Unit name | Topics in Modern Geometry 34 |
---|---|

Unit code | MATHM0008 |

Credit points | 10 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 1A (weeks 1 - 6) |

Unit director | Dr. Jordan |

Open unit status | Not open |

Pre-requisites |
MATH20200 (Metric Spaces) and MATH21800 (Algebra 2). MATH33300 (Group Theory) is helpful but not essential. Students may not take this unit if they have taken the corresponding Level H/6 unit Topics in Modern Geometry 3. |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Unit aims

To provide an introduction to various types of geometries which are all central to modern research. The unit will look at non-Euclidean geometries and also at algebraic geometry and in particular will look at areas very relevant to recent research.

General Description of the Unit

Geometry is a very significant part of several areas of mathematics and also has important applications to physics. The development of different geometries has been an important theme throughout the history of mathematics and is very relevant to current research. 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 unit 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.

Relation to Other Units

The course expands ideas introduced in Metric Spaces (Math 20200) and Algebra 2 (Math21800), and has relations to Group Theory (Math 33300) and Algebraic Topology (Math M1200).

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Learning Objectives

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 Lie groups;
- be familiar with aspects of algebraic curves.

By pursuing an individual project on a more advanced topic students should have:

- developed an awareness of a broader literature,
- gained an appreciation of how the basic ideas may be further developed,
- learned how to assimilate material from several sources into a coherent document.

Lectures, including examples and revision classes, supported by lecture notes with problem sets and model solutions. Self-study with directed reading based on recommended material.

80% Examination and 20% Coursework.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

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.
- Undergraduate algebraic geometry, Miles Reid, Cambridge University Press, 1988.
- Lie groups: an introduction through linear groups, Wulf Rossmann, Oxford University Press, 2002.
- Introduction to metric and topological spaces, W.A. Sutherland, Oxford University Press, 1975.
- General topology, S. Willard, Dover 1970.
- Differential Forms and Connections, R. W. R. Darling, CUP 1994.
- Lie Groups, Lie Algebras and Representations, B. C. Hall, Springer
- Graduate texts in mathematics.