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 to various types of geometries which are all central to modern research. The unit will look at basic concepts in algebraic geometry which is a requirement for research projects in areas of geometry, number theory, and advanced algebraic geometry.

Unit description

The aim of this course is to develop basic geometric tools to explore properties of systems of polynomial equations and varieties. The unit will start by giving the key definitions of affine varieties, the Zariski topology and manifolds with several examples given to illustrate the definitions. The unit will provide an introduction to algebraic curves, smoothness and tangent spaces of varieties.

Relation to Other Units

The course expands ideas introduced in MATH21800 Algebra 2, and has relations to MATH20200 Metric Spaces, MATH33300 Group Theory and MATHM1200 Algebraic Topology.

Additional unit information can be found at http://www.maths.bristol.ac.uk/study/undergrad/current_units/index.html

Students who successfully complete the unit should:

- be able to clearly define topological groups, discrete groups and manifolds
- be familiar with examples of all three;
- be able to use techniques from abstract algebra and mathematical analysis to solve problems in geometry;
- be familiar with aspects of one of differentiable manifolds, projective space and algebraic curves or hyperbolic geometry.
- have developed an awareness of a broader literature,
- have gained an appreciation of how the basic ideas may be further developed,
- be able 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.

Coursework will be 4 homework sheets during the lectures weeks 1-6 each worth 2.5% and then a sheet with questions from the whole unit set in weeks 7-10 worth 10%

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.

Recommended:

Harris, Algebraic Geometry: A First Course

Smith et al., An Invitation to Algebraic Geometry

Reid, Undergraduate Algebraic Geometry

Gathmann, Algebraic Geometry class notes