2nd degree (Master)
English and Hebrew
Coordinator Office Hours:
Prof Eran Nevo
Polytopes have fascinated humans since antiquity and are related to many areas of modern mathematics. We will study polytopes, focusing on connections between
their geometric and combinatorial properties.
Learning outcomes - On successful completion of this module, students should be able to:
Deduce combinatorial properties of polytopes from their geometry and convexity. To give a lecture to peers.
Teaching arrangement and method of instruction:
1. Faces of polytopes:
the face lattice, polarity, simple and simplicial
polytopes, projective transformations.
basic constructions (e.g. product, join, cyclic polytope, Gale’s evenness condition).
2. Graphs of polytopes:
Tell a simple polytope from its graph –
Kalai’s proof, Balinski's theorem, refinement theorems, the Hirsch conjecture on diameter and Santos' counterexample.
3. Schlegel diagrams.
4. Gale duality.
5. f-vectors of simplicial polytopes: Dehn-Sommerville relations, McMullen's upper bound theorem and shellability; Barnette's lower bound theorem and rigidity; the g-theorem.
6. Fiber polytopes: the associahedron and the permutohedron.
7. Realization spaces of polytopes.
8. Subfamilies: centrally symmetric polytopes, cubical polytopes, balanced polytopes.
G¨unter Ziegler, Lectures on Polytopes
Additional Reading Material:
Branko Gr¨unbaum, Convex Polytopes
Igor pak, Lectures on Discrete and Polyhedral Geometry
End of year written/oral examination 0 %
Presentation 80 %
Participation in Tutorials 20 %
Project work 0 %
Assignments 0 %
Reports 0 %
Research project 0 %
Quizzes 0 %
Other 0 %