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Undergraduate Study

MATH3061 Geometry and Topology

General Information

This page contains information on the senior mainstream unit of study MATH3061.

  • Taught in Semester 2.
  • Credit point value: 6.
  • Classes per week: Three lectures and one tutorial.
  • Lecturer(s): Anne Thomas and Andrew Mathas .

Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the handbook entry for MATH3061 for further information relating to MATH3061.

You may also view the description of MATH3061 and the description of in the University's course search database.

For enrolled students or other authorized people only, here is a link to the Canvas page for MATH3061.

Students have the right to appeal any academic decision made by the School or Faculty: see sydney.edu.au/students/academic-appeals.html.

Consultation

LecturerAvailabilityTimeLocation
Anne ThomasWeeks 1 – 6Thursday 9amzoom
Andrew MathasWeeks 7 – 12Thursday 9amzoom

Assessment

Date*DescriptionBetter markWeighting
23:59 September 10 Geometry Assignment 10%
9:00 September 16 Geometry Quiz 10%
23:59 October 22 Topology Assignment 10%
9:00 November 4 Topology Quiz 10%
Exam 60%
All dates are given in Sydney time.

References

  • Nigel O'Brian and Jonathan Hillman, Lecture notes for MATH3061: Geometry and Topology
  • (G) Judith N. Cederberg,, A course in modern geometries. New York: Springer-Verlag, c1989, 513 73.
  • (G) George E. Martin, Transformation geometry: an introduction to symmetry. New York: Springer-Verlag, c1982, 516.55 8.
  • (G) William P. Thurston, The geometry and topology of three-manifolds. Princeton, NJ: Princeton University, 1979
  • (T) Colin Adams, The knot book : an elementary introduction to the mathematical theory of knots. Providence, R.I., American Mathematical Society, 2004, 514.2242 3
  • (T) Donald W.Blackett, Elementary topology: a combinatorial and algebraic approach. London; New York: Academic Press, 1982, 513.83 102
  • (T) P.A. Firby and C.F. Gardiner, Surface topology. New York: Ellis Horwood, 1991, 513.83 110 A.

Check your marks EdStem Lecture recordings Online resources References

Enquiries

All enquiries about this unit of study should be directed to MATH3061@sydney.edu.au. Any mathematical questions sent to this email address will be redirected to the EdStem forum (NOTE: to register for this course on EdStem you first need to do it through Canvas). Please give your name and SID when emailing us. We reserve the right not to reply to anonymous emails. We will use Canvas for assignment submission.

If you experience problems reading pdf files online, here are some useful tips.

Unit outline

Canvas webpage

The MATH3061 Canvas webpage contains details about consultations, assessment, textbooks, objectives, and learning outcomes for MATH3061.

Geometry
This part of the course will explore geometric transformations and properties that are invariant under these transformations.

Week 1
Linear algebra review. The Euclidean plane. Transformations. Isometries.
Week 2
Transformation groups. Reflections. Fixed points. Rotations.
Week 3
Involutions. Glide-reflections. Classification of isometries. Parity.
Week 4
Symmetry groups. Affine transformations. Derivative of an isometry.
Week 5
The projective plane. Projective lines. Collineations.
Week 6
Conics. Classification of Conics.
Topology
Topology is the study of surfaces under continuous deformation. That is, we allow ourselves to stretch surfaces but not to tear them.

Week 7
Graphs, subdivision, sums of degrees = twice the number of edges, connectedness, circuits, trees.
Week 8
Disc, annulus, torus, Möbius band, Klein bottle, sphere, projective planes, homeomorphism, stereographic projection.
Week 9
Triangulated surfaces, Euler characteristic, invariance under subdivision, cutting, pasting, boundaries, orientation, edge equation.
Week 10
Classification of surfaces, genus, oriented closed surfaces in three dimensions, handles, crosscaps.
Week 11
Platonic surfaces. Graphs on surfaces: K5 is not planar. Map colouring: the five colour theorem, the Heawood estimate for maps on surfaces.
Week 12
Knots: Polygonals knots, knots diagrams, the unknot, trefoil knots, figure eight knots, knot colouring. Knot determinants, n-colourings, Seifert surfaces, and knot genus.

Textbook and references

Lecture notes will be published in the resource table below.

Supplementary notes for both parts of math3061 are available online: These notes are also available as one book from Kopystop, 36 Mountain Street, Broadway.

See the course Canvas webpage for additional textbook references.

Further reading and resources (topology)

  1. Topological equivalence of a torus and a coffee cup
  2. The Klein bottle (YouTube video)
  3. Gluing a torus bottle (YouTube video)
  4. An Introduction to Topology, E. C. Zeeman
  5. Platonic solids (Wikipedia)
  6. The Four-Color Problem: Concept and Solution, Steven G. Krantz.
  7. The Rolfsen Knot Table
  8. Torus knots
  9. Seifert surface

Online resources

Lectures Tutorials Assessment
Week 1
9/8-13/8
Geometry 1A
Geometry 1B
Geometry 1C
Geometry 1D
Geometry 1E
No tutorials in week 1
Week 2
16/8-20/8
Geometry 2A
Geometry 2B
Geometry 2C
Geometry 2D
Geometry 2E
Tutorial 1 questions
Tutorial 1 solutions
Week 3
23/8-27/8
Geometry 3A
Geometry 3B
Geometry 3C
Geometry 3D
Geometry 3E
Tutorial 2 questions
Tutorial 2 solutions
Week 4
30/8-3/9
Geometry 4A
Geometry 4B
Geometry 4C
Geometry 4D
Geometry 4E
Tutorial 3 questions
Tutorial 3 solutions
Geometry Assignment - questions
Week 5
6/9-10/9
Geometry 5A
Geometry 5B
Geometry 5C
Geometry 5D
Geometry 5E
Tutorial 4 questions
Tutorial 4 solutions
Geometry Assignment (10%)
Due 23:59 September 10
Week 6
13/9-17/9
Geometry 6A
Geometry 6B
Geometry 6C
Geometry 6D
Geometry 6E
Tutorial 5 questions
Tutorial 5 solutions
Geometry Quiz (10%)
9:00 September 16
Week 7
20/9-24/9
Topology 7A
Topology 7A unfilled
Topology 7B
Topology 7B unfilled
Topology 7C
Topology 7C unfilled
Topology 7D
Topology 7D unfilled
Topology 7E
Topology 7E unfilled
Tutorial 6 questions
Tutorial 6 solutions
Geometry Assignment - solutions
Mid-semester break
Week 8
4/10-8/10
Labour Day
Topology 8A unfilled
Topology 8B
Topology 8B unfilled
Topology 8C
Topology 8C unfilled
Topology 8D
Topology 8D unfilled
Topology 8E
Topology 8E unfilled
Tutorial 7 questions
Tutorial 7 solutions
Labour Day
Week 9
11/10-15/10
Topology 9A
Topology 9A unfilled
Topology 9B
Topology 9B unfilled
Topology 9C
Topology 9C unfilled
Topology 9D
Topology 9D unfilled
Topology 9E
Topology 9E unfilled
Tutorial 8 questions
Tutorial 8 solutions
Topology Assignment - questions
Week 10
18/10-22/10
Topology 10A
Topology 10A unfilled
Topology 10B
Topology 10B unfilled
Topology 10C
Topology 10C unfilled
Topology 10D
Topology 10D unfilled
Topology 10E
Topology 10E unfilled
Tutorial 9 questions
Tutorial 9 solutions
Topology Assignment (10%)
Due 23:59 October 22
Week 11
25/10-29/10
Topology 11A
Topology 11A unfilled
Topology 11B
Topology 11B unfilled
Topology 11C
Topology 11C unfilled
Topology 11D
Topology 11D unfilled
Topology 11E
Topology 11E unfilled
Tutorial 10 questions
Tutorial 10 solutions
Week 12
1/11-5/11
Topology 12A
Topology 12A unfilled
Topology 12B
Topology 12B unfilled
Topology 12C
Topology 12C unfilled
Topology 12D
Topology 12D unfilled
Topology 12E
Topology 12E unfilled
Tutorial 11 questions
Tutorial 11 solutions
Topology Assignment - solutions
Topology Quiz (10%)
9:00 November 4
Week 13
8/11-12/11
Tutorial 12 questions
Tutorial 12 solutions
Topology Quiz - solutions

Timetable

 

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