Incidences between points and lines in three dimensions

We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no plane contains more than $s$ lines, is $$O\left(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + n\right)$$ (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between $m$ and $n$). This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step in their solution of Erd{\H o}s's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth~\cite{Gu14}. The present paper presents a different and simpler derivation, with better bounds than those in \cite{Gu14}, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions

The flecnode polynomial: a central object in incidence geometry

We give a brief exposition of the proof of the Cayley-Salmon theorem and its recent role in incidence geometry. Even when we don't use the properties of ruled surfaces explicitly, the regime in which we have interesting results in point-line incidence problems often coincides with the regime in which lines are organized into ruled surfaces.Comment: 12 pages. An expository note submitted to ICM proceeding

Incidences with Curves in ℝ^d

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R^d is I(P, C) = O (mdk-d+1/^k +ε_n^(dk)−d−+1/dk-d + ∑^(d-1)/_(j=2) m/jk−j+1+/k ε_n d(j−1)(k−1)/(d−1)(jk−j+1) q_j /(d−j)(k−1) (d−1)(jk−j+1) +m + n), where the constant of proportionality depends on k, ε and d, for any ε > 0, provided that no j-dimensional surface of degree c_j (k, d, ε), a constant parameter depending on k, d, j, and ε, contains more than q_j input curves, and that the q_j ’s satisfy certain mild conditions. This bound generalizes a recent result of Sharir and Solomon [20] concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth [8] (as well as the earlier bound of Guth and Katz [10]) in three dimensions (Guth’s three-dimensional bound has a better dependency on q). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl [7], in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi [4] and by Hablicsek and Scherr [11] concerning rich lines in high-dimensional spaces

Distinct Distance Estimates and Low Degree Polynomial Partitioning

We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155–190, 2015): we prove that if L is a set of L lines in R[superscript 3] with at most L[superscript 1/2] lines in any low degree algebraic surface, then the number of r-rich points of is L is ≲ L[superscript (3/2) + ε] r[superscript -2]. This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (2015). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of N points in R[superscript 2] c[subscript ε]N[superscript 1-ε] distinct distances