Convex Polytopes
Time: Fri 2013-01-18 13.15 - 15.00
Location: Room 3721, Lindstedtsvägen 25, 7th floor, Department of mathematics, KTH
Brief Description
The main objective of the course is to give an introduction to the fascinating mathematics of convex polytopes. A polytope is a fundamental geometric object which most easily can be described as the convex hull of a finite set of points in a finite dimeionsional Euclidean space. Polytopes are interesting objects from a pure mathematical perspective in combinatorics and algebra. They are also of interest in e.g. optimization since the optimum of a linear optimization problem is obtained in a vertex of a polytope.
In this course the emphasis will be on the combinatorial properties of polytopes. We will discuss basic properties of polytopes; e.g. vertex/facets incidences, the f-vector, the i-skeleton, and methods to study them, e.g. projections, Schlegel diagrams, shelling, Gale diagrams. An important part of the combinatorial theory of polytopes are constructions and special classes of polytopes; the cyclic poltyopes, Birkhoff polytope, zonotopes (oriented matroids), Minkovskisum, 0/1-polytopes, the permutaeder, associaeder etc. There will also be a possibility for the participants to influence part of the curriculum.
The theory of convex polytopes is a highly active research area. One of the main conjectures (the so called Hirsch conjecture) was disproved two years ago by Francisco Santos. We will at the end of the course discuss this very interesting development and other research problems, e.g what can the combinatorial structure of a 4-dimensional polytope look like?
Prerequisites
Linear algebra and SF1630 (or SF1631, SF2736) Diskret Matematik or the corresponding knowledge.