Courses:

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After a historical motivation of the definition of a Lie group we shall develop basic manifold theory, vector fields, tangent spaces leading to the Exponential mapping for an affine connection. The exponential mapping for a Lie group emerges as a special case. The relationship between Lie subgroups and Lie subalgebras via the exponential mapping will be developed in detail. Lie's three fundamental theorems relating all finite-dimensional Lie algebras with local Lie groups will be proved via Maurer-Cartan forms and the global version by means of the Levi-decomposition. We conclude with a study of invariant differential forms on Lie groups and coset spaces, ending with some results about cohomology of Lie groups.

Textbook

Helgason, Sigurdur. Differential Geometry, Lie Groups, and Symmetric Spaces. Providence, R.I.: American Mathematical Society, 2001. ISBN: 0821828487.

Prerequisites

These are 18.100B (Analysis and Metric Spaces) and 18.700 (Linear Algebra). The course 18.101 (Calculus in Several Variables) would be useful and some familiarity with topological groups is also helpful.

Grading

There are no scheduled tests and no final exam. The course is graded on the basis of homework assignments. Many other problems are suggested during the term.