MATH 540 Modern Geometries (Graduate)

A review of the basic techniques of straight edge and compass construction, including constructions requiring propositions from Book III of Euclid’s Elements (ca. 300 BC), gives students an intuitive foundation necessary for the approaches that follow. First is an overview of the nature of postulates and propositions in Euclid, which are radically different from their modern counterparts. From there, the bulk of the course is devoted to geometry based in Hilbert’s axioms (1899), where the theory of plane geometry is built with an eye toward understanding the dilemma of Euclid’s parallel postulate. The historical development leading from Euclid to hyperbolic and elliptic geometry is then situated in this mathematical context.

Prerequisites: Calculus I, II, and linear algebra, and an introductory course on mathematical proofs.