Lie Groups are of central importance in Mathematics and required background for every research mathematician and theoretical physicist. At the nexus of Analysis, Geometry, and Topology, Lie Groups have essential applications to Algebraic and Differential Geometry, Differential Equations, Mathematical Physics, Topology, and many other areas of Mathematics and Physics. This course will give a broad introduction to Lie Groups, with minimal prerequisites. Topics will include:
1. Classical matrix Lie groups and Lie algebras
2. Topological, smooth, and analytic manifolds
3. Group actions on manifolds
4. Vector fields andthe exponential map
5. Peter-Weyl Theorem
6. Lie groups and Lie algebras
7. Baker-Campbell-Hausdorff formula
8. Structure and representation of complex semi-eimple Lie algebras and Lie Groups
9. Fundamental gorup of a Lie group
10. Principle bundles
The course will draw primarily on the following references: • T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer, 1985. • D. Bump, Lie groups, Springer, 2013. • S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001. • J. Hilgert and K-H. Neeb, Structure and Geometry of Lie Groups, Springer 2012. • V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer, 1984.
Real Analysis, Linear Algebra, and Elementary Topology or permission of the instructor.