Week 1: Geometry of complex numbers. The quaternions. Motivating examples: Space rotations and Rotations of a sphere.
Week 2: The general linear groups. Conjugation and Change of basis. . All matrix groups are real matrix groups.
Week 3: The Unitary groups. The Euclidean isometry group. Low dimensional examples.
Week 4: Topology of matrix groups. Open sets. Continuity. Connected sets. Compact sets.
Week 5: Lie algebras. Examples. Lie algebras as vector fields. The Lie algebras of orthogonal groups
Week 6: Matrix Exponentiation. Properties of Matrix Exponentiation. One parameter subgroups
Week 7: Analysis background. Differentiation. Chain rules. Inverse and Implicit function theorems.
Week 8: Restriction of exponential map to Lie algebras. Realization of matrix groups as smooth manifolds
Week 9: The Lie bracket. Adjoint representation. Example of Adjoint representation
Week 10: The double cover Sp(1) → SO(3), Some other double covers. Sketch of the Lie group-Lie algebra correspondence.
Week 11: Maximal torus. Center of compact matrix groups. Conjugates of maximal tori
Week 12: Introduction to smooth manifolds and Lie groups. Example that not all Lie groups are matrix groups. Applications of Lie groups in Natural sciences.
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