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 V-representations. Fourier-Motzkin 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: Dehn-Somerville equations. Affine hull of f-vectors for simplicial polytopes. Shellings of polytopal complexes.
• Mar 10: Shellings (cont'd). Euler-Poincare 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 d-polytopes are infinitesimally rigid. Overview of characterizations and representations of 3-polytopes: Steinitz, Tutte, (primal-dual) circle packing (Koebe, Brightwell and Scheinerman), Schramm's midscribe representation, and related results.
• Apr 2: proof of (primal-dual) circle packing.
• Apr 7: Representation of 3-polytopes cont'd.
• Apr 9: class cancelled.
• Apr 14: Hirsch conjecture. Kalai-Kleitman upper bound. 0-1 polytopes.
• Apr 16: Characterization of inscribable 3-polytopes 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 Loewner-John 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 3-polytopes.
• 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.