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This subject extends the methods of calculus and linear algebra to study the geometry and topology of higher dimensional spaces. The ideas introduced are of great importance throughout mathematics, physics and engineering. This subject will cover basic material on the differential topology of manifolds including integration on manifolds, and give an introduction to Riemannian geometry. Topics include: Differential Topology: smooth manifolds, tangent spaces, inverse and implicit function theorems, differential forms, bundles, transversality, integration on manifolds, de Rham cohomology; Riemanian Geometry: connections, geodesics, and curvature of Riemannian metrics; examples coming from Lie groups, hyperbolic geometry, and other homogeneous spaces.
Intended learning outcomes
After completing this subject, students will gain:
- an understanding of the basic notions of Differential Topology, including smooth manifolds, vector bundles, differential forms and integration on manifolds;
- an understanding of the basic notions of Riemannian Geometry, including connections, curvature and geodesics;
- the ability to work with smooth manifolds, smooth maps, differential forms and Riemannian metrics;
- the ability to do geometric calculations in local coordinates;
- a knowledge of important examples of Lie groups and symmetric spaces;
- the ability to pursue further studies in this and related areas.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 2 December 2019