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MA 3330 Non - Uniform Geometries

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

Course Objectives:

  1. Explore the content of Euclid’s Elements.
  2. Use analytic geometry to gain an understanding of hyperbolic, elliptic, and Euclidean geometry, and how they are related.
  3. Use the concept of curvature to unify the various geometries and extend to non-uniform spaces.
  4. Relate the angle sum of a triangle to curvature and non-Euclidean geometries. (Gauss-Bonnet theorem)
  5. Review trigonometry and explore in terms of non-Euclidean geometry.
  6. Explore Descartes’ exterior angle theorem, and its relation to curvature and Euler’s theorem.
  7. Explore topologies of closed surfaces.

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.)