Course Catalogue

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This course will study differential topology and geometry of smooth manifolds. The course will introduce the tools used to study the topology of smooth manifolds and Riemannian geometry on such manifolds (the mathematics behind Einstein's theory of General Relativity).

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Mandatory Entry Requirements

Recommended Entry Requirements

4H Differential Geometry

4H Algebraic and Geometric Topology

Assessment

100% Final Exam

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

This course will study differential topology and geometry of smooth manifolds. These are the spaces of greatest interest in topology - examples include the three-dimensional physical universe and four-dimensional space-time we inhabit as well as spheres and tori (in any dimension) and two-dimensional surfaces. The start of the course will introduce tools to study the topology of smooth manifolds. The course will then study Riemannian geometry (the mathematics which enabled Einstein's General Relativity theory). A main goal of the course will be to understand how geometry and topology interact on manifolds, and how information about one can be used to understand the other.

At the end of this course students should be able to:

• Demonstrate knowledge of the central definitions and facts of selected topics in differential topology and Riemannian geometry and use these to solve problems of a numerical or logical nature. Specific topics which may be covered are:

Differential topology:

1. Definitions and examples of manifold, tangent bundle, differential forms,

tensor fields

2. Definition, properties and examples of integration on manifolds

3. Statement and applications of Stokes' theorem

4. Definition, properties and examples of de Rham cohomology.

5. Definition and examples of bundles and connections.

Riemannian Geometry: 

1. Definition, properties and examples of metrics, curvature, geodesics, exponential map.

2. Statement and applications of Gauss-Bonnet theorem.

3. Statement and application of Myers' theorem on topology of positively curved manifolds.

4. Statement and application of Cheeger-Gromoll splitting theorem.

5. Understand the basic outline of Ricci flow and Perelman's proof of the Poincar´e conjecture (and possibly also Geometrisation) for 3-manifolds.

•Reproduce the proofs of the main theorems covered in the course.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.