MATH3061: Semester 2, 2021

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Units MATH3061 Semester 2 2021 [Normal day]

Unit of study_

MATH3061: Geometry and Topology

Semester 2, 2021 [Normal day] - Camperdown/Darlington, Sydney

Overview

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

Unit code	MATH3061
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH3001 or MATH3006
Prerequisites ?	12 credit points of MATH2XXX
Corequisites ?	None
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	Anne Thomas, anne.thomas@sydney.edu.au
Lecturer(s)	Anne Thomas, anne.thomas@sydney.edu.au
	Milena Radnovic, milena.radnovic@sydney.edu.au
	Andrew Mathas, andrew.mathas@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Record+)	Final exam Online final exam	60%	Formal exam period	2 hours
Outcomes assessed:
Assignment	Geometry assignment Geometry assignment	10%	Week 05 Due date: 10 Sep 2021 at 23:59 Closing date: 17 Sep 2021	n/a
Outcomes assessed:
Online task	Quiz 1 (Geometry) Online quiz	10%	Week 06 Due date: 16 Sep 2021 at 09:00 Closing date: 16 Sep 2021	50 minutes
Outcomes assessed:
Assignment	Topology assignment Topology assignment	10%	Week 10 Due date: 22 Oct 2021 at 23:59 Closing date: 29 Oct 2021	n/a
Outcomes assessed:
Online task	Quiz 2 (Topology) Online quiz	10%	Week 12 Due date: 02 Nov 2021 at 12:00 Closing date: 02 Nov 2021	50 minutes
Outcomes assessed:
= Type B final exam ?

Assessment summary

Detailed information for each assessment can be found on Canvas.

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.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

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	1. Introduction; 2. Linear algebra review; 3. The Euclidean plane	Lecture (3 hr)
Week 02	1. Transformations; 2. Isometries; 3. Translations; 4. Reflections	Lecture (3 hr)
Week 02	1. Introduction; 2. Linear algebra review; 3. The Euclidean plane	Tutorial (1 hr)
Week 03	1. Fixed points; 2. Rotations; 3. Glide-reflections; 4. Classification of isometries	Lecture (3 hr)
Week 03	1. Transformations; 2. Isometries; 3. Translations; 4. Reflections	Tutorial (1 hr)
Week 04	1. Conjugation; 2. Parity; 3. Affine transformations	Lecture (3 hr)
Week 04	1. Fixed points; 2. Rotations; 3. Glide-reflections; 4. Classification of isometries	Tutorial (1 hr)
Week 05	1. The derivative of an isometry; 2. The projective plane	Lecture (3 hr)
Week 05	1. Conjugation; 2. Parity; 3. Affine transformations	Tutorial (1 hr)
Week 06	1. Collineations; 2. Conics	Lecture (3 hr)
Week 06	1. The derivative of an isometry; 2. The projective plane; 3. Collineations	Tutorial (1 hr)
Week 07	1. Graphs, subdivision; 2. Trees; 3. Eulerian circuits	Lecture (3 hr)
Week 07	1. Conics ; 2. Graphs, subdivision; 3. Trees; 3. Eulerian circuits	Tutorial (1 hr)
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)
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	Tutorial (1 hr)
Week 09	1. Invariance under subdivision; 2. Cutting and pasting; 3. Boundaries; 4. Orientation; 5. Edge equation	Lecture (3 hr)
Week 09	1. Invariance under subdivision; 2. Cutting and pasting; 3. Boundaries; 4. Orientation; 5. Edge equation	Tutorial (1 hr)
Week 10	1. Classification of surfaces: genus, oriented closed surfaces in three dimensions, handles, and crosscaps; 2. Platonic solids	Lecture (3 hr)
Week 10	1. Classification of surfaces: genus, oriented closed surfaces in three dimensions, handles, and crosscaps; 2. Platonic solids	Tutorial (1 hr)
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)
Week 11	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)
Week 12	Knots: polygonals knots, knots diagrams, the unknot, trefoil knots, figure eight knots, knot colouring, knot determinants, Seifert surfaces	Lecture (3 hr)
Week 12	Knots: polygonals knots, knots diagrams, the unknot, trefoil knots, figure eight knots, knot colouring, knot determinants, Seifert surfaces	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

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

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 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
Learning outcomes

Responding to student feedback

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

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

Additional information

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.

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

Additional information

Additional information

Disclaimer

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