• Lecture I: Lie groups and Lie algebras
• Lecture II: Invariant measures
• Lecture III: Finite-dimensional represenations
• Lecture IV: Algebraic groups
• Lecture V: Lattices in Lie groups
• Lecture VI: Space of Euclidean lattices and other arithmetic constructions

Outline of the course:

• Basic notions of Lie theory:
  • Basic examples and definitions of Lie theory
  • Lie algebra, exponential map, and Lie correspondence
  • Solvable and semisimple groups
  • Finite-dimensional represenations
  • Integration and invariant measures
• Introduction to algebraic groups
  • Algebraic varieties and algebraic groups
  • Semisimple/unipotent elements and Jordan decomposition
  • Rigidity of algebraic actions
• Lattices and arithmetic groups
  • Hyperbolic space and geometric constructions of lattices
  • Mahler compactness criteria
  • Arithmetic constructions of lattices
  • Borel density theorem

References:

• W. Rossmann, Lie Groups: An Introduction Through Linear Groups
• J.-P. Serre, Lie Algebras And Lie Groups
• J. S. Milne, Algebraic Groups, Lie Groups, and their Arithmetic Subgroups
• D. Witte Morris, Introduction to Arithmetic Groups