Unit information: Introduction to Geometry in 2020/21

Unit name	Introduction to Geometry
Unit code	MATH20004
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Mackay
Open unit status	Not open
Pre-requisites	MATH11005 Linear Algebra and Geometry, MATH10012 ODEs, Curves and Dynamics, and MATH10011 Analysis
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit Aims

The aim of this unit is to introduce fundamental concepts in geometry in a hands-on and rigorous way, focused on curves and surfaces, and to lay the foundations for more advanced courses in later years.

Unit Description

Geometry is central to mathematics, both as a subject in its own right and as an essential viewpoint on nearly every area of pure and applied mathematics. This course focuses on the geometry of curves and surfaces in R² and R³, continuing on from calculus.

A key concept is the curvature of a surface, which describes locally how it bends, whether it is flat like Euclidean space, positively curved like a sphere, or, less-familiarly, negatively curved like a saddle. A major goal is to prove the Gauss-Bonnet theorem, which links the curvature of a surface to its overall shape.

Relation to Other Units

The unit complements MATH20901 Multivariable Calculus and MATH20006 Metric Spaces, and leads into later units in geometry such as MATH30018 Fields, Forms & Flows and MATH30001 Topics in Modern Geometry.

Intended Learning Outcomes

By the end of the course the students should

• have developed an understanding of basic notions and results in differential geometry, like tangents, normals, curvature and Gauss-Bonnet
• know and be able to use basic properties of Euclidean, spherical and hyperbolic geometries
• understand, be able to prove and apply the methods in the course to the description and solution of problems from pure and applied mathematics.

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References

Recommended

• Barrett O'Neill, Elementary Differential Geometry, Academic Press, 2006
• Manfredo Perdigao do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976
• Andrew Pressley, Elementary Differential Geometry, Springer-Verlag, 2010
• Dirk J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley, 1961
• P.M.H. Wilson, Curved Spaces, Cambridge University Press, 2008