Junior Seminar: Hyperbolic geometry
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Fall 2008
Tuesday 5:30-
601 Fine
Books
- J. W. Anderson, Hyperbolic geometry. Easy treatment of the plane hyperbolic geometry. It covers around half of the topics of this seminar.
- H. Meschkowski, Non-Euclidean geometry. From histroical and logical point of view. Through parallel postulate.
- S. Katok, Fuchsian groups. We will more or less cover the first four chapters of this book.
- B. Iversen, Hyperbolic geometry. Unfortunately it is out of stock. I have the library's copy. You can borrow it from me.
- A. F. Beardon, The geometry of discrete groups. The first five weeks, we more or less follow this book. Chapters 7, 8, and parts of 9, 10.
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DATE |
PRESENTER |
TOPIC |
Sep 23 |
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Fundamental concepts
Parallel postulate. Different models. Hyperbolic metric.
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Sep 30 |
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Hyperbolic area and trigonometry Gauss-Bonnet, Angle of parallelism, The sine rule, The cosine rule I, II. |
Oct 7 |
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Polygons Area of a polygon, Convex polygon, Quadrilaterals, Pentagons, Hexagons. |
Oct 14 |
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The geometry of geodesics Distance from a line, Perpendicular bisector, Common orthogonal of disjoint geodesics, Pencils of geodesics. |
Oct 21 |
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Geometry of isometries Classification of isometries, Displacement function, Canonical region. |
Nov 4 |
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Fuchsian groups Discreteness criteria (GDG) or (HG'), Algebraic properties (FG), Elementary Fuchsian groups (FG), Jorgensen inequality (FG). |
Nov 11 |
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Fundamental domains Drichlet domain, Modular group, Locally finite domain. |
Nov 18 |
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A work of Siegel Some remarks on discontinuous groups, The Annals of Mathematics, Second Series, Vol. 46, No. 4, (Oct., 1945), pp. 708-718. |
Nov 25 |
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Either study signature of a Fuchsian group, or reserve this time to catch up with the schedule.
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Dec 2 |
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Continued fraction C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80.
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Dec 9 |
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Uniformization theorem Hyperbolic surface, Hopf-Rinow theorem, Uniformization theorem. (HG') |
Jan ? |
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Monodromy theorem Geodesic lifting property, Monodromy theorem. (HG') |
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