Polytopes are simple objects, yet they have a rich and continuously developing theory. They have many applications both inside and outside of combinatorics, including applications to computational algebra, symplectic geometry, and optimization.

In this course, we will concentrate on developing and applying certain geometric aspects of polytopes. Topics include the following:

1. 3-dimensional polytopes: Steinitz' theorem and its generalizations.
2. Gale transform and various counterexamples in the 4-dimensional world.
3. (Very) high dimensional polytopes and their applications:
   1. The weak perfect graph conjecture
   2. The Brunn-Minkowski inequality
   3. Volumes of polytopes
   4. Approximations of convex bodies by ellipsoids
   5. Counting faces of centrally symmetric polytopes
   6. Almost spherical sections of cubes, octahedrons, and general centrally symmetric convex bodies.

Prerequisites: No specific graduate backgroun is assumed, but students are expected to be familiar with teh basics of linear and abstract algebra, and of n-dimensional real analysis, as well as some very basic notions from probability.