Department of Mathematics
MATH300 - Geometry and
Topology
Unit Syllabus
Geometry:
- Projective geometry: the real projective
plane considered as the set of 1-dimensional subspaces of 3-dimensional
euclidean space, collinearity and linear dependence, the collinearity
lemma, the theorems of Desargues and Pappus, cross-ratios, harmonic
conjugates, perspectivities and projectivities, the fundamental
theorem of projective geometry, finite projective planes and
combinatorial problems.
- Symmetry: isometries in 2- and 3-dimensional
euclidean space, groups of symmetries and the classification
of frieze and wallpaper patterns.
- Ruler and compass constructions, including
the impossibility of trisecting an arbitrary angle.
- Isoperimetric problems.
- Non-euclidean planes.
Topology:
- Surfaces: an informal introduction to
topological spaces and continuous functions and homeomorphisms,
(compact) surfaces including possible boundaries considered as
polygons with some or all edges identified in pairs, surgery,
Euler characteristic, orientability, the classification theorem
for surfaces, embedability of graphs in surfaces, maps on surfaces,
chromatic number, Heawood's formula, the chromatic number of
a graph.
- Knots and links: Equivalence of knots
and links, Reidemeister moves, the Alexander group and number,
the Alexander module over the ring of Laurent polynomials and
the Alexander polynomial, describing knots combinatorially.
