This is a schedule of the what we have covered so far and the plan for
the next few lectures.

Feb 5: Introduction

Feb 10: Polytopes, polyhedra and cones. Equivalence between H and
Vrepresentations. FourierMotzkin elimination, double description
method.

Feb 12: Farkas lemma, Caratheodory's theorem. Colorful version.

Feb 17: no classes (and Feb 18: cancelled due to snow storm)

Feb 19: Faces. Defs + basic properties.

Feb 24: Face lattice. Polarity. Combinatorial equivalence.

Feb 26: Simplicial and simple polytopes. Cyclic polytopes.

Mar 3: Reconstructing a simple polytope from its
graph.

Mar 5: DehnSomerville equations. Affine hull of fvectors for
simplicial polytopes. Shellings of polytopal complexes.

Mar 10: Shellings (cont'd). EulerPoincare formula. Upper bound theorem.

Mar 12: Upper bound theorem.

Mar 17: Class cancelled.

Mar 19: Lower bound theorem (via rigidity).

Mar 24: Spring break.

Mar 26: Spring break.

Mar 31: Lower bound theorem cont'd: simplicial dpolytopes are
infinitesimally rigid. Overview of characterizations and representations of
3polytopes: Steinitz, Tutte, (primaldual) circle packing (Koebe, Brightwell
and Scheinerman), Schramm's midscribe representation, and related
results.

Apr 2: proof of (primaldual) circle packing.

Apr 7: Representation of 3polytopes cont'd.

Apr 9: class cancelled.

Apr 14: Hirsch conjecture. KalaiKleitman upper bound. 01 polytopes.

Apr 16: Characterization of inscribable 3polytopes by Rivin.

Apr 21: Patriot's day. No classes.

Apr 23: Proof of Rivin's characterization.

Apr 28: Volume of a polytope. Approximating using the LoewnerJohn
ellipsoids. Hardness of approximations for deterministic algorithms in
separation oracle model.

Apr 30: Computing the volume exactly. Gram's formula. Lawrence's
signed decomposition formula. Filliman duality.
Some of the topics that wemay not have time to cover.

Rationality of 3polytopes.

Relationship (lower bounds) between number of faces and facets for centrally
symmetric polytopes. Number of facets for round polytopes.
 Volume of polytopes. Reconstructing a polytope from its
facet volumes and facet normals.

Polyhedral combinatorics.