Introductory description

Many problems in modern pure Mathematics require a combination of algebraic and topological methods. One such famous problem, solved finally by Adams in 1960, was in which dimensions one could have division algebras. The module will explain the problem and its solution to Year 4 MMaths students, preparing them for research in pure Mathematics.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Cayley-Dickson Process and Octonions
Review of Representation Theory of Finite Groups
Hurwitz Theorem about Composition Algebras
Parallelisability of Spheres and Division Algebras
Foundations of Topological K-Theory
Lambda-operations and Adams Operations
Hopf Invariant

Learning outcomes

By the end of the module, students should be able to:

• Develop a sophisticated command of the many facets of a major branch of algebra with important applications across the whole of mathematics.
• Master an advanced command of octonions and their algebraic properties.
• Learn topological K-theory.
• Apply advanced notions of Topology such as Adams operations and Hopf invariant, both in theoretical arguments and in practical applications.

Indicative reading list

Number by Ebbinghaus, et al, Graduate Texts in Maths volume 123
Rings that are nearly associative by Zhevlakov et al., Pure and Applied Maths volume 104
Noncommutative rings by Herstein, Carus Maths Monographs volume 15
Vector bundles in K-theory by Hatcher, online book in progress

Subject specific skills

Working knowledge of composition algebras and topological K-theory
Ability to apply K-theory methods to a variety of problems. including those in Algebra, Geometry and Topology.

Transferable skills

Ability to translate scientific ideas into mathematical language.
Ability to communicate complex ideas and mathematical results clearly.
Ability to analyse and solve abstract mathematical problems.