### Askold Khovanski (University of Toronto)

*The theory of fewnomials*

*Fields Institute, Stewart Library*

October 2, 2-3:30 pm

October 5, 11:30 -1 pm

October 8, 10- 11:30 am

1. What the theory is about.

2. Generalizations of Rolle's Theorem and two simple versions
of the theory

3. Theorems on Real and Complex Fewnomials (with proofs)

*Fields Institute, Stewart Library*

**October 14, 11:30 am- 1 pm**

October 16, 2-3:30 pm

### Saul Schleimer (University of Warwick)

*Introduction to three-manifolds, triangulations, and normal
surfaces*

*Fields Institute, Room 230*

**November 6, 9, 13, 23 (2-3:30 pm)**

***November 27 (1-2:30 pm)***

1. Three-manifolds, important examples (S^3, T^3, PHS^3, Seifert-Weber
space, handlebodies), the fundamental group, incompressible
surfaces, Haken manifolds, the loop theorem, sphere decomposition
and connect sum, torus decomposition

2. Thurston geometries, examples (Seifert fibred spaces, quotients
of S^3, E^3, H^3), the geometrization theorem and the Poincare
conjecture, implications for the homeomorphism problem, recognition
of: three-sphere, unknot, Haken manifolds, H^3/\Gamma (Jason
Manning's algorithm)

3. (Pseudo) Triangulations of three-manifolds, ideal triangulations,
examples, Weeks' SnapPea program, normal and almost normal surfaces,
Haken sum and geometric consequences, incompressible surfaces
normalize, Jaco-Tollefsen algorithm (sphere decomposition),
finding compressions, unknot recognition lies in NP, Jaco-Oertel
algorithm

(Haken recognition)

4. Sweep-outs, normalization of almost normal surfaces, Rubinstein-Thompson
algorithm for three-sphere recognition

5. Crushing triangulations along two-spheres, Casson's algorithm
to recognize the three-sphere, Agol-Hass-Thurston algorithm,
Three-sphere

recognition lies in NP