MATH3061 Geometry and Topology
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.
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.
|Anne Thomas||Weeks 1 – 6||Thursday 9am||zoom|
|Andrew Mathas||Weeks 7 – 12||Thursday 9am||zoom|
|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%|
- 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.
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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The MATH3061 Canvas webpage contains details about consultations, assessment, textbooks, objectives, and learning outcomes for MATH3061.
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 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.
Lecture notes will be published in the resource table below.Supplementary notes for both parts of math3061 are available online:
See the course Canvas webpage for additional textbook references.
Further reading and resources (topology)
- Topological equivalence of a torus and a coffee cup
- The Klein bottle (YouTube video)
- Gluing a torus bottle (YouTube video)
- An Introduction to Topology, E. C. Zeeman
- Platonic solids (Wikipedia)
- The Four-Color Problem: Concept and Solution, Steven G. Krantz.
- The Rolfsen Knot Table
- Torus knots
- Seifert surface
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