# MATH3061 Geometry and Topology

## General Information

This page contains information on the Senior mainstream Unit of Study MATH3061 Geometry and Topology.

- Taught in Semester 2.
- Credit point value: 6.
- Classes per week: Three lectures and one tutorial.
- Lecturer(s): Boris Lishak 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 MATH3061 handbook entry for further information relating to MATH3061.

You may also view the description of MATH3061 in the central units of study database.

Students have the right to appeal any academic decision made by the School or Faculty. For further information, see the Science Faculty web site.

Mathematics and Statistics Honours Information Session:

**Thursday 13 September, 1–2pm,
Room 361, Carslaw Building.**

Refreshments will be provided.

*
This event is part of the
Postgraduate Study & Research Week, 10–14 September.
*

Check your marks EdStem Lecture recordings Online resources References Timetable

## 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. Please give your name and SID when emailing us. We reserve the right not to reply to anonymous emails.

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

## Information about the exam

For general information about exams, and the exam timetable, please see the University's examinations web page.

The MATH3061 exam is scheduled for 13:50-16:00 on Saturday November 24, 2016. The exam has six questions, three questions for each of the geometry and topology components of the course. Each question may be attempted. The exam is worth 66% of your final assessment.

*Please note that the questions for the geometry and topology
sections of the course must be answered in separate booklets*.

The exam is based on all the material covered in lectures, tutorials, assignments, and the quizzes. The best way to prepare for the exam is to read through and think about your lecture notes and then to try to solve as many questions as possible (without looking at the solutions) from the lectures, tutorials, and assignments. You might like to look at the 2013 MATH3061 exam. We strongly recommend that you attempt these questions before looking at the solutions, which will be made available from this web page before November 16. Here are the solutions to this exam:

Geometry Solutions Topology Solutions.

*We recommend that you do not look at these solutions until you have first attempted to do all of the questions in the exam.*

In addition, here are the questions for the 2014 exam together with the solutions for the topology questions.

## Pre-exam consultation times

Boris Lishak and Andrew Mathas will be available for pre-exam consultation at the following times:

- Boris Lishak
- Carslaw 534: 10-11am, Thursday November 15 and 22
- Andrew Mathas
- Carslaw 718: 10-11am, Wednesday November 14 and 21

## Unit outline

** Information Sheet **

The MATH3061 Information Sheet contains details of the lecturers and lecture times, consultation times, tutorials, assessment, textbooks, objectives, and learning outcomes for MATH3061.

**Geometry**

The course will closely follow the "Geometry" lecture notes by O'Brian.

- Week 1
- Introduction. Linear Algebra review. The Euclidean Plane. Reading: pages 1–13. [For the tutorial problem: the image of a point (x,y) under a halfturn about point P(p,q) is (2p−x,2q−y).]
- Week 2
- Transformations. Isometries. Translations. Reflections. Reading: pages 13–27 (skip the section on symmetries).
- Week 3
- Fixed points. Rotations. Glide-reflections. Classification of isometries. Reading: pages 28–40.
- Week 4
- Parity. Affine transformations. Reading: pages 51–58.
- Week 5
- The derivative of an isometry. The projective plane. Collineations. Reading: pages 67, 97–103.
- Week 6
- Collineations. Conics. Reading: pages 103–107, 113–115.
- Week 7
- Classification of Conics. Review. Reading: pages 116–121.

**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: K
_{5}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.
- Week 13
- Knot determinants, n-colourings, Seifert surfaces, and knot genus.

## Textbook and references

Lecture notes for both parts of MATH3061 are available for purchase as a book from Kopystop at 36 Mountain Street, Broadyway.

See the course information sheet 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

## Online resources

**Check your marks**

MATH3061 students: please check that your marks for Geometry Assignment, Geometry Quiz, Topology Assignment and Topology Quiz have been recorded correctly by pressing the "Check marks" button.

Please note that any corrections to Geometry Assignment marks must be made by Friday, Sep 21.

Please note that any corrections to Geometry Quiz marks must be made by Friday, Sep 21.

Please note that any corrections to Topology Assignment marks must be made by Friday, Nov 23.

Please note that any corrections to Topology Quiz marks must be made by Friday, Nov 23.

## Timetable

Show timetable / Hide timetable.