Textbooks

R. Rimanyi: Euclidean and Non-Euclidean Geometries

This is the text for a one-semester undergraduate geometry course, as I taught it a few times at UNC Chapel Hill. The treatment of Euclidean and Spherical geometries are (minimally) vector based. Hyperbolic geometry is introduced via its hyperboloid model, and the vocabulary of Special Relativity is used moderately. Students with or without high school geometry background can take this course, but some mathematical maturity is assumed, typically acquired in a Discrete Math course and a few semesters of Calculus. Allusions to Analysis, Group Theory, Physics, Topology, (Multi-)Linear Algebra, Differential Geometry appear throughout the text.

R. Rimanyi: Graduate Topology Coloring Book

This series of lectures contains the material for the class Math 681, Graduate Topology, as it was taught in Fall 2021—a.k.a. the δ semester—at the University of North Carolina at Chapel Hill. It is called a Coloring Book, because numerous arguments, indicated by the sign ♠, that were presented in the class are not typed in. In fact, those arguments are deliberately left out of this text: reading those arguments would have no educational value for the reader. Figuring out those arguments (“coloring between the contours”) does. Hence, whenever the reader meets a ♠ sign, they should stop and fill in the missing proof. Such “coloring” of this Coloring Book is an essential part of learning the subject.