**Abstract:** We will discuss hyperbolic geometry in dimensions two and three at an introductory level, starting with the basics of geometry in the hyperbolic plane and hyperbolic 3-space. The intrinsic geometry of closed hyperbolic surfaces will be emphasized, with detailed investigation of specific examples. We will also give a flavor of the vast field of hyperbolic 3-manifolds through gluings of polyhedra and link complements.

**Outline of topics to be covered:
**

- 1.
- The hyperbolic plane, its metric, various models thereof
- 2.
- Hyperbolic geodesics, angles, areas, polygons
- 3.
- The group of hyperbolic isometries, classification
- 4.
- Construction of hyperbolic surfaces
- 5.
- Comments on Hyperbolic vs. Riemann surfaces vs. algebraic curves
- 6.
- Geodesics and isometries of hyperbolic surfaces
- 7.
- Examples of hyperbolic surfaces
- 8.
- Hyperbolic 3-space, isometries, geodesics, hyperplanes
- 9.
- The sphere at infinity, connection with conformal geometry
- 10.
- Examples of hyperbolic 3-manifolds

**Some possible student project topics:
**

- Construction of hyperbolic polyhedra (Andreev's theorem)
- Tilings of the hyperbolic plane and their symmetry groups
- Geodesics on hyperbolic punctured tori
- Hyperbolic surfaces with many automorphisms
- Volumes of knot complements
- Convex geometry in hyperbolic space
- Hyperbolic manifolds with boundary

**Prerequisites:** Some familiarity with topology and differential geometry of smooth manifolds (especially closed surfaces). Previous exposure to Riemannian geometry would be useful, but is not necessary.

**As summer tutorials are typically small, there is a great opportunity to tailor the level, pace, and content to the common background of a group of students, working from the basic outline and ideas above. Interested students concerned about prerequisites should contact me.
**

**References:**

__Three dimensional geometry and topology__, W. Thurston, Princeton Mathematical Series, 35. Princeton University Press, 1997__Lectures on hyperbolic geometry__, R. Benedetti and C. Petronio, Universitext, Springer-Verlag, 1991__Foundations of hyperbolic manifolds__, J. Ratcliffe, Graduate texts in mathematics 149, Springer-Verlag, 1994

**David Dumas
ddumas@math.harvard.edu
**