Polytopes, linear programming, the Hirsch conjecture, and open mathematical collaboration
Event details
| Date | 18.11.2010 |
| Hour | 17:15 |
| Speaker | Gil Kalai |
| Location | |
| Category | Conferences - Seminars |
Polytopes attracted mathematicians since ancient times. The ancient Egyptians knew quite a bit of geometry of polytopes, and the pyramids are, of course, a special type of polytopes. The ancient Greek discovered the five platonic solids. Euler, who can be regarded as the father of modern graph theory proved a remarkable formula which explains the relation between combinatorics and polytopes: for every polytope in space with V vertices, E edges and F faces,
V-E+F=2
For example, for the cube, V=8, E=12, and F=6 and indeed 8-12+6=2.
Polytopes in higher dimension than three were studied since the 19th century. The first rigorous proof of Euler's formula for higher dimension was obtained by Poincare. Poincare used tools from algebraic topology, a new subject of study that he himself developed. It turns out that Euler's formula is closely related to topology, an important part of geometry. Linear programming which is one of the main application of mathematics also leads to exciting problems about polytopes. On of these problems, The Hirsch conjecture was solved recently by Franciscos Santos after 53 years. I will describe some results about polytops I will discuss also a joint collective effort "polymath3" to attack a weak version of Hirsch conjecture.
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- General public
- Free