- Study the slicing of a hypercube.
- Relate the Vandermonde determinant to the q-analogue of simplices.
- Relate Schur functions and simplices by way of the Vandermonde determinant.
- Relate polytopes and convexity.
Lecture 1: Course is a combinatorial focus on convex polytopes and hyperplane arrangements.
- Euler's theorem: 1 - v + e - f + 1 = 0
- Steinitz's theorem: A 3-polytope exists iff 1 - v + e - f + 1 = 0, v <= 2f - 4, f <= 2v - 4.
- Regular polytopes.
Lecture 2: Definition of polytope P as convex hull of vertices. Sum of lambda x vertex where lambdas are nonnnegative and sum to 1.
Lecture 3: Intersections and products of polytopes are also polytopes.
- Main theorem of polytopes: Polytopes = convex hulls of finitely many points = bounded intersections of halfspaces.
Lecture 4: V-polyhedron intersected with affine plane is a V-polyhedron.
Lecture 5 and 6: Farkas' lemma versions 1 to 4.
Lecture 7: Faces of polytopes. Face of P in direction c is the set of all x in P where c-x is maximal.
- Affine spaces.
- Dim(face) = Dim(Aff(face))
- f-vector gives the number of faces in each dimension
- f-poly the generated function where the lowest coefficient gives the number of vertices and the highest coefficient gives the volume
Lecture 8: Construction of faces
- Building polytopes: pyramids (adding the center).
- Vertex figures (converting the vertices to faces).
- Face lattice
Lecture 9: Face lattice
- P and Q are combinatorially isomorphic if their face lattices are isomorphic.
- Polar (Dual) polytopes. Dual = c in dual space where c x <=1 for all x in P.
- If 0 is in P, then P equals its dual's dual.
- Dual faces.
- The face lattices of P and its dual are opposites.
- Simple and simplicial polytopes.
- P is simple iff its dual is simplicial.
Lecture 11: The cyclic polytope.
Lecture 12: Graphs of polytopes
Lecture 13: How good is linear programming?
- Hirsch conjecture is false.
Lecture 14: Balinski's theorem: P is a d-polytope implies G(P) is d-connected.
Lecture 15: If P is simple, then G(P) determines P combinatorially.
Lecture 16: Complexes, subdivisions, triangulations.
- Every P has a triangulation.
Lecture 17: Triangulation of d-cross-polytope.
Lecture 18: Counting lattice points in polytopes.
Lecture 19: Partition functions.
Lecture 20: Generating functions for cones.