**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