[RG #109] Combinatorics of Polytopes and Complexes: Relations with Topology and Algebra

March 1 - August 31, 2007

Organizer:

Gil Kalai (The Hebrew University of Jerusalem)

Polytopes have intrigued mathematicians since ancient times. The ancient Egyptians knew quite a bit of the geometry of polytopes, and the pyramids are, of course, a special type of polytopes. The ancient Greeks discovered the five platonic solids. The five platonic solids: note that the Icosahedron is dual to the Dodecahedron, the Cube is dual to the Octahedron and the Tetrahdron is self-dual.

Euler, who can be regarded as the father of modern graph theory, proved a remarkable formula that explains the relationship between combinatorics and polytopes.  Euler's formula asserts that: 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 + 16 = 2.

Euler's formula is one of the most important formulae is mathematics and can be regarded as a starting point for topology.

Polytopes in dimensions higher than three have been studied since the 19th century. The first rigorous proof of an extension of Euler's formula for higher dimension was obtained by Poincaré. Poincaré used tools from algabraic 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.

The research group will explore the following topics: the important and mysterious notion of "duality" between polytopes; the notion of "valuations" of convex sets; random polytopes and complexes; the relationships between combinatorics and topology; the "rigidity" of graphs; and metric aspects of polytopes.