Project III (MATH3382) 20192020
Geometries
Anna Felikson
Description:For centuries people were convinced that the only "correct geometry"  geometry related to the real word  is the Euclidean geometry. However in XIX century it was discovered that other geometries may be as "true" as the Euclidean one and, moreover, that they may be related to the real word as well. The main examples of nonEuclidean geometry are spherical geometry (in particular, geometry of the globe) and hyperbolic geometry (used for example as Minkowski timespace in general relativity). All three geometries (hyperbolic, Euclidean and spherical) share a number of properties, but in the other aspects they are very different. For example, a nonEuclidean triangle is determined (up to isometry) by three angles, i.e. AAA congruence of triangles holds for nonEuclidean triangles. One can consider also other geometries such as Möbius geometry, affine geometry, projective geometry, finite geometry... In this project we will follow Klein's famous idea that geometry is actually a set considered together with a group of symmetries acting on it. We will think about different groups acting on various spaces and mesuring how symmetric is the space. We will also think how these geometries are related to each other. Various topics may include:

Prerequisites: Algebra II, Complex analysis II, willingness to learn geometry. 
Resources:One can start with the following books: You can also look up the wikipedia article on nonEuclidean geometry here.email: Anna Felikson  