Math 210A: Topics in Geometry
Math 210A: Topics in Geometry
Fall 2005
Fuchsian Groups

Instructor: Moon Duchin
Office: Kerr 673, 4-8158
Office Hours: by appointment (mduchin@math)
Course meets: Wellman 125, Tuesdays and Thursdays, 4:40-6pm ....[no class Thursday November 3]
Announcements and Handouts and Whatnot

Text Svetlana Katok, Fuchsian Groups,
Chicago Lectures in Mathematics
Background Katok writes that the book is intended for a one-quarter graduate course, assuming only "a basic knowledge of real and complex analysis and abstract algebra." In fact, I think this overstates the prerequisites for reading most of the book-- for at least the first several chapters, what is required is calculus, comfort manipulating complex numbers, group theory, and familiarity with matrices. Despite its humble starting point, the text is a great segue into quite sophisticated mathematics.
Outline Text
I. Hyperbolic Geometry
II. Fuchsian Groups
III. Fundamental Regions
IV. Geometry of Fuchsian Groups
V. Arithmetic Fuchsian Groups
Supplementary Topics
-History of non-Euclidean geometry; axiomatics
-Flat surfaces
-Curvature and the Gauss-Bonnet Theorem
-Flat structures on hyperbolic surfaces
-Lattices
-Three-manifolds and Kleinian groups; Geometrization
Expectations At the beginning, there will be a small number of required problems to make sure that everyone can work with the basic objects comfortably. After we have covered the core material of chapters 1-3, the emphasis will shift to projects. The last several course meetings will be devoted to presentations.

Links

Supplementary Sources