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Syllabus polytopes - 80679

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Last update 13-08-2019
HU Credits: 2

Degree/Cycle: 2nd degree (Master)

Responsible Department: Mathematics

Semester: 1st Semester

Teaching Languages: English and Hebrew

Campus: E. Safra

Course/Module Coordinator: Eran Nevo

Coordinator Email: nevo@math.huji.ac.il

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, Gales evenness condition).
2. Graphs of polytopes:
Tell a simple polytope from its graph
Kalais 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:
Gunter Ziegler, Lectures on Polytopes

Additional Reading Material:
Branko Grunbaum, Convex Polytopes

Igor pak, Lectures on Discrete and Polyhedral Geometry

Course/Module evaluation:
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 %

Additional information:
 
Students needing academic accommodations based on a disability should contact the Center for Diagnosis and Support of Students with Learning Disabilities, or the Office for Students with Disabilities, as early as possible, to discuss and coordinate accommodations, based on relevant documentation.
For further information, please visit the site of the Dean of Students Office.
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