# Gwybodaeth Modiwlau

Module Identifier
MAM7120
Module Title
Modern Differential Geometry
2020/2021
Co-ordinator
Semester
Intended for use in future years
Mutually Exclusive
Pre-Requisite
Pre-Requisite
Pre-Requisite

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   conventional examination  100%
Supplementary Assessment 2 Hours   conventional examination  100%

### Learning Outcomes

On successful completion of this module students should be able to:

1. illustrate the basic theory of differential calculus in R^n;

2. compare the generalization of these concepts to the setting of differential manifolds;

3. define vector and tensor fields, and perform algebraic and analytic computions with them;

4. synthesise mathematical idenities in an intrinsic setting;

5. formulate and analyze physical models in terms of exterior calculus;

6. identify basic examples of (matrix) Lie groups;

### Aims

IMAPS wishes to introduce new level 3 modules reflecting research interests and expertise of new staff, thereby rectifying the previous problem of very limited range of modules for final year students. This module introduces the important area of differential geometry, MAM has some common lectures but extends the applications. It is intended to offer this module in alternate years. Differential geometry is currently unrepresented in the curriculum and we would like to introduce it as an option.

### Brief description

Differential Geometry is an important subject in modern mathematics, essential in the description of several areas of mathematical physics, and engineering.

### Content

Differential Calculus on R^n
Functions, vector fields, tensors. Curvilinear coordinates and hypersurfaces.
Differentiable Manifolds
Introduction of the notion of manifolds and generalization of the above concepts to this situation. Lie brackets of vector fields.
Connections
Affine connections, Koszul connections, torsion. Riemannian manifolds and the Levi-Civita connection. Geodesics.
Differential Forms
Introduction to exterior algebra and calculus
Lie groups
Lie groups and Lie algebras.

### Module Skills

Skills Type Skills details
Application of Number Throughout the module.
Communication Students will be expected to submit written worksheet solutions
Improving own Learning and Performance Feedback via tutorials
Information Technology Extensive use of spreadsheets.
Personal Development and Career planning Students will be exposed to an area of application that they have not previously encountered.
Problem solving All situations considered are problem-based to a greater or lesser degree.
Research skills Students will be encouraged to consult various books and journals for examples of application.
Subject Specific Skills Using differential geometric techniques in modeling. Understanding relevance to modern mathematical applications.
Team work N/A

### Notes

This module is at CQFW Level 7