Students taking this course will not only aquire further mathematical knowledge, but they will deepen and better understand the interrelations between the Algebra and Geometry courses they took in the first two years of the undergraduate program. In addition, they will familiarize with the interdisciplinary approach to mathematical problems and enforce their mathematical perception. The techniques used in the course are fundamental for many mathematical disciplines but are also used extensively for many applications of any kind of motion simulation and graphic design.
Course Content (Syllabus)
(Note: Special attention will be given to the cases n=2,3.)
The group Aff(n). Short reminder on the isometries of the plane and space. The group ISO(n). Subgroups of Isometries (discrete, finite, fixed point). Circle and the group SO(2). Spherical geometry (spherical coordinates, triangles, great circles). Isometries of the sphere, the groups O(3), SO(3). Stereographic projection, real projective line, Mobius tranformations. SL(2,R) and RP(1), the group PSL(2,R). Complex projective line, SL(2,C) and action on CP(1), Riemann sphere, the group PSL(2,C). Hyperbolic plane, real projective plane and SL(3,R).
Additional bibliography for study
1) Vaughn Climenhaga, Anatole Katok, From Groups to Geometry and Back, Student Mathematical Library, Vol. 81, A.M.S. 2017.
2) David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, Cambridge University Press, 2012.
3) Kristopher Tapp, Matrix Groups for Undergraduates, Student Mathematical Library, Vol. 79, A.M.S. 2016.