Speaker: Noam Solomon (TAU)

Title: Incidences between points and lines in three and four dimensions

Abstract:
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We show that the number of incidences between m distinct points
and n distinct lines in R^4 is

O(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) +
m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n),

for a suitable absolute constant c,
provided that no 2-plane contains more than s input lines, and no
hyperplane or quadric contains more than q lines.
The bound holds without the factor 2^{c\sqrt{\log m}} when
m \le n^{6/7} or m \ge n^{5/3}. Except for this factor, the
bound is tight in the worst case.

This result generalizes to four dimensions the classical
Szemer\'edi-Trotter theorem for incidences in the plane, and the
groundbreaking paper of Guth and Katz from 2010 for incidences in
R^3 (using this bound they solved Erd\"os's distinct distances problem).

The talk will include a review on techniques for deriving incidence
bounds in two and three dimensions, including the polynomial
partitioning method of Guth and Katz and more classical methods, and
will explain the various issues that arise when extending the analysis
to four dimensions.

We will also sketch a new proof for the Guth-Katz bound in three
dimensions, and a new incidence bound for points and lines contained
in a two-dimensional surface.

Joint work with Micha Sharir.