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Incidences & Co. (a love story between discrete objects)

Wednesday, July 19th, 2017 16:10 (notice the date in July)

Schreiber 309

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Incidences & Co. (a love story between discrete objects)

Noam Solomon, TAU

Abstract: 

 

In 1983, Szemer\'edi and Trotter proved a tight bound for the number of incidences between points and lines in the plane. Ever since then, incidence geometry has been a very active research area, bridging between computer science and mathematics, with many connections to diverse topics, from range searching algorithms (Computational Geometry) to the Kakeya problem (Harmonic Analysis).

Over 25 years later, Guth and Katz proved a tight incidence bound for points and lines in three dimensions. Their proof introduced methods from advanced algebra and especially from algebraic geometry which were not used in combinatorics before. This enabled Guth and Katz to (almost) settle the Erd\"os distinct distances problem - a problem which stubbornly stood open for over 60 years, despite very brave attempts to solve it. The work of Guth and Katz has given significant added momentum to incidence geometry, making many problems, deemed hopeless before the breakthrough, amenable to the new techniques.

In this talk I will present the area of incidence geometry, before and after, highlighting the basics of the new "algebraic" approach, and will survey the results in my thesis, conducted under the supervision of Micha Sharir. Among these results are bounds on (i) the number of incidences between points and lines in four dimensions, (ii) the number of incidences between points and lines that lie on special surfaces and (iii) the number of incidences between points and general algebraic curves and surfaces. We will then talk about applications of these results, and discuss interesting open related questions.

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