 ## Discrete Geometry (201-2-0191 ). Spring 2018

#### Ben Gurion University of the Negev, Department of Mathematics

Instructor: Prof. Shakhar Smorodinsky
• Office: Math building (58), room 208
• Tel.: (08) 6461604
• E-mail: "My first name" at math and you know whats next"
Time and place of lectures:
• Tuesday 12-14 building 58 (math) room 201
• Wednesday 14-16 building 58 (math) room 201

### About the course:

The course is intended for 3rd year undergraduate (with my permisiion) as well as M.Sc. and Ph.D. students both in computer science and/or mathematics. We will touch main topics in the area of discrete geometry. Some of the topics are motivated by the analysis of algorithms in computational geometry, wireless and sensor networks. Some other beautiful and elegant tools are proved to be powerful in seemingly non-related areas such as additive number theory or hard Erdos problems. The course does not require any special background except for basic linear algebra, probability and combinatorics.

• Probability.

homework

### Syllabus

• " Fundamental theorems and basic definitions: Convex sets, convex combinations, Separation Helly's theorem, fractional Helly, Radon's theorem, Caratheodory's theorem, centerpoint theorem. Tverberg's theorem. Planar graphs. Koebe's Theorem. A geometric proof of Lipton-Tarjan separator theorem for planar graphs.
• " Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Incidences between points and lines and points and unit circles. Repeated distance problem, distinct distances problem. Selection lemmas: points and discs, points and simplexes. upper bounds on k-sets in R2 and R3. An application of incidences to additive number theory. Coloring and hiting problems for geometric hypergraphs : VC-dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. Conflict-free colorings .
• " Arrangements and Davenport Schinzel sequences. Geometric permutations.
• " Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.

#### RECOMMENDED BOOKS:

• J. Matousek, Lectures on Discrete Geometry, GTM 212, Springer 2002.
• J. Pach and P.K. Agarwal, Combinatorial Geometry, John Wiley & Sons, New-York, NY, 1995.