大象传媒

Prerequisites: MAT220 is essential, and it is possible to lower the introduction to depend solely on this. For the algebraic variant indicated above, you must have MAT224. For the topological variant you must either have taken MAT243 or follow it in parallel with the project.

Description: This title covers a wide range of projects that are relevant to the methods that are used by the members of the Topology Group. K-theory is based on the banal fact that there is no canonical choice of basis: the quagmire of linear algebra. Despite, or perhaps because of, K-theory's trivial origin, the theory has been an important - and elegant - tool to manage particularly complex mathematical systems.

The modern method of attack is to take the combinatorial problem and encode it in a space. This space's properties reflect the original problem, but are open to deformations of the space - without losing information! In this way, we can obtain results that the original context was too "stiff" to allow us to reach.

The precise formulation of the project will depend on the student, and can move in the algebraic or topological direction. In the algebraic direction, the focus will be on number theory; in the topological direction, the focus will be on bundles.

What you will learn:

1. In the algebraic direction: basic algebraic K-theory with particular focus on elementary K-theory and applications in algebra.
2. In the topological direction: you are introduced to a powerful tool for studying topological spaces, and special bundles over these.

References:

[1] In the algebraic direction: Milnor, John . Annals of Mathematics 大象传媒, No. 72. ; University of Tokyo Press, Tokyo, 1971. xiii+184 pp.
[2] In the topological direction: Atiyah, M.F. . Notes by D. W. Anderson. Second edition. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. xx+216 pp. ISBN: 0-201-09394-4.