MATH31442 - 2006/2007
General Information
- Title: Introduction to Algebraic Geometry
- Unit code: MATH31442
- Credits: 10
- Prerequisites: 212, 252 (ex-UMIST), MT2212, MT2262 (ex-VUM)
- Co-requisite units:
- School responsible: Mathematics
- Member of staff responsible: Dr Gabor Megyesi (Mathematics and Social Sciences P.7, Tel: 63644)
Specification
Aims
Brief Description of the unit
Algebraic geometry studies objects called varieties defined by polynomial equations. A very simple example is the hyperbola defined by the equation xy = 1 in the plane. There is a way of associating rings to varieties, and then the geometric properties can be studied using algebra, for example points correspond to maximal ideals, or the geometry of the variety can give information about certain algebraic properties of the ring. Algebraic geometry originated in nineteenth century Italy, but it is still a very active area of research. It has close connections with algebra, number theory, topology, differential geometry and complex analysis.
Learning Outcomes
Future topics requiring this course unit
Syllabus
- Affine and projective spaces.
- Affine and projective varieties.
- Co-ordinate rings.
- Function fields.
- Morphisms and rational maps.
- Hilbert's Nullstellensatz.
- The classification of curves.
- The group law on the points of an elliptic curve.
Textbooks
Teaching and learning methods
Two lectures per week plus one weekly examples class.