MTH5111 · Differential geometry

Overview

Manifolds are topological spaces that are locally homeomorphic to Euclidean space. A differentiable structure on a manifold makes it possible to generalize many concepts from calculus in Euclidean spaces to manifolds. This is a course on differentiable manifolds and related basic concepts, which are the common ground for differential geometry, differential topology, and geometric analysis. Foundational topics covered in the unit include: Smooth manifolds and coordinate systems, submanifolds, tangent and cotangent bundles, tensor bundles, tensor fields, Lie derivatives and tensor derivations. This unit will also cover advanced topics and applications such as: degree theory, de Rham cohomology, symplectic geometry and classical mechanics, Riemmanian geometry, comparison geometry, Lie groups and homogeneous spaces.

Areas of study: Master of Mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

Learning outcomes

1. Apply expert differential geometric techniques to solve problems that arise in pure and applied mathematics.
2. Construct coherent and precise logical arguments.
3. Develop and extend current techniques in differential geometry so that they can be applied to new situations in novel ways.
4. Communicate complex ideas effectively.
5. Independently learn and assimilate new mathematical ideas and techniques.

Workload

• Three hours of seminars; • One hour of applied class and • Eight hours of independent study per week

Student experience, not official rules. Nothing here changes what the Handbook says.