What is the course about?
Spatial Languages is a series of courses in the Humanities department at City Lit. Each aims to unpack a mathematical theory in a way that is accessible and relevant to philosophers, creative artists and others. Mathematics offers great conceptual clarity and intuitive pictures that can be valuable when forming views about the visual.
In mathematics, space has traditionally been studied by geometry, whose characteristic gesture is measurement. Although it didn’t start out that way, geometry has therefore become a quantitative subject and answers to its questions are usually numerical: lengths, areas, angles and so on.
Topology takes a very different approach. It disregards everything about a space that can be measured in order to focus more closely on its qualitative properties, in particular the way it is connected together. The London tube map gets the geometry of its subject all wrong, but captures its topology perfectly.
This course presents ideas from topology in an accessible manner. Whenever possible we solve problems by making models and drawings rather than doing calculations. In fact, even in advanced topology one rarely sees calculation playing any role at all, and most numbers that appear are doing nothing more complicated than counting things.
What will we cover?
• Definitions of general topological spaces and the tame examples of them called “manifolds”
• 2D manifolds such as the torus, Möbius strip, Klein bottle and “projective plane”.
• Homeomorphisms, the continuous and reversible transformations that define the subject.
• Knots, links and braids, culminating in the practical construction of Seifert surfaces for knots.
• Technical notions of basic topology such as closure, compactness and the separation axioms, including weird spaces that arise from these.
• The “fundamental group” and covering spaces.
• Triangulation and basic ideas of (co)homology, with reference to 3D computer graphics.
• Intuitive pictures relating to intersection theory, Morse theory and differential geometry (without any technicalities)
• Proofs of some technical results in topology, and a discussion of the nature and role of proof in mathematics generally.
What will I achieve?
By the end of this course you should be able to...
• Give precise definitions of the key terms and ideas of topology.
• Describe the topological properties of manifolds, knots and links, and of some “degenerate” spaces.
• Pose questions about a topological space and explore various possible ways to approach answering them.
• Provide visual interpretations of these definitions and results and explore them through model-making and drawing.
• Identify (in many cases) when two spaces are topologically the same and explain why or why not.
• Apply the language of topology to a wide variety of problems and identify when this language is at work in different contexts.
What level is the course and do I need any particular skills?
This course has no specific prerequisites – in particular, no mathematical skills or knowledge will be assumed.
How will I be taught, and will there be any work outside the class?
We will use a mixture of presentation and discussion in class. Some optional reading for between classes may be provided.
Are there any other costs? Is there anything I need to bring?
No, all required materials will be supplied during the course.
When I've finished, what course can I do next?
Please visit our website at www.citylit.ac.uk for more information about courses currently on offer which explore mathematics.