Unit information: Topics in Modern Geometry 3 in 2013/14

Unit name	Topics in Modern Geometry 3
Unit code	MATH30001
Credit points	10
Level of study	H/6
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Jordan
Open unit status	Not open
Pre-requisites	Math 20200 (Metric spaces) and Math 21800 (Algebra 2). Students may not take this unit if they have taken the corresponding Level M/7 unit Topics in Modern Geometry 34.
Co-requisites	Math 33300 (Group Theory) is helpful but not essential.
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Geometry is a very significant part of several areas of mathematics and also has important applications to physics. The unit will start by giving the key definitions of topological groups, discrete groups and manifolds, with several examples given to illustrate the definitions. The course will then look at spherical geometry and hyperbolic geometry, as illustrations of non-Euclidean geometries. Finally the unit will give an introduction to algebraic curves and introduce the concept of a Lie group, both of these will be illustrated with several examples.

The aims of this new unit are to give students an introduction to selected areas of modern geometry and in particular to make students familiar with several examples of discrete groups and actions of discrete groups. A recurring theme in the unit will be gaining an understanding of abstract definitions via concrete examples.

Note that although there there may be a slight overlap in topics between this unit and the proposed new unit 'Introduction to Lie Groups, Lie Algebras and their Representations', the treatments will will be given from very different points of view and will be complementary rather than overlapping.

Intended Learning Outcomes

Students who successfully complete the unit should:

• be able to clearly define topological groups, discrete groups and manifolds and be familiar with examples of all three;
• use techniques from abstract algebra and mathematical analysis to solve problems in geometry;
• be familiar with aspects of hyperbolic geometry;
• be familiar with aspects of algebraic curves.

Teaching Information

Lectures, including examples and revision classes, supported by lecture notes with problem sets and model solutions.

Assessment Information

Formative assessment will be provided by problem sheets with questions that will be set and marked through the course.

The final assessment mark for the unit will be based on a 1½-hour written examination (100%).

Reading and References

Lecture notes and handouts will be provided covering all the main material.

The following supplementary texts provide additional background reading:

• Hyperbolic geometry, James W. Anderson, Springer, 2007.
• The geometry of discrete groups, Alan F. Beardon, Springer, 1983.
• Fuchsian groups, Svetlana Katok, University of Chicago Press, 1992.
• Complex algebraic curves, Frances Kirwan, Cambridge University Press, 1992.
• Algebraic curves: an introduction to algebraic geometry, William Fulton, Benjamin/Cummings Publishing, 1969.
• Lie groups: an introduction through linear groups, Wulf Rossmann, Oxford University Press, 2002.