Algebra, or more accurately abstract algebra, extends the ideas of multiplication and addition in sets of matrices or real numbers to a more general setting. Abstraction allows us to view the results of specific calculations in a more generic setting. This is a core objective of pure mathematics. It means that a single theorem can be applied in many different mathematical situations.
The primary algebraic objects are groups, rings and fields and these will be the main players in the algebra part of the module. This second module builds on the first year algebra course, extending results and producing new ideas which help us gain a deeper understanding of the algebraic structures which govern mathematics. The fundamental notions which will be introduced are substructures, structure preserving functions and quotient structures and the course will introduce these ideas illustrating them with numerous examples.

Combinatorics studies discrete mathematical structures. These structures are very simple themselves, but they often give rise to incredibly complex problems that are beyond the capacity of current computers to solve. Combinatorics is an essential component of many mathematical areas and also has important applications in Computer Science, Physics, Economics and Biology.

The Combinatorics part of the module consists of three topics. The first discusses advanced counting arguments, illustrating links to other areas of Mathematics. The second consists of topics in Graph Theory. There are many beautiful results in this area (e.g. the four colour theorem). These results are easily accessible but often require surprising ideas. The third topic builds on the previous two and deals with Combinatorial Algorithms and their efficiency, thus emphasizing links between Combinatorics and Computer Science.

Learning Outcomes

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

Recognise when a set equipped with one or two binary operations satisfies the axioms for a group, a ring or a field;

Understand and prove elementary theorems about groups rings and fields;

Determine when a map is a ring or group homomorphism;

Construct quotient structures and work with them;

Give non trivial examples of groups, rings and fields;

Apply a variety of advanced counting techniques and prove combinatorial identities;

Understand certain fundamental concepts and results in Graph Theory (including graph colourings, trees and Euler circuits);

Understand basic notions of Complexity Theory and analyse basic algorithms for combinatorial problems;

Students taking the module at Level H will explore the subject beyond the taught syllabus.