MATH312 - Geometry and Topology

Year

Credit points

Campus offering

Prerequisites

MATH104 Differential and Integral Calculus and MATH205 Geometry

Teaching organisation

Unit rationale, description and aim

Forming the last unit in the calculus sequence of units, and starting the study of analysis, this unit introduces topology from a geometric viewpoint. Geometric properties that do not depend upon length are first investigated, followed by a consideration of the basic definitions of topology. Several problems that can be solved using these ideas will be considered.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Demonstrate an understanding of the motivation for studying spherical and hyperbolic geometries (GA5, GA8) 

LO2 - Demonstrate a basic understanding of projective geometry (GA5) 

LO3 - Describe the ideas underlying topology (GA5, GA9) 

LO4 - Explain the differences between various simple topological surfaces (GA5) 

LO5 - Describe the connection between topology and graphs (networks) (GA5, GA9) 

LO6 - Define the map colouring problem and prove the Six-Colour Theorem (GA5) 

LO7 - Describe some basic properties of knots and knot polynomials (GA5, GA8, GA9, GA10) 

LO8 - Solve problems involving topological ideas (GA5, GA8). 

Graduate attributes

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA9 - demonstrate effective communication in oral and written English language and visual media 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics may include: 

1. Revision of Euclidean geometry and introduction to spherical and hyperbolic geometries 
2. Introduction to projective geometry. 
3. Some theorems of projective geometry 
4. Introduction to topology 
5. Topology of surfaces 
6. Introduction to graphs (networks) 
7. Connectedness and planarity, Euler’s formula 
8. Map colouring 
9. Introduction to knots, definition, equivalence 
10. Minimal surfaces, perimeters, the isoperimetric and related problems. 

Learning and teaching strategy and rationale

This unit includes 4 contact hours per week over 12 weeks, comprising 2 hours of lectures and 2 hours of tutorials each week. Use will be made of both electronic resources, including online resources, as well as hands-on experience of the objects studied — in particular topological surfaces, map colouring, knots and minimal surfaces. 

Assessment strategy and rationale

A range of assessment procedures will be used to meet the unit learning outcomes and develop graduate attributes consistent with University assessment requirements. Such procedures may include, but are not limited to: essays, reports, examinations, student presentations or case studies. 

Consideration should be given to including student presentations in the assessment schedule for this unit. 

Overview of assessments

Representative texts and references

Brannan, D., Esplen, M., & Gray, J. (1999). Geometry. Cambridge: Cambridge University Press. 

Coxeter, H. (1989). Introduction to geometry, New York: John Wiley & Sons. 

Coxeter, H., & Greitzer, S. (1996). Geometry revisited. Washington DC: The Mathematical Association of America. 

Flegg, H. (2003). From geometry to topology. New York: Dover. 

Kelley, J., & Sloan, S. (2008). General topology. Mountain View, CA: Ishi Press. 

Marcus, D. (2008). Graph theory: A problem oriented approach. Washington DC: The Mathematical Association of America. 

Reid, M., & Szendroi, B. (2005). Geometry and topology. Cambridge: Cambridge University Press. 

Richeson, D. (2008). Euler’s gem: The polyhedron formula and the birth of topology. Princeton: Princeton University Press 

Stahl, S. (2010). Geometry from Euclid to knots. Mineola, NY: Dover Publications. 

Wilson, R. (2010). Introduction to graph theory. Pearson.