Manifolds and Lie Groups

Instructor: Guanghua JI

Classroom and Time: C701, THU 9-11

Prerequisites: General topology, functional analysis, group theory.

Grading: The course grade will be based homework(50%), final exam(50%).

References:

1, J. Hilgert, K.-H. Neeb, Structure and Geometry of Lie Groups, Springer, 2012.

2, J. M. Lee, Introduction to Smooth Manifolds, GTM 218.

3, N. R. Wallach, Harmonic Analysis on Homogeneous Spaces, 1973.

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Syllabus:

Chapter I, Smooth Manifolds

1.1 Smooth Maps in Several Variables

1.2 Smooth Manifolds and Smooth Maps

1.3 The Tangent Bundle

1.4 Vector Fields

1.5 Integral Curves and Local Flows

1.6 Submanifolds

Chapter II, Basic Lie Theory

2.1 Lie Groups and Their Lie Algebras

2.2 The Exponential Function of a Lie Group

2.3 Closed Subgroups of Lie Groups and Their Lie Algebras

2.4 Constructing Lie Group Structures on Groups

2.5 Covering Theory for Lie Groups

Chapter III, Smooth Actions of Lie Groups

3.1 Homogeneous Spaces

3.2 Frame Bundles

3.3 Integration on Manifolds

3.4 Invariant Integration

3.5 Integrating Lie Algebras of Vector Fields

Chapter IV, Elementary Representations Theory

4.1 Representations

4.2 Finite Dimensional Representations

4.3 Induced Representations

4.4 The Regular Representation

4.5 The Peter-Weyl Theorem

4.6 Characters and Orthogonality Relations

Chapter V, Structure of Compact Lie Groups

5.1 Maximal Tori and Cartan subalgebra

5.2 Cartan's Theorem

5.3 The Weyl Groups

5.4 Structure of Compact Lie Groups

5.5 The Weyl Integration Formula