# Mathematics

## ECM3732 - Applications of Geometry and Topology (2018)

MODULE TITLE CREDIT VALUE Applications of Geometry and Topology 15 ECM3732 Prof Mitchell Berger (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 51
DESCRIPTION - summary of the module content

On this module, you will have the opportunity to study mathematical topics involving geometry, topology, and their applications in science and technology.  Firstly, you will become familiar with the mathematical description of curves and surfaces, and the idea of topological equivalence. Secondly, you will learn about various topics from geometry and topology, such as knot theory, classification of surfaces, and the shape of bubbles and soap films. You will then learn about the applications of geometry and toplogy, including the geometry of DNA molecules, the shape of the universe, and the topology of magnetic fields.

Prerequisite module: ECM1706, ECM2706 or equivalent

AIMS - intentions of the module

This module intends to develop your sense of shape, geometry, and topology. By taking it, you will gain a better understanding of possible geometrical structures and their mathematical description. The module covers some applications of knot theory and braid theory in detail. You will also have the opportunity to learn about current cosmological speculations concerning the shape of the universe.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate a working knowledge of the mathematical representation of geometrical objects.
Discipline Specific Skills and Knowledge:
2 reveal an understanding of the key concepts of geometry and topology, and appreciate their relevance to many areas of mathematics.
Personal and Key Transferable/ Employment Skills and  Knowledge:
3 display enhanced problem-solving skills.
4 show competence in modelling geometric objects in computer graphics.
5 demonstrate self management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- curves and surfaces: parameterised curves and surfaces, manifolds, connectivity, genus, Euler’s Formula;

- basic geometry and topology of curves: tangent, normal, and binormal vectors, curvature and torsion, geometrical phase, linking number and crossing number;

- ribbons: twist, writhe. DNA geometry;

- knots: knots and links, knot invariants, DNA topology;

- braids: the braid group, relation to knots and links;

- applications: mixing theory, dynamics, solar flares;

- bubbles: surface curvature, minimal surfaces.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures Guided independent study 117 Coursework preparation, private study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Not applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours - Summer Exam Period All None
Coursework assignment 1: Basic geometrical ideas 10 10 hours All Written and verbal
Coursework assignment 2: Knots, links and Braids 10 10 hours All Written and verbal

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Banchoff, Thomas F, Lovett, Stephen Differential geometry of curves and surfaces A K Peters 2010 978-1568814568 [Library]
Extended Oprea, John The mathematics of soap films: explorations with Maple 1 AMS Bookstore 2000 0821821180 [Library]
Extended Carlson, Stephan C Topology of surfaces, knots and manifolds: a first undergraduate course 1 Wiley, New York 2001 0471355445 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES ECM1706, ECM2706
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Thursday 06 July 2017 Wednesday 27 February 2019
KEY WORDS SEARCH Geometry; topology; knot theory.