Department of Mathematical Sciences
Binghamton University

Math 580A, Convex Polytopes
Syllabus

Fall 2012

Textbook

• First "Month":
  Chapter 0. Introduction and Examples.
  • Affine vs. linear (vector) geometry.
  • Intuitive discussion of convex polytopes and polyhedra. Types of polytopes.
  • Cyclic polytopes.
  • The permutahedron, and the acyclotope of a graph.
    (The acyclotope is not in the book; I've written notes on the acyclotope [incomplete].)
  • Combinatorial polytopes: 0/1 polytopes. Linear programming.
  • Reading: Whatever else is in the chapter.


• Second "Month":
  Chapter 1. Polytopes, Polyhedra, and Cones.
  • H-polyhedra vs. V-polyhedra.
  • Fourier-Motzkin elimination.
  • The Farkas lemmata.
  • Recession cone.
  • Carathéodory's theorem.


• Third "Month":
  Chapter 2. Faces of Polytopes.
  • Faces.
  • The face lattice.
  • Polar duality.
  • Simple vs. simplicial polytopes (reading).
  • Projective transformations (reading).


• Fourth "Month":
  Chapter 6 (partial). Duality, Gale Diagrams, and Applications.
  • Gale transforms and Gale diagrams.
  • Signed dependencies and signed circuits; signed separations and signed cocircuits; cycles and cocycles.
  • Oriented matroids.
  • Duality of matroids vs. Gale transforms.


• Fifth "Month":
  Chapter 7 (selections). Fans, Arrangements [of Hyperplanes], Zonotopes, and Tilings.
  The treatment does not follow the book closely.
  • Zonotopes, arrangements of hyperplanes, and their dual relationship.
  • Acyclic orientations, vertices, and regions.
  • The number of acyclic orientations, vertices, and regions.