Geometry and Groups

Course specifics: 5,5 ECTS, 3h x 13 weeks

Schedule for Spring Semester 2023: Wed 2-5pm

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).

• Matrix Lie groups of small dimension. SL(2), SO(2), SO(3), SU(2), Sp(1).

•  Spherical geometry, isometries of the sphere.

• 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.

References

• Vaughn Climenhaga, Anatole Katok, From Groups to Geometry and Back, Student Mathematical Library, Vol. 81, A.M.S. 2017.

• David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, Cambridge University Press, 2012.

• Kristopher Tapp, Matrix Groups for Undergraduates, Student Mathematical Library, Vol. 79, A.M.S. 2016.