18.755 Fall 2008

18.755, Fall 2008, Sigurdur Helgason

18.755: Introduction to Lie Groups
Manifolds and Lie groups. Examples of applications to differential equations. The exponential mapping and the connection to structure of Lie algebras, the adjoint groups and the automorphism groups. Discussion of Lie transformation groups with special emphasis on semisimple Lie groups and symmetric spaces. Invariant differential forms with Haar measure, integration on homogeneous spaces, cohomology of Lie groups and homogeneous spaces. Familiarity with topological group helpful. 18.101 is recommended but not required. Text: S. Helgason, Differential Geometry, Lie groups and Symmetric Spaces.

After a discussion of Lie groups as tools for differential equation theory, I will discuss basic manifold theory, differentiable functions, vector fields, tangent spaces , submanifolds, and integral curves to vector fields. This is contained in Ch.I of the text but after discussing integral curves to a vector field we proceed directly to Lie groups and will cover most of Chapter II. I will then dicuss semisimple Lie groups and Symmetric Spaces and solvable groups Then differential forms from Chapter and their role in he theory of Lie groups and homogeneous spaces.