This module starts the study of metric spaces, focusing on their fundamental properties and applications. Students will gain a deep understanding of the concepts and techniques used to analyse metric spaces. This will cover key topics such as convergence, continuity, compactness, connectedness and completeness. The module will change focus to explore the geometry of curves and surfaces in 2- and 3-dimensional Euclidean space. Students will investigate the properties and behaviour of curves and surfaces, with a focus on curvature and geometric transformations.
Learning Outcomes
By the end of the module students should be able to:
Understand the definitions of metric and topological spaces and verify these definitions in examples.
Understand convergence and continuity and determine when sequences converge and when functions are continuous in examples.
Understand the definitions of compactness, connectedness, completeness and separation axioms, and prove basic results involving these properties.
Describe and analyse the geometry of curves and surfaces in 2- and 3-dimensional Euclidean space.
Compute and interpret various types of curvature for curves and surfaces.
Apply geometric transformations to solve problems in differential geometry.
Assessment
Assessment Methods & Exceptions
Assessment:
2hr examination (80%) In-course assessment (20%) (including a variety of assessment possibly including problem sheets, class tests, online quizzes and group projects)
