| Text | Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics |
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| 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. | |
