MATH 323S: Euclidean and non-Euclidean Geometry
The primary focus of this course in Fall 2024 will be on understanding how problems in geometry led to the development of axiomatic reasoning and how pursuing that careful reasoning led to the discovery of counter-intuitive notions of geometry that did not agree with the classical development since the time of Euclid (hence, 'non-Euclidean geometry'). The lectures and class discussion will introduce a version of Hilbert's axioms for geometry in a logically coherent sequence and study their implications and significance. This will lead to a much more rigorous standard of proof, as well as a way to understand how different 'models' of geometry can be used to discover and illuminate flaws in the classical proofs and ways to improve arguments and discover new results. Finally, the course will introduce and develop non-Euclidean geometry in several ways, which will lead to a study of how Einstein's rejection of an essentially Euclidean view of geometry allowed him to develop his theory of relativity.