeomorphism; 9. Stereographic projection; 10. Triangulated surfaces; 11. Euler characteristic Lecture (3 hr) LO7 LO8
1. Disc; 2. Annulus; 3. Torus; 4. Möbius band; 5. Klein bottle; 6. Sphere; 7. Projective planes; 8. Homeomorphism; 9. Stereographic projection; 10. Triangulated surfaces; 11. Euler characteristic Tutorial (1 hr) LO7 LO8
Week 09 1. Invariance under subdivision; 2. Cutting and pasting; 3. Boundaries; 4. Orientation; 5. Edge equation Lecture (3 hr) LO7 LO8 LO9
1. Invariance under subdivision; 2. Cutting and pasting; 3. Boundaries; 4. Orientation; 5. Edge equation Tutorial (1 hr) LO7 LO8 LO9
Week 10 1. Classification of surfaces: genus, oriented closed surfaces in three dimensions, handles, and crosscaps; 2. Platonic solids Lecture (3 hr) LO7 LO8 LO9
1. Classification of surfaces: genus, oriented closed surfaces in three dimensions, handles, and crosscaps; 2. Platonic solids Tutorial (1 hr) LO7 LO8 LO9
Week 11 1. Graphs on surfaces: K5 is not planar; 2. Map colouring: the five colour theorem, the Heawood estimate for maps on surfaces Lecture (3 hr) LO10 LO11
1. Graphs on surfaces: K5 is not planar; 2. Map colouring: the five colour theorem, the Heawood estimate for maps on surfaces Tutorial (1 hr) LO10 LO11
Week 12 Knots: polygonals knots, knots diagrams, the unknot, trefoil knots, figure eight knots, knot colouring, knot determinants, Seifert surfaces Lecture (3 hr) LO12
Knots: polygonals knots, knots diagrams, the unknot, trefoil knots, figure eight knots, knot colouring, knot determinants, Seifert surfaces Tutorial (1 hr) LO12

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

Lecture notes will be posted on the webpage. We will also post tutorials and solutions, quiz information, assignments and solutions.

Supplementary notes for both parts of Math 3061 are available as one book from Kopystop, 36 Mountain Street, Broadway. These notes are optional.

Reference Books for Geometry

1. William P. Thurston, The geometry and topology of three-manifolds, Princeton, NJ: Princeton University, 1979.
2. George E. Martin, Transformation geometry: an introduction to symmetry, New York: Springer-Verlag, c1982, 516.55 8.
3. Judith N. Cederberg, A course in modern geometries, New York: Springer-Verlag, c1989, 513 73.

Reference Books for Topology

1. P.A. Firby and C.F. Gardiner, Surface Topology, New York: Ellis Horwood, 1991, 513.83 110 A.

2. Donald W.Blackett, Elementary topology: a combinatorial and algebraic approach, London; New York: Academic Press, 1982, 513.83 102.

3. The knot book : an elementary introduction to the mathematical theory of knots, Providence, R.I., American Mathematical Society, 2004, 514.2242 3.

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.

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

  • LO1. demonstrate knowledge of the definitions and various properties of isometries, affine transformations and collineations, and the relationships between them
  • LO2. describe the relationship between the Euclidean plane and the real projective plane, and use homogeneous coordinates and vector/matrix notation in various types of calculations in these planes
  • LO3. classify an isometry into one of four categories using various properties (parity, fixed point set etc), write an isometry as a composite of one, two or three reflections, and interpret a composite of reflections as a specific isometry (e.g. by successfully applying the strategy of replacement of pairs of reflections);
  • LO4. recognise a transformation as affine (and find its derivative) from its coordinate formula and calculate the coordinate formula of a specific affine transformation which maps a given triangle to another;
  • LO5. find a collineation which maps a given quadrangle to another quadrangle, and a given conic to another conic in the projective plane;
  • LO6. construct simple proofs of propositions dealing with isometries, affine transformations, collineations and the objects they act upon
  • LO7. apply the notion of the topological equivalence of graphs and surfaces under cutting and pasting
  • LO8. work with two ways of representing surfaces, in polygonal form or as a possible punctured sphere with handles and cross-caps attached
  • LO9. classify surfaces and determine their standard forms
  • LO10. investigate map colouring problems on surfaces
  • LO11. determine possible regular polygonal decompositions of surfaces
  • LO12. perform Reidemeister moves on knots and distinguish between knots using knot invariants

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
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

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

No changes have been made since this unit was last offered.

General announcements for MATH3061 will appear on the EdStem and on Canvas.

Consultations with lecturers will be online, times will be announced on EdStem and on Canvas.

Tutorials start in week 2.

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.