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HU Credits:
2
Degree/Cycle:
2nd degree (Master)
Responsible Department:
Mathematics
Semester:
2nd Semester
Teaching Languages:
English
Campus:
E. Safra
Course/Module Coordinator:
Eran Nevo
Coordinator Office Hours:
Teaching Staff:
Prof Eran Nevo
Course/Module description:
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.
Course/Module aims:
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.
Attendance requirements(%):
Teaching arrangement and method of instruction:
Course/Module Content:
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.
Required Reading:
G¨unter Ziegler, Lectures on Polytopes
Additional Reading Material:
Branko Gr¨unbaum, Convex Polytopes
Igor pak, Lectures on Discrete and Polyhedral Geometry
Grading Scheme :
Presentation / Poster Presentation / Lecture 80 %
Attendance / Participation in Field Excursion 20 %
Additional information:
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