MA 3329 Uniform Geometries

MA 3329 Uniform Geometries

Course Goals: To help students to develop logical reasoning skills and knowledge in understanding the two-dimensional geometry with a focus on the axiomatic development of geometry.

Course Objectives:

  1. Explore the content of Euclid’s Elements.
  2. Introduce Hilbert’s axiom system.
  3. Introduce other Euclidean geometries (e.g. analytic geometry, transformational geometry, etc.)
  4. Explore variations of Euclid’s Parallel Postulate.
  5. Relate the angle sum of a triangle to curvature and non-Euclidean geometries. (Gauss-Bonnet theorem)
  6. Review trigonometry and explore in terms of Euclidean and non-Euclidean geometry.

Student Learning Outcomes: Students will be able to:

  1. Cite basic definitions.
  2. Prove typical geometry proofs. (Pythagorean theorem, alternate interior angles, propositions in the Elements, etc. )
  3. Compute geometric quantities in Euclidean and non-Euclidean spaces.
  4. Prove geometric theorems in non-Euclidean spaces (e.g. angle sum theorem on the sphere.)
  5. Prove basic manifold theorems (e.g. Euler characteristic of a torus is zero.)