Geometry and Topology (MATH3061)

UNIT OF STUDY

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.

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Further unit of study information

Classes

Three 1 hour lectures and one 1 hour tutorial per week.

Assessment

One 2 hour exam, tutorial tests, assignments (100%)

Faculty/department permission required?

No

Unit of study rules

Prerequisites and assumed knowledge

12 credit points of Intermediate Mathematics

Prohibitions

MATH3006, MATH3001

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