# MANIFOLDS AND TOPOLOGY - 2024/5

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 the 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.

MAT2047 Curves and Surfaces is a recommended prior module, but is not pre-requisite. Students will also find the material in MAT3044 Riemannian Geometry complementary.

### Module provider

Mathematics & Physics

### Module Leader

VYTNOVA Polina (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 6

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

## Overall student workload

Independent Learning Hours: 64

Lecture Hours: 33

Tutorial Hours: 5

Guided Learning: 15

Captured Content: 33

## Module Availability

Semester 2

## 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 and vector fields. Exterior algebra and calculus on manifolds. Differential forms. The inner derivative, coordinate-independence, integration of differential forms and 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 |
---|---|---|

School-timetabled exam/test | In-semester test (50 min) | 20 |

Examination | End-of-Semester Examination (2 hours) | 80 |

## Alternative Assessment

NA

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

- Understanding of subject knowledge, and recall of key definitions and properties of topological spaces and manifolds.
- The ability to identify and use the appropriate techniques to solve geometric problems relating to manifolds and topology.

Thus, the

__summative assessment__for this module consists of:

- One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcome 1 and 2.
- A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 3.

Formative assessment

There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in tutorials and office hours.

## Module aims

- Introduce students to topological spaces, and ideas such as Hausdorff 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 Poincare Lemma and the Mayer-Vietoris sequence.

## Learning outcomes

Attributes Developed | ||

001 | Students will demonstrate an understanding of topological spaces and smooth manifolds, properties of differential forms and the action of the exterior derivative and wedge product. | KC |

002 | Students will be able to calculate the homotopy operator for closed differential forms, solve certain classes of partial differential equations, and use Stokes's Theorem to determine whether certain closed differential forms are exact. | KC |

003 | Students will be able to construct the Mayer-Vietoris sequence for a large class of manifolds and use the associated techniques to calculate the corresponding Betti numbers. Students will 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:

Introduce students 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:

- Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.
- Five biweekly one-hour tutorials per semester. These tutorials provide an opportunity for students to gain feedback and assistance with the exercises which complement the module notes.
- Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.
- Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

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.

## Reading list

https://readinglists.surrey.ac.uk

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

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

**Digital Capabilities**: The SurreyLearn page for MAT3009 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

**Employability**: The module MAT3009 equips students with skills which significantly enhance their employability. Students will learn to visualise geometric problems and formulate these problems mathematically using tools from differential geometry. Students will learn to evaluate complex geometric problems, break them into manageable components, and apply their knowledge and logical reasoning to arrive at solutions. These are highly sought after skills in many professions.

**Global and Cultural Capabilities:** Student enrolled in MAT3009 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

**Resourcefulness and Resilience**: MAT3009 is a module which demands the ability to visualise geometric problems, and formulate and solve these problems using the abstract mathematics of differential geometry. This rigorous branch of mathematics blends abstract mathematics with geometry, in an approach which stretches students’ understanding and to which they will learn to adapt. Students will gain skills in analysing geometric problems using lateral thinking, and will complete assessments which challenge them and build resilience.

**Sustainability:** Differential geometry and topology lie in the foundation of computed tomography which has a broad variety of applications from geology to medicine. One or more case studies will be included in the module to illustrate applications in the context of sustainable decision-making.

## Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Economics and Mathematics BSc (Hons) | 2 | 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 2024/5 academic year.