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

Lecturer	Availability	Time	Location
Anne Thomas	Weeks 1 – 6	Thursday 9am	zoom
Andrew Mathas	Weeks 7 – 12	Thursday 9am	zoom

Assessment

Date^*	Description	Weighting
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.

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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:

Nigel O’Brian's Geometry lecture notes.
Jonathan Hillman's Topology lecture notes.

These notes are also available as one book from Kopystop, 36 Mountain Street, Broadway.

See the course Canvas webpage for additional textbook references.

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

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