Instructor: Danny Calegari 432 Science Center

• Half-course (spring term 2001). TT 11:30-1 in Room 101B Science Center
• Office hours 11:30-1 Friday
• Prerequisite: Mathematics 21 ab, 113, 138 or by consent of instructor
• Assessment: homework 50%, take-home final 50%
• Email dannyc

Course Assistant: Kathy Paur 321b Science Center

• Office hours 1-2 Thursday and by appointment
• Section 6 Monday 507 Science Center
• Email paur

A continuation of the study of spherical, Euclidean and especially hyperbolic geometry in two and three dimensions begun in Mathematics 138. The emphasis will be on the relationship with topology, and the existence of metrics of constant curvature on a vast class of two and three dimensional manifolds.

We will concentrate mainly on a detailed study of examples, and we will try to be as explicit and as elementary as possible. Topics to be covered might include: uniformization for surfaces, shapes and volumes of hyperbolic polyhedra, circle packing and Andreev's theorem, and hyperbolic structures on knot complements.

• Third version of class notes; available as a .tar-red .tex file with .eps figures or .ps file. Contains section on hyperbolic 3-manifolds and Poincare's polyhedron theorem; still incomplete.
• Homework 1 - Scissors congruence and equidecomposability is available as .tex or .pdf.
• Homework 2 - Euclidean, hyperbolic and conformal geometry is available as .tex or .pdf.
• Homework 3 - Hyperbolic geometry continued is available as .tex or .pdf.
• Homework 4 - Heegaard decompositions and link surgery is available as a .tar-red .tex file with .eps figures or .ps file.
• Homework 5 - Fundamental groups and covering spaces is available as .tex or .pdf.
• Homework 6 - Hyperbolic structures on 3-manifolds is available as a .tar-red .tex file with .eps figures or .ps file.
• The final exam is available as a .tex or .pdf file.

Answers to homework problems should take the form of complete sentences. It's important not merely to find the solution to a problem but also to communicate that solution effectively.

• Three dimensional geometry and topology W. Thurston Princeton Mathematical Series, 35. Princeton University Press, 1997
• Complex analysis, 3rd ed. L. Ahlfors McGraw-Hill, 1979
• Classical tessellations and three-manifolds J. Montesinos Universitext, Springer-Verlag, 1987 out of print

• Thurston's notes (unpublished) are available from the MSRI here.
• Lectures on hyperbolic geometry R. Benedetti and C. Petronio Universitext, Springer-Verlag, 1991
• Invitation to geometry H. Coxeter John Wiley and sons, 2nd ed. 1989
• Foundations of hyperbolic manifolds J. Ratcliffe Graduate texts in mathematics 149, Springer-Verlag, 1994
• Hyperbolic manifolds and discrete groups M. Kapovich Progress in mathematics 183, Birkhauser, 2000
• Algebraic topology M. Greenberg and J. Harper Perseus Books, 1982
• Hatcher's notes on 3-manifold topology are available from his home page. (Look under book projects for "3-manifolds")
• The SnapPea home page is here. SnapPea is a computer program for finding hyperbolic structures on manifolds presented as surgery on a link in the three-sphere.

last updated: 16th March 2001 If you have any comments on the material in this page, or if you wish to comment on the material in the course, contact Danny Calegari via email. The image to the right is a symmetric tiling of Euclidean 3-space. It is a still from the video Not Knot produced by the (now defunct) Geometry Center. Danny Calegari's home page