By the end of this course, students will be able to:

• Define and explain the structure and properties of Lie groups and Lie algebras.
• Demonstrate understanding of principal fiber bundles, including their local trivializations, transition functions, and structure groups.
• Construct and analyze connections and curvature on principal bundles, and understand their geometric significance.
• Identify and distinguish almost complex and complex manifolds, and determine integrability conditions using tools such as the Nijenhuis tensor.
• Define and work with Hermitian metrics and Kähler structures, and compute associated geometric quantities such as the Kähler form.
• Apply differential and complex geometric concepts to study the local and global structure of complex manifolds.
• Describe and analyze the geometry of symmetric spaces, including their classification and curvature properties.