Unit of study_

# MATH3061: Geometry and Topology

The aim of the unit is to expand visual/geometric ways of thinking. The Geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasising the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The Topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). Topics include the classification of surfaces, map colouring, decomposition of knots and knot invariants.

Code MATH3061 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? 12 credit points of MATH2XXX None MATH3001 or MATH3006 Theory and methods of linear transformations and vector spaces, for example MATH2061, MATH2961 or MATH2022

At the completion of this unit, you should be able to:

• LO1. demonstrate knowledge of the definitions and various properties of isometries, affine transformations and collineations, and the relationships between them
• LO2. describe the relationship between the Euclidean plane and the real projective plane, and use homogeneous coordinates and vector/matrix notation in various types of calculations in these planes
• LO3. classify an isometry into one of four categories using various properties (parity, fixed point set etc), write an isometry as a composite of one, two or three reflections, and interpret a composite of reflections as a specific isometry (e.g. by successfully applying the strategy of replacement of pairs of reflections);
• LO4. recognise a transformation as affine (and find its derivative) from its coordinate formula and calculate the coordinate formula of a specific affine transformation which maps a given triangle to another;
• LO5. find a collineation which maps a given quadrangle to another quadrangle, and a given conic to another conic in the projective plane;
• LO6. construct simple proofs of propositions dealing with isometries, affine transformations, collineations and the objects they act upon
• LO7. apply the notion of the topological equivalence of graphs and surfaces under cutting and pasting
• LO8. work with two ways of representing surfaces, in polygonal form or as a possible punctured sphere with handles and cross-caps attached
• LO9. classify surfaces and determine their standard forms
• LO10. investigate map colouring problems on surfaces
• LO11. determine possible regular polygonal decompositions of surfaces
• LO12. perform Reidemeister moves on knots and distinguish between knots using knot invariants

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.