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HU Credits:
6
Degree/Cycle:
2nd degree (Master)
Responsible Department:
Mathematics
Semester:
1st Semester
Teaching Languages:
English
Campus:
E. Safra
Course/Module Coordinator:
Eran Nevo
Coordinator Office Hours:
By appointment
Teaching Staff:
Prof Eran Nevo
Course/Module description:
The course focuses on basic notions and techniques in the field of Discrete Geometry, regarding point configurations and polytopes.
The techniques include algebraic, topological, geometric and combinatorial methods.
Details on the selected topics appear below.
Course/Module aims:
Learning outcomes - On successful completion of this module, students should be able to:
Students will know fundamental results in Discrete Geometry and be able use algebraic, topological, geometric and combinatorial methods to address problems in the field.
Attendance requirements(%):
The course will include also short student presentations.
Teaching arrangement and method of instruction:
Short pre-recorded lectures; live online discussions on the recorded material; short student presentations.
Course/Module Content:
Radon's lemma, Helly's theorem, centerpoints, colorful Caratheodory theorem
Euler's formula, crossing numbers, ampli cation through probabilistic method, Szemeredi-
Trotter theorem, applications to sum-product estimates
Unit distances problem, distinct distances, Erdos-Szekeres theorem via hypergraph Ramsey
theory
Number of joints via polynomial method
Polytopes and polyhedra, Minkowski-Weyl theorem, Steinitz' theorem
Balinksi's theorem, Hirsch conjecture, vertex-decomposibility
Gale duality, non-rational polytopes, oriented matroids and their realizability
Neighborly, cyclic, stacked polytopes, f-vectors, Dehn-Sommerville relations, shellability,
upper bound theorem
Triangulations, Voronoi and Delaunay, the associahedron.
Other or additional topics may be studied
Required Reading:
* J. Matousek. Lectures on discrete geometry. Vol. 212. Springer Science & Business Media, 2013.
* G.M. Ziegler. Lectures on polytopes. Vol. 152. Springer Science & Business Media, 2012.
Additional Reading Material:
-- N. Alon and J. Spencer. The probabilistic method. John Wiley & Sons, 2016.
-- R. Graham, B. Rothschild, and J. Spencer. Ramsey theory. Wiley Series in Discrete Mathematics
and Optimization Vol. 20, John Wiley & Sons, 1990.
-- L. Guth. Polynomial methods in combinatorics. University Lecture Series, Vol. 64, American Math-
ematical Society, 2016.
Course/Module evaluation:
End of year written/oral examination 100 %
Presentation 0 %
Participation in Tutorials 0 %
Project work 0 %
Assignments 0 %
Reports 0 %
Research project 0 %
Quizzes 0 %
Other 0 %
Additional information:
Joint course with Berlin Free University (FUB). Lecturers: Christian Haase (FUB), Florian Frick (FUB and Carnegie Mellon University), Eran Nevo (HUJI).
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