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Module

MAS8715 : Algebraic Topology & Galois Theory

  • Offered for Year: 2021/22
  • Module Leader(s): Dr Alina Vdovina
  • Lecturer: Dr David Stewart
  • Owning School: Mathematics, Statistics and Physics
  • Teaching Location: Newcastle City Campus
Semesters
Semester 1 Credit Value: 20
Semester 2 Credit Value: 10
ECTS Credits: 15.0

Aims

To give an introduction to how discrete, algebraic methods can be used to solve continuous, topological problems.

To present an introduction to Galois theory, bringing together ideas from several other modules (linear algebra, algebra, group theory). To develop the theory of field extensions and soluble groups, culminating in the Galois correspondence. To apply the theory to find solutions of polynomial equations, to explain why and when these exist, and to solve problems concerning constructibility of geometric figures.

Module Summary

This course is an introduction to general topology, that is, the study of shape. Topology is sometimes referred to as the geometry of rubber sheets. One of the main questions is: given two shapes, can we stretch and shrink one of the shapes so that it becomes the shape of the other? To show that this is possible, we merely need to exhibit a deformation which transforms one into the other. However, to prove that this is not possible, we have to find something about shape which remains unaltered by any deformation. This course introduces one of the simplest yet most powerful of these invariants, the fundamental group. This measures the number and position of loops which can be drawn on the surface. As part of the course we shall develop Group Theory appropriate to Algebraic Topology.

Galois theory originates from the work of Evariste Galois, on the existence of formulas for the solutions of polynomial equations. The standard formula for the roots of a quadratic equation can be generalised to give a formula for roots of a polynomial of degree 3 or of degree 4. However, as Abels showed at the beginning of the 19th century, no such formula can exist for polynomials of degree 5 or more. To explain this, Galois’ idea was to consider “symmetries” between the solutions of a polynomial equation. Nowadays, these symmetries are described in terms of group theory, a subject which can be viewed as beginning in the work of Galois. In modern terminology, Galois demonstrated how solutions of a polynomial equation correspond to certain groups of symmetries of a field. Moreover, properties of solutions of an equation may be read off from properties of the group and vice-versa. The module covers the field and group theory necessary to establish this theory, and applies the theory to study the existence and computation of solutions of polynomials; and to the construction of regular polygons, using ruler and compasses.

Outline Of Syllabus

Review of topology. Quotient spaces. Connectivity and path connectivity. Homotopy and path homotopy. Composition of paths. Fundamental group. Covering spaces. Fundamental group of the circle. Retractions. Deformation retraction. The fundamental groups of some surfaces. Abelian groups, free groups and free products of groups. The Seifert-Van Kampen Theorem. Fundamental groups of a wedge of circles, the Torus and the Dunce cap. Classification of compact surfaces. Introduction to homology: simplicial complexes, chain complexes, simplicial homology.

Fields and polynomials: field extensions and their degrees; algebraic and transcendental extensions. Ruler and compass constructions: squaring the circle, regular polygons. Soluble groups: derived series, simple groups. Galois group of a field extension: Fundamental Theorem of Galois Theory. Solutions of equations by radicals.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture501:0050:0020 online 30 present in person
Scheduled Learning And Teaching ActivitiesLecture31:003:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture101:0010:00Problem Classes – On-Line
Guided Independent StudyAssessment preparation and completion501:0050:00Completion of in course assessments
Guided Independent StudyIndependent study1871:00187:00Preparation time for lectures, background reading, coursework review
Total300:00
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M4written exercise 1
Written exercise1M4written exercise 2
Written exercise2M4written exercise 3
Written exercise2M4written exercise 4
Written exercise2M4written exercise 5
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

Timetable