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Incidence Theorems and Their Applications

Author(s): Dvir, Zeev

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Abstract: 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 multisource 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. We will discuss these questions both over the reals and over finite fields and see how they come up in the theory of randomness extractors. 3. Sylvester–Gallai type problems: In this type of problems, one is presented with a configuration of points that contain many 'local' dependencies (e.g., three points on a line) and is asked to derive a bound on the dimension of the span of all points. We will discuss several recent results of this type, over various fields, and see their connection to the theory of locally correctable error-correcting codes. Throughout the different parts of the survey, two types of techniques will make frequent appearance. One is the polynomial method, which uses polynomial interpolation to impose an algebraic structure on the problem at hand. The other recurrent techniques will come from the area of additive combinatorics.
Publication Date: 2012
Citation: Dvir, Zeev. "Incidence Theorems and Their Applications." Foundations and Trends® in Theoretical Computer Science 6, no. 4 (2012): pp. 257-393. doi:10.1561/0400000056
DOI: 10.1561/0400000056
ISSN: 1551-305X
EISSN: 1551-3068
Pages: 257 - 393
Type of Material: Journal Article
Journal/Proceeding Title: Foundations and Trends® in Theoretical Computer Science
Version: Author's manuscript

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