January 30, 2023

Thematic Program on the Foundations of Computational Mathematics
September-December, 2009

Short Course Series

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)

Ketan Mulmuley (University of Chicago)
On P vs. NP, geometric complexity theory, explicit proofs and the complexity barrier

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

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