Topics in Modern Geometry 34

Unit aims

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.

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.

Learning objectives

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.

Transferable skills

Using an abstract framework to better understand how to attack a concrete problem.

Syllabus

Affine and projective varieties

The Zariski topology

Hilbert's Basis Theorem

Hilbert's Nullstellensatz

Families & parameter spaces

Smoothness and tangent spaces

Bertini's Theorem

• Harris, Algebraic Geometry: A First Course
• Smith et al., An Invitation to Algebraic Geometry
• Gathmann, Algebraic Geometry class notes

Unit code: MATHM0008
Level of study: M/7
Credit points: 10
Teaching block (weeks): 1 (1-6)
Lecturers: Dr Tom Ducat and Dr Gene Kopp

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 MATH30001 Topics in Modern Geometry 3.

None

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 50.

The final mark is calculated as follows:

• 80% from a 1 hour 30 minute exam in January
• 20% from a written project*

* Project will involve independent study, an awareness of a broader literature and a critical evaluation of how the basic ideas may be further developed.

NOTE: Calculators are not allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.