Mathematics 139  Classical Geometry and LowDimensional
Topology; Spring 2001

Instructor: Danny Calegari 432 Science Center
 Halfcourse (spring term 2001). TT 11:301 in Room 101B Science Center
 Office hours 11:301 Friday
 Prerequisite: Mathematics 21 ab, 113, 138 or by consent of instructor
 Assessment: homework 50%, takehome final 50%
 Email dannyc
Course Assistant: Kathy Paur 321b Science Center
 Office hours 12 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
.tarred .tex file with .eps figures or
.ps file.
Contains section on hyperbolic 3manifolds 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 .tarred .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 3manifolds is
available as a .tarred .tex file with .eps
figures or .ps file.
 The final exam is available as a .tex or
.pdf file.
The final exam was posted on this website at noon on Wednesday,
May 9th, and is due in my mailbox at 3pm on Tuesday, May 15th.
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.
Recommended texts for course

 Three dimensional geometry and topology W. Thurston Princeton
Mathematical Series, 35. Princeton University Press, 1997
 Complex analysis, 3rd ed. L. Ahlfors McGrawHill, 1979
 Classical tessellations and threemanifolds J. Montesinos
Universitext, SpringerVerlag, 1987 out of print
 Thurston's notes (unpublished) are available from the MSRI
here.
 Lectures on hyperbolic geometry R. Benedetti and C. Petronio
Universitext, SpringerVerlag, 1991
 Invitation to geometry H. Coxeter
John Wiley and sons, 2nd ed. 1989
 Foundations of hyperbolic manifolds J. Ratcliffe
Graduate texts in mathematics 149, SpringerVerlag, 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 3manifold topology are available from his
home page. (Look under
book projects for "3manifolds")
 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 threesphere.
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
3space. It is a still from the video Not Knot produced
by the (now defunct) Geometry Center.
Danny Calegari's home page

