Topics in Modern Geometry
To provide an introduction 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.
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.
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 their problem solving skills and writing skills,
- have gained an appreciation of connections between areas.
Using an abstract framework to better understand how to attack a concrete problem.
Affine and projective varieties
The Zariski topology
Hilbert’s Basis Theorem
Families and parameter spaces
Smoothness and tangent spaces
Reading and References
- Kirwan, Complex Algebraic Curves
- Harris, Algebraic Geometry: A First Course
- Smith et al., An Invitation to Algebraic Geometry
- Reid, Undergraduate Algebraic Geometry
- Gathmann, Algebraic Geometry class notes
MATH20200 Metric Spaces and MATH21800 Algebra 2. MATH33300 Group Theory is helpful but not essential.
Methods of teaching
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.
Methods of Assessment
The pass mark for this unit is 40.
The final mark is calculated as follows:
- 80% Exam
- 20% Coursework
NOTE: Calculators are not allowed in the examination.
For information resit arrangements, please see the re-sit page on the intranet.
Further exam information can be found on the Maths Intranet.