Discrete mathematics and algebra

Описание мероприятия

Для кого эта программа

The course is designed to enable you to:

• obtain general knowledge about the areas of discrete mathematics and algebra
• understand a variety of methods used to construct mathematical proofs
• acquire an insight into applications such as coding and design

Описание программы

This course is intended to give an introduction to the areas of mathematics known as discrete mathematics and the study of modern algebra. A key aim is to provide an insight into the interactions between these areas, in particular to modern applications such as coding and cryptography.

Учебный план:

Counting: selections, inclusion-exclusion, partitions and permutations, Stirling numbers, generating functions, recurrence relations.

Graph Theory: basic concepts (graph, adjacency matrix, etc.), walks and cycles, trees and forests, colourings.

Set Systems: matching, finite geometries, block designs.

Abstract groups: revision of key concepts such as cyclic groups, subgroups, homomorphisms and Lagrange’s theorem. Conjugation and normal subgroups. Group actions.

Applications of algebra to discrete mathematics I: permutations, orbits and stabilisers, the orbit-stabiliser theorem; applications to counting problems.

Rings and polynomials: the Euclidean algorithm for polynomials, integral domains, ideals, factor rings, fields, field extensions.

Finite fields: construction, the primitive element theorem, and finite linear algebra.

Applications of algebra to discrete mathematics II: finite Geometry: designs, affine and projective planes.

Error-correcting codes: linear codes, cyclic codes, perfect codes.

Результат обучения:

At the end of the course and having completed the essential reading and activities, you should be able to:

• demonstrate knowledge definitions, concepts and methods in the topics covered and how to apply these
• find and formulate simple proofs
• model situations in a mathematical way and derive useful results.

Требования к поступающим:

If taken as part of a BSc degree, courses which must be passed before this courses may be attempted:

• MT2116 Abstract mathematics.