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

Explain the historical and logical foundations of non-Euclidean geometry, including the role of the parallel postulate and why its modification leads to alternative geometries.

Describe and analyze various models of the hyperbolic plane, such as the Poincaré disk and upper half-plane models, and translate between them.

Understand the structure and action of the Möbius group on the hyperbolic plane, including identification and classification of Möbius transformations.

Apply classical geometric notions (e.g., length, distance, isometry, parallelism, convexity, area) within the context of hyperbolic geometry.

Analyze group actions on the hyperbolic plane, and understand the significance of such actions in hyperbolic geometry.

Identify and construct fundamental domains for group actions on the hyperbolic plane.