Course Content (Syllabus)
Crash courses on Group Theory. Complex numbers and rotations of the plane. Quaternions. Rotations of the 2- sphere, rotations of R^3 , rotations of R^4. Reflections.
Isometry group of of R^2 and R^3. Isometry subgroups (discrete, finite, fixed point).
Circle and SO(2). Spherical geometry, isometries of the sphere, the groups O(3), SO(3).
Stereographic projection, real projective line, Mobius transformations, SL(2,r) and action on 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. Inversion.
Real projective plane and SL(3,R). Groups of matrices and Topology. Rudiments of Lie groups.
Additional bibliography for study
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.