Geometric structures and Lie groups

This mini-course’s aim is to introduce some concepts of Differential Geometry centred around Riemannian and complex manifolds defined with reference to the action of a Lie group. The approach will be elementary and previous knowledge of manifolds - albeit useful - will not be essential. Through examples we will familiarise ourselves with several important properties and results.

Depending on time and the participants’ interest, we could address some of the following:

1. Matrix groups: revisiting classical examples, properties, the concept of a Lie group.

2. Lie algebras and differential forms: definition of Lie algebra via the exterior derivative, nilpotent algebras, computing cohomology by invariant forms.

3. Metric structures on Lie groups: invariant and bi-invariant Riemannian metrics, curvature.

4. Transformation groups: group actions, examples of homogeneous spaces, the Erlangen Programme.

5. Structures arising from closed forms: symplectic forms, the Kähler property, existence and non-existence theorems.

6. Other geometric applications.

Reference textbooks:
• B.Hall, Lie algebras, Lie groups and representations, Springer, 2015
• J.Stillwell, Naive Lie theory, Springer, 2008
• S.Chiossi, Lectures on Lie theory: the geometry of Lie groups, Lie algebras and their representations, in preparation (version 2026).

Schedule:

Tuesday 12 May, 10:30-12:30 Room L + 16:30-17:30 Room F

Wednesday 13 May, 14:30-17:30 Room L

Tuesday 19 May, 10:30-12:30 Room L + 12:30-13:30 Room E

Wednesday 20 May, 14:30-17:30 Room L