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In addition to the table of contents of the required textbook, given below is a list of additional readings for the course.
Required Textbook
Helgason, Sigurdur. Differential Geometry, Lie Groups, and Symmetric Spaces. Providence, R.I.: American Mathematical Society, 2001. ISBN 0821828487.
Table of Contents
Chapter I: Elementary Differential Geometry

Manifolds

Tensor Fields

Vector Fields and 1Forms

Tensor Algebra

The Grassman Algebra

Exterior Differentiation

Mappings

The Interpretation of the Jacobian

Transformation of Vector Fields

Effect on Differential Forms

Affine Connections

Parallelism

The Exponential Mapping

Covariant Differentiation

The Structural Equations

The Riemannian Connection

Complete Riemannian Manifolds

Isometries

Sectional Curvature

Riemannian Manifolds of Negative Curvature

Totally Geodesic Submanifolds

Appendix

Topology

Mappings of Constant Rank
Chapter II: Lie Groups and Lie Algebras
 The Exponential Mapping
 The Lie Algebra of a Lie Group
 The Universal Enveloping Algebra
 Left Invariant Affine Connections
 Taylor's Formula and the Differential of the Exponential Mapping
 Lie Subgroups and Subalgebras
 Lie Transformation Groups
 Coset Spaces and Homogeneous Spaces
 The Adjoint Group
 Semisimple Lie Groups
 Invariant Differential Forms
 Perspectives
Chapter III: Structure of Semisimple Lie Algebras

Preliminaries

Theorems of Lie and Engel

Cartan Subalgebras

Root Space Decomposition

Significance of the Root Pattern

Real Forms

Cartan Decompositions

Examples. The Complex Classical Lie Algebras
Chapter IV: Symmetric Spaces

Affine Locally Symmetric Spaces

Groups of Isometries

Riemannian Globally Symmetric Spaces

The Exponential Mapping and the Curvature

Locally and Globally Symmetric Spaces

Compact Lie Groups

Totally Geodesic Submanifolds. Lie Triple Systems
Chapter V: Decomposition of Symmetric Spaces

Orthogonal Symmetric Lie Algebras

The Duality

Sectional Curvature of Symmetric Spaces

Symmetric Spaces with Semisimple Groups of Isometries

Notational Conventions

Rank of Symmetric Spaces
Chapter VI: Symmetric Spaces of the Noncompact Type

Decomposition of a Semisimple Lie Group

Maximal Compact Subgroups and Their Conjugacy

The Iwasawa Decomposition

Nilpotent Lie Groups

Global Decompositions

The Complex Case
Chapter VII: Symmetric Spaces of the Compact Type

The Contrast between the Compact Type and the Noncompact Type

The Weyl Group and the Restricted Roots

Conjugate Points. Singular Points. The Diagram

Applications to Compact Groups

Control over the Singular Set

The Fundamental Group and the Center

The Affine Weyl Group

Application to the Symmetric Space U/K

Classification of Locally Isometric Spaces

Geometry of U/K. Symmetric Spaces of Rank One

Shortest Geodesics and Minimal Totally Geodesic Spheres

Appendix. Results from Dimension Theory
Chapter VIII: Hermitian Symmetric Spaces

Almost Complex Manifolds

Complex Tensor Fields. The Ricci Curvature

Bounded Domains. The Kernel Function

Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type

Irreducible Orthogonal Symmetric Lie Algebras

Irreducible Hermitian Symmetric Spaces

Bounded Symmetric Domains
Chapter IX: Structure of Semisimple Lie Groups

Cartan, Iwasawa, and Bruhat Decompositions

The RankOne Reduction

The SU (2,1) Reduction

Cartan Subalgebras

Automorphisms

The Multiplicities

Jordan Decompositions
Chapter X: The Classification of Simple Lie Algebras and of Symmetric Spaces

Reduction of the Problem

The Classical Groups and Their Cartan Involutions

Some Matrix Groups and Their Lie Algebras

Connectivity Properties

The Involutive Automorphisms of the Classical Compact Lie Algebras

Root Systems

Generalities

Reduced Root Systems

Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams

The Nonreduced Root Systems

The Highest Root

Outer Automorphisms and the Covering Index

The Classification of Simple Lie Algebras over C

Automorphisms of Finite Order of Semisimple Lie Algebras

The Classifications

The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV

The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV

Irreducible Hermitian Symmetric Spaces

Coincidences between Different Classes. Special Isomorphisms
Additional Readings
The first two papers below are quite elementary and nontechnical and are passed out at the very beginning of the course. They serve as motivation. The third one is more technical and is passed out near the end of the course.

Helgason, Sigurdur. "Sophus Lie, the mathematician" (
PDF  1.2 MB). O.A. Laudal and B. Jahren (eds.)
The Sophus Lie Memorial Conference. Oslo 1992 Proceedings , Oslo: Universitetsforlaget (Scandinavian University Press) 1994.

Sophus Lie and the Role of Lie Groups in Mathematics (
PDF). Opening lecture by Sigurdur Helgason at a Nordic Teachers Conference in Reykjavik 1990.

Reprinted with permission from Springer. Helgason, Sigurdur. "A Centennial: Wilhelm Killing and the Exceptional Groups." (
PDF  1.2MB)
The Mathematical Intelligencer 12, no. 3 (1990).
For a thorough treatment of the history of the subject, see

Hawkins, Thomas. The Emergence of the Theory of Lie Groups. New York: Springer, 2000. ISBN: 0387989633.

Borel, Armand.
Essays in the History of Lie Groups and Algebraic Groups. Providence, R.I.: American Mathematical Society, 2000. ISBN: 0821802887.