MANIFOLDS AND TOPOLOGY - 2022/3

Module code: MAT3009

Module Overview

This module introduces students to topological spaces and manifolds. Topology is the study of properties of spaces which are invariant under continuous transformations, and forms one of the cornerstones of pure mathematics. Study of topology leads to a deeper understanding of concepts of continuity and connectedness. After introducing these key ideas, with appropriate examples, smooth manifolds are introduced. Differential forms and cohomology are developed, which enables the classification of manifolds using topologically invariant numbers.

Module provider

Mathematics

GRANT James (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 6

JACs code: G100

Module cap (Maximum number of students): N/A

Independent Learning Hours: 95

Lecture Hours: 11

Seminar Hours: 11

Guided Learning: 11

Captured Content: 22

Module Availability

Semester 1

Prerequisites / Co-requisites

None

Module content

Indicative content includes:

• Introduction to Manifolds and Topology, point set topology, homeomorphisms and diffeomorphisms, coordinate patches, the definition of a manifold with boundary, orientation

• Vector Fields and Differential Forms: vector spaces and their duals (revision), the summation convention, tangent vectors, vector fields, exterior algebra and calculus, differential forms, the inner derivative, coordinate-independence, integration of differential forms, Stokes' Theorem.

• De Rham Cohomology: sequences, exactness, the de Rham complex, homotopy, the Poincaré Lemma, de Rham cohomology, invariance under homotopy, Brouwer's fixed-point theorem

• Algebraic Topology: the Mayer-Vietoris sequence, Betti numbers, classification of manifolds, generators of cohomology groups; short exact sequences, maps in cohomology.

Assessment pattern

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test ONLINE TEST 20
Examination Online ONLINE EXAM 80

Alternative Assessment

NA

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

• The ability to understand and formulate statements about the topology and geometry of various spaces.

• A knowledge of the subject, important definitions, and theorems with proofs

• The ability to apply this knowledge to the analysis of unseen problems in the tests and the examination.

Thus, the summative assessment for this module consists of:

• One two hour examination (the best three answers out of five questions contribute to the exam mark) at the end of Semester 2; worth 80% of the module mark.

• One in-semester test, worth 20% of the module mark.

Formative assessment and feedback

There is written feedback from marked coursework assignments over an 11 week period.
Verbal feedback is provided by the lecturer during seminars, and also in office hours.

Module aims

• Introduce students to topological spaces, and ideas such as Haudsorff topological spaces, homeomorphisms, connectedness and compactness.
• Introduce students to the core ideas of differential geometry, such as smooth manifolds, vector fields, differential forms and operators such as the exterior derivative and the wedge product.
• Develop ideas of integration over manifolds, Stokes's Theorem and its applications, culminating in an investigation of de Rham cohomology, including the Poincarè Lemma and the Mayer-Vietoris sequence.

Learning outcomes

Attributes Developed
1 Demonstrate understanding of topological spaces and smooth manifolds, properties of differential forms and the action of the exterior derivative and wedge product. K
2 Apply these techniques in calculating the homotopy operator for closed differential forms, solving certain classes of partial differential equations, and using Stokes's Theorem to determine whether certain closed differential forms are exact. KCT
3 Construct the Mayer-Vietoris sequence for large classes of manifolds and use the associated techniques to calculate the corresponding Betti numbers. Understand how this construction can be used to distinguish between topologically distinct spaces. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Be an introduction to the core ideas in topology and differential geometry, concentrating in particular on differential forms and how their properties can be used to classify different types of manifolds.

The learning and teaching methods include:

• 2x1 and 1x1 hour lectures per week, for 11 weeks, with typeset notes to complement the course, and Q+A opportunities for students.

• 4 additional seminars to discuss aspects of the course, such as the example
sheets provided to the students.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT3009

Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2022/3 academic year.