Understand
We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics.
- The three main types of problems we will discuss are: (1) Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc..), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the Szemeredi-Trotter theorem, over the reals and over finite fields and discuss their applications in combinatorics (e.g., in the recent solution of Guth and Katz to Erdos' distance problem) and in computer science (in explicit constructions of multi-source extractors).
- (2) Kakeya type problems: These problems deal with arrangements of lines that point in different directions.
- The goal is to try and understand to what extent these lines can overlap one another.